[math-fun] Terminology: "Characteristic Generating Function"?
"Indicator Generating Function"? "Predicate Generating Function"? What is the name for the G.F. that encodes an integer sequence in the exponents with coefficients in {0,1} ? E.g., the OGF for the squares is Sum[n^2*x^n, {n, ∞}] == -((x*(1 + x))/(-1 + x)^3), but the thing I'm looking for is Sum[x^n^2,{n,0,∞}]==(1/2)*(1 + EllipticTheta[3, 0, x]). (Years ago I sent some of you the GF for expressions for k as the sum of n (optionally distinct) members of such a set, in terms of the GF predicating membership in the set. It turned out to be a superspecial case of a formula in Generatingfunctionology that was so general I didn't even recognize it as the same problem.) The reason I ask is that Neil just came up with 1/2 (-1+EllipticTheta[2,0,x]/x^(1/4)+EllipticTheta[3,0,x]) predicating the quartersquares<http://en.wikipedia.org/wiki/Multiplication_algorithm#Quarter_square_multiplication> A002620 <https://oeis.org/A002620>, from which he got (1/(2 (-1 + x)^2))((-1 + x) x^(3/4) EllipticTheta[2, 0, x] + x (3 - x + (-1 + x) EllipticTheta[3, 0, x])) generating what we think is the number of volumes you can pump with n of the recently discussed rotors, arranged optimally. --rwg
A248333 ! {0, 0, 0, 0, 1, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 16, 16, 17, 18, 19, 20, 20, 21, 22} So now we have a generating function. I wonder if there's a conjectural closed form resembling In[482]:= Table[Floor[1/(E - (1 + 1/n)^n)], {n, 22}] Out[482]= {1, 2, 2, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 10, 11, 12, 13, 13, 14, 15, 16, 16} or Richard R. Forberg's empirical formula in A002620. --rwg Max # of Dots&Boxes boxes constructible with n dots. Apropos the original subject, I still don't know the term. NeilB recently read g'f'ology. May he noticed an apt phrase. The distinction is between, e.g., 0,1,4,9,16,... and 1,1,0,0,1,0,0,0,0,1,0,... . On 2012-03-21 20:13, Bill Gosper wrote:
"Indicator Generating Function"? "Predicate Generating Function"? What is the name for the G.F. that encodes an integer sequence in the exponents with coefficients in {0,1} ? E.g., the OGF for the squares is Sum[n^2*x^n, {n, ∞}] == -((x*(1 + x))/(-1 + x)^3), but the thing I'm looking for is Sum[x^n^2,{n,0,∞}]==(1/2)*(1 + EllipticTheta[3, 0, x]). (Years ago I sent some of you the GF for expressions for k as the sum of n (optionally distinct) members of such a set, in terms of the GF predicating membership in the set. It turned out to be a superspecial case of a formula in Generatingfunctionology that was so general I didn't even recognize it as the same problem.)
The reason I ask is that Neil[B] just came up with 1/2 (-1+EllipticTheta[2,0,x]/x^(1/4)+EllipticTheta[3,0,x]) predicating the quartersquares<http://en.wikipedia.org/wiki/Multiplication_algorithm#Quarter_square_multiplication> A002620 <https://oeis.org/A002620>, from which he got (1/(2 (-1 + x)^2))((-1 + x) x^(3/4) EllipticTheta[2, 0, x] + x (3 - x + (-1 + x) EllipticTheta[3, 0, x])) generating what we think is the number of volumes you can pump with n of the recently discussed rotors, arranged optimally. --rwg
These generating functions are very common, and widely used in number theory (and here I respectfully disagree with Warren's comment that they aren't useful). Suppose S = s_0, s_1, ... is any set of nonnegative numbers, and define "x^S" to be G = Sum_i x^(s_i). Then the coefficient of x^n in G^m is the number of ways of writing n as a sum of m terms of S. See any book on number theory, especially in connection with the "circle method". But I've never seen a special name for this kind of g.f. Mel Nathanson, Additive Number Theory: The Classical Bases, page 122, just refers to G as "the generating function" for S. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Thu, Oct 15, 2015 at 1:37 AM, rwg <rwg@sdf.org> wrote:
A248333 ! {0, 0, 0, 0, 1, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 16, 16, 17, 18, 19, 20, 20, 21, 22} So now we have a generating function. I wonder if there's a conjectural closed form resembling In[482]:= Table[Floor[1/(E - (1 + 1/n)^n)], {n, 22}]
Out[482]= {1, 2, 2, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 10, 11, 12, 13, 13, 14, 15, 16, 16}
or Richard R. Forberg's empirical formula in A002620. --rwg Max # of Dots&Boxes boxes constructible with n dots.
