I see the coefficient of n^7 in the summation of 12th powers can be well approximated by pi :) On Sat, Jan 26, 2019 at 8:40 PM Bill Gosper <billgosper@gmail.com> wrote:
This message was received and forwarded without spurious linebreaks. If illegible, another gift from xmission.com. (But you can tediously delete the extra linebreaks.) —rwg (To certain eavesdroppers this is a SPOILER.) ---------- Forwarded message --------- From: Bill Gosper <billgosper@gmail.com> Date: Sat, Jan 26, 2019 at 5:33 PM Subject: Faulhaber 12 in 4 iterations To: Bill Gosper <billgosper@gmail.com>
Shaved Pascal: Array[Binomial[#1, #2 - 1] Boole[#2 <= #1] &, {13, 13}]; 1 0 0 0 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 0 1 3 3 0 0 0 0 0 0 0 0 0 0 1 4 6 4 0 0 0 0 0 0 0 0 0 1 5 10 10 5 0 0 0 0 0 0 0 0 1 6 15 20 15 6 0 0 0 0 0 0 0 1 7 21 35 35 21 7 0 0 0 0 0 0 1 8 28 56 70 56 28 8 0 0 0 0 0 1 9 36 84 126 126 84 36 9 0 0 0 0 1 10 45 120 210 252 210 120 45 10 0 0 0 1 11 55 165 330 462 462 330 165 55 11 0 0 1 12 66 220 495 792 924 792 495 220 66 12 0 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13
The only divisions are in the initialization: Boole[#1 == #2]/#1
MatrixForm /@ NestList[2 # - #.%.# &, Array[Boole[#1 == #2]/#1 &, {13, 13}], 4]
1 0 0 0 0 0 0 0 0 0 0 0 0 0 1/2 0 0 0 0 0 0 0 0 0 0 0 0 0 1/3 0 0 0 0 0 0 0 0 0 0 0 0 0 1/4 0 0 0 0 0 0 0 0 0 0 0 0 0 1/5 0 0 0 0 0 0 0 0 0 0 0 0 0 1/6 0 0 0 0 0 0 0 0 0 0 0 0 0 1/7 0 0 0 0 0 0 0 0 0 0 0 0 0 1/8 0 0 0 0 0 0 0 0 0 0 0 0 0 1/9 0 0 0 0 0 0 0 0 0 0 0 0 0 1/10 0 0 0 0 0 0 0 0 0 0 0 0 0 1/11 0 0 0 0 0 0 0 0 0 0 0 0 0 1/12 0 0 0 0 0 0 0 0 0 0 0 0 0 1/13
1 0 0 0 0 0 0 0 0 0 0 0 0 -1/2 1/2 0 0 0 0 0 0 0 0 0 0 0 -1/3 -1/2 1/3 0 0 0 0 0 0 0 0 0 0 -1/4 -1/2 -1/2 1/4 0 0 0 0 0 0 0 0 0 -1/5 -1/2 -2/3 -1/2 1/5 0 0 0 0 0 0 0 0 -1/6 -1/2 -5/6 -5/6 -1/2 1/6 0 0 0 0 0 0 0 -1/7 -1/2 -1 -5/4 -1 -1/2 1/7 0 0 0 0 0 0 -1/8 -1/2 -7/6 -7/4 -7/4 -7/6 -1/2 1/8 0 0 0 0 0 -1/9 -1/2 -4/3 -7/3 -14/5 -7/3 -4/3 -1/2 1/9 0 0 0 0 -1/10 -1/2 -3/2 -3 -21/5 -21/5 -3 -3/2 -1/2 1/10 0 0 0 -1/11 -1/2 -5/3 -15/4 -6 -7 -6 -15/4 -5/3 -1/2 1/11 0 0 -1/12 -1/2 -11/6 -55/12 -33/4 -11 -11 -33/4 -55/12 -11/6 -1/2 1/12 0 -1/13 -1/2 -2 -11/2 -11 -33/2 -132/7 -33/2 -11 -11/2 -2 -1/2 1/13
