A141697 T(n,k) = (q*Sum_{j=0..k+1} (-1)^j*binomial(n+1, j)*(k+1-j)^n - p*binomial(n-1, k))/2 where p=12 and q=14.
1, 1, 1, 1, 16, 1, 1, 59, 59, 1, 1, 158, 426, 158, 1, 1, 369, 2054, 2054, 369, 1, 1, 804, 8247, 16792, 8247, 804, 1, 1, 1687, 29925, 109123, 109123, 29925, 1687, 1, 1, 3466, 102088, 617302, 1092910, 617302, 102088, 3466, 1, 1, 7037, 334664, 3185840, 9171722, 9171722, 3185840, 334664, 7037, 1
Offset: 1
Examples
Triangle begins: 1; 1, 1; 1, 16, 1; 1, 59, 59, 1; 1, 158, 426, 158, 1; 1, 369, 2054, 2054, 369, 1; 1, 804, 8247, 16792, 8247, 804, 1; 1, 1687, 29925, 109123, 109123, 29925, 1687, 1;
Links
- G. C. Greubel, Rows n = 1..100 of triangle, flattened
- Thomas Baruchel, A conjectured formula for the polylogarithm of a negative integer order, Mathematics Stack Exchange question, Jun 04 2018.
Programs
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Magma
[ 7*(&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k+1]]) - 6*Binomial(n-1,k): k in [0..n-1], n in [1..10]]; // G. C. Greubel, Nov 13 2019
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Maple
T:= proc(n, k): 7*add((-1)^j*binomial(n+1, j)*(k-j+1)^n, j = 0..k+1) - 6*binomial(n-1, k); end proc; seq(seq(T(n,k), k=0..n-1), n=1..10); # G. C. Greubel, Nov 13 2019
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Mathematica
i=12; l=14; Table[Table[(l*Sum[(-1)^j*Binomial[n+1, j](k+1-j)^n, {j, 0, k+1}] - i*Binomial[n-1, k])/2, {k,0,n-1}], {n,10}]//Flatten -
PARI
T(n,k) = 7*sum(j=0, k+1, (-1)^j*binomial(n+1,j)*(k-j+1)^n) - 6* binomial(n-1,k); for(n=1,10, for(k=0,n-1, print1(T(n,k), ", "))) \\ G. C. Greubel, Jun 03 2018
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PARI
row(n) = Vec(7*(1 - x)^(n+1)*polylog(-n,x)/x - 6*(1 + x)^(n-1)); \\ Michel Marcus, Jun 08 2018
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Sage
[[ 7*sum( (-1)^j*binomial(n+1,j)*(k-j+1)^n for j in (0..k+1)) - 6*binomial(n-1,k) for k in (0..n-1)] for n in (1..10)] # G. C. Greubel, Nov 13 2019
Formula
p=12; q=14; T(n,k) = (q*Sum_{j=0..k+1} (-1)^j*binomial(n+1, j)*(k+1-j)^n - p*binomial(n-1, k))/2.
Extensions
Edited by G. C. Greubel, Nov 13 2019
Comments