Suggested tweaks to the combinatorial interpretations of A003114: Also [number of partitions with all differences > 1, thus by Ferrers transposition,] number of partitions of n such that if k is the largest part, then each of {1, 2, ..., k-1} occur at least twice. Example: a(9)=5 because we have [3, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1]. (Schur's remark in http://mathworld.wolfram.com/Rogers-RamanujanIdentities.html. Caution: Identity (7) is not true for general a !) Also (http://en.wikipedia.org/wiki/Glaisher%27s_theorem). Also (www.macalester.edu/~bressoud/pub/fpipc.tex) number of partitions into distinct parts with every even part > twice the number of odd parts. In[524]:= Table[ Length[Select[IntegerPartitions[k], FreeQ[Differences[#], 0] && Min[Select[#, EvenQ]] > 2*Length[Select[#, OddQ]] &]], {k, 0, 22}] Out[524]= {1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, 31, 35, 41} --rwg NJAS> Joerg, Yes, certainly! Thank you! Neil On Mon, Oct 8, 2012 at 10:25 AM, Joerg Arndt <arndt@jjj.de <http://gosper.org/webmail/src/compose.php?send_to=arndt%40jjj.de>> wrote:
* Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> [Oct 08. 2012 08:02]:> > (For R-R, special-case a:=q)> >> > [...]> >> > [ inf 2 ]> > [ ==== n n ]> > [ \ a q ]> > [ 0 a > -------------- ]> > inf [ / qpoch(q, q, n) ]> > /===\ [ ==== ]> > | | [ 0 a ] [ n = 0 ]> > (d44) | | [ ] = [ ]> > | | [ k ] [ inf 2 ]> > k = 0 [ q 1 ] [ ==== n n - n ]> > [ \ a q ]> > [ 0 > -------------- ]> > [ / qpoch(q, q, n) ]> > [ ==== ]> > [ n = 0 ]> >> > [...]>> I toyed around a bit (note my start k=1 of the product):>>>> [ inf 2 ]> [ ==== n ]> [ \ q ]> [ 0 > -------------- ] (UR = H(q) => A003106)> inf [ / qpoch(q, q, n) ]> /===\ [ ==== ]> | | [ 0 1 ] [ n = 0 ]> (dxx) | | [ ] = [ ]> | | [ k ] [ inf 2 ]> k = 1 [ q 1 ] [ ==== n + n ]> [ \ q ]> [ 0 > -------------- ] (LR= G(q) => A003114)> [ / qpoch(q, q, n) ]> [ ==== ]> [ n = 0 ]>> I.e., R-R proper.> I am much tempted to put that into both A003106 and A003114> (obviously with attribution to RWG). OK, Neil ?>>>> [ inf 2 ]> [ ==== n - n ]> [ \ q ]> [ 0 q > -------------- ] (UR=A006141)> inf [ / qpoch(q, q, n) ]> /===\ [ ==== ]> | | [ 0 q ] [ n = 0 ]> (dxx) | | [ ] = [ ]> | | [ k ] [ inf 2 ]> k = 1 [ q 1 ] [ ==== n ]> [ \ q ]> [ 0 > -------------- ] (LR= H(q) => A003106)> [ / qpoch(q, q, n) ]> [ ==== ]> [ n = 0 ]>> inf> /===\> | | [ 0 q^n ] [ 0 D(q) ] (D=A000009)> (dxx) | | [ ] = [ ]> | | [ k ] [ 0 D(q) ] (D=A000009)> k = 1 [ 1+q 1 ]>>>> inf> /===\> | | [ 0 -q^n ] [ 0 E(q) ] (E=A010815 = 1/A000041)> (dxx) | | [ ] = [ ]> | | [ k ] [ 0 E(q) ] (E=A010815 = 1/A000041)> k = 1 [1-q 1 ]>>> ASCII for president!> SHHHHH! Rich said no politics.