1 0 0 0 0 0 0 0 0 0 0 0 0 -1/2 1/2 0 0 0 0 0 0 0 0 0 0 0 1/6 -1/2 1/3 0 0 0 0 0 0 0 0 0 0 0 1/4 -1/2 1/4 0 0 0 0 0 0 0 0 0 -23/15 0 1/3 -1/2 1/5 0 0 0 0 0 0 0 0 -25/4 -23/6 0 5/12 -1/2 1/6 0 0 0 0 0 0 0 -1573/84 -75/4 -23/3 0 1/2 -1/2 1/7 0 0 0 0 0 0 -301/6 -1573/24 -175/4 -161/12 0 7/12 -1/2 1/8 0 0 0 0 0 -5759/45 -602/3 -1573/9 -175/2 -322/15 0 2/3 -1/2 1/9 0 0 0 0 -639/2 -5759/10 -602 -1573/4 -315/2 -161/5 0 3/4 -1/2 1/10 0 0 0 -52223/66 -3195/2 -5759/3 -1505 -1573/2 -525/2 -46 0 5/6 -1/2 1/11 0 0 -1958 -52223/12 -11715/2 -63349/12 -3311 -17303/12 -825/2 -253/4 0 11/12 -1/2 1/12 0 -2211491/455 -11748 -52223/3 -35145/2 -63349/5 -6622 -17303/7 -2475/4 -253/3 0 1 -1/2 1/13
1 0 0 0 0 0 0 0 0 0 0 0 0 -1/2 1/2 0 0 0 0 0 0 0 0 0 0 0 1/6 -1/2 1/3 0 0 0 0 0 0 0 0 0 0 0 1/4 -1/2 1/4 0 0 0 0 0 0 0 0 0 -1/30 0 1/3 -1/2 1/5 0 0 0 0 0 0 0 0 0 -1/12 0 5/12 -1/2 1/6 0 0 0 0 0 0 0 1/42 0 -1/6 0 1/2 -1/2 1/7 0 0 0 0 0 0 0 1/12 0 -7/24 0 7/12 -1/2 1/8 0 0 0 0 0 -2363/15 0 2/9 0 -7/15 0 2/3 -1/2 1/9 0 0 0 0 -12285/4 -7089/10 0 1/2 0 -7/10 0 3/4 -1/2 1/10 0 0 0 -4729715/132 -61425/4 -2363 0 1 0 -1 0 5/6 -1/2 1/11 0 0 -326865 -4729715/24 -225225/4 -25993/4 0 11/6 0 -11/8 0 11/12 -1/2 1/12 0 -14093005307/5460 -1961190 -4729715/6 -675675/4 -77979/5 0 22/7 0 -11/6 0 1 -1/2 1/13
1 0 0 0 0 0 0 0 0 0 0 0 0 -1/2 1/2 0 0 0 0 0 0 0 0 0 0 0 1/6 -1/2 1/3 0 0 0 0 0 0 0 0 0 0 0 1/4 -1/2 1/4 0 0 0 0 0 0 0 0 0 -1/30 0 1/3 -1/2 1/5 0 0 0 0 0 0 0 0 0 -1/12 0 5/12 -1/2 1/6 0 0 0 0 0 0 0 1/42 0 -1/6 0 1/2 -1/2 1/7 0 0 0 0 0 0 0 1/12 0 -7/24 0 7/12 -1/2 1/8 0 0 0 0 0 -1/30 0 2/9 0 -7/15 0 2/3 -1/2 1/9 0 0 0 0 0 -3/20 0 1/2 0 -7/10 0 3/4 -1/2 1/10 0 0 0 5/66 0 -1/2 0 1 0 -1 0 5/6 -1/2 1/11 0 0 0 5/12 0 -11/8 0 11/6 0 -11/8 0 11/12 -1/2 1/12 0 -691/2730 0 5/3 0 -33/10 0 22/7 0 -11/6 0 1 -1/2 1/13
In[35]:= Sum[k^12, {k, n - 1}] // Expand
Out[35]= -691 n/2730 + 5 n^3/3 - 33 n^5/10 + 22 n^7/7 - 11 n^9/6 + n^11 - n^12/2 + n^13/13 —rwg (Probably don't need 5th iteration 'til Faulhaber 16. There's a bit to be gained by not recomputing the already-converged upper rows.) _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun