|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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.
|
|
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
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|