Apropos the original subject, I still don't know the term. NeilB recently read g'f'ology. May he noticed an apt phrase. The distinction is between, e.g., 0,1,4,9,16,... and 1,1,0,0,1,0,0,0,0,1,0,... .
On 2012-03-21 20:13, Bill Gosper wrote:
"Indicator Generating Function"? "Predicate Generating Function"? What is the name for the G.F. that encodes an integer sequence in the exponents with coefficients in {0,1} ? E.g., the OGF for the squares is Sum[n^2*x^n, {n, ∞}] == -((x*(1 + x))/(-1 + x)^3), but the thing I'm looking for is Sum[x^n^2,{n,0,∞}]==(1/2)*(1 + EllipticTheta[3, 0, x]). (Years ago I sent some of you the GF for expressions for k as the sum of n (optionally distinct) members of such a set, in terms of the GF predicating membership in the set. It turned out to be a superspecial case of a formula in Generatingfunctionology that was so general I didn't even recognize it as the same problem.)
The reason I ask is that Neil[B] just came up with 1/2 (-1+EllipticTheta[2,0,x]/x^(1/4)+EllipticTheta[3,0,x]) predicating the quartersquares< http://en.wikipedia.org/wiki/Multiplication_algorithm#Quarter_square_multipl...
A002620 <https://oeis.org/A002620>, from which he got (1/(2 (-1 + x)^2))((-1 + x) x^(3/4) EllipticTheta[2, 0, x] + x (3 - x + (-1 + x) EllipticTheta[3, 0, x])) generating what we think is the number of volumes you can pump with n of the recently discussed rotors, arranged optimally. --rwg
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the thing I'm looking for is Sum[x^n^2,{n,0,∞}]
The wikipedia page on Lambert series has "For Liouville's function \lambda(n): \sum_{n=1}^{\infty} \lambda(n) \frac{q^n}{1-q^n} = \sum_{n=1}^{\infty} q^{n^2} " On Wed, Oct 14, 2015 at 10:37 PM, rwg <rwg@sdf.org> wrote:
A248333 ! {0, 0, 0, 0, 1, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 16, 16, 17, 18, 19, 20, 20, 21, 22} So now we have a generating function. I wonder if there's a conjectural closed form resembling In[482]:= Table[Floor[1/(E - (1 + 1/n)^n)], {n, 22}]
Out[482]= {1, 2, 2, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 10, 11, 12, 13, 13, 14, 15, 16, 16}
or Richard R. Forberg's empirical formula in A002620. --rwg Max # of Dots&Boxes boxes constructible with n dots.
Apropos the original subject, I still don't know the term. NeilB recently read g'f'ology. May he noticed an apt phrase. The distinction is between, e.g., 0,1,4,9,16,... and 1,1,0,0,1,0,0,0,0,1,0,... .
On 2012-03-21 20:13, Bill Gosper wrote:
"Indicator Generating Function"? "Predicate Generating Function"? What is the name for the G.F. that encodes an integer sequence in the exponents with coefficients in {0,1} ? E.g., the OGF for the squares is Sum[n^2*x^n, {n, ∞}] == -((x*(1 + x))/(-1 + x)^3), but the thing I'm looking for is Sum[x^n^2,{n,0,∞}]==(1/2)*(1 + EllipticTheta[3, 0, x]). (Years ago I sent some of you the GF for expressions for k as the sum of n (optionally distinct) members of such a set, in terms of the GF predicating membership in the set. It turned out to be a superspecial case of a formula in Generatingfunctionology that was so general I didn't even recognize it as the same problem.)
The reason I ask is that Neil[B] just came up with 1/2 (-1+EllipticTheta[2,0,x]/x^(1/4)+EllipticTheta[3,0,x]) predicating the
quartersquares<http://en.wikipedia.org/wiki/Multiplication_algorithm#Quarter_square_multiplication> A002620 <https://oeis.org/A002620>, from which he got (1/(2 (-1 + x)^2))((-1 + x) x^(3/4) EllipticTheta[2, 0, x] + x (3 - x + (-1 + x) EllipticTheta[3, 0, x])) generating what we think is the number of volumes you can pump with n of the recently discussed rotors, arranged optimally. --rwg
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rwg