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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.
2
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
1,5
COMMENTS
Row n is made of coefficients from 7*(1 - x)^(n+1) * polylog(-n,x)/x - 6*(1 + x)^(n-1). - Thomas Baruchel, Jun 03 2018
LINKS
Thomas Baruchel, A conjectured formula for the polylogarithm of a negative integer order, Mathematics Stack Exchange question, Jun 04 2018.
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.
a(n) = 3*A168524(n) - 2*A154337(n). - Thomas Baruchel, Jun 08 2018
EXAMPLE
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;
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
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
PROG
(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
(PARI) row(n) = Vec(7*(1 - x)^(n+1)*polylog(-n, x)/x - 6*(1 + x)^(n-1)); \\ Michel Marcus, Jun 08 2018
(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
(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
CROSSREFS
Cf. Eulerian numbers (A008292) and Pascal's triangle (A007318).
Cf. A141696.
Sequence in context: A245910 A133824 A154228 * A202750 A177823 A142462
KEYWORD
nonn,tabl,easy
AUTHOR
Roger L. Bagula, Sep 11 2008
EXTENSIONS
Edited by G. C. Greubel, Nov 13 2019
STATUS
approved