OFFSET
0,8
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
FORMULA
T(n, k) = Sum_{j=0..k} Stirling1(n, n-j)*binomial(n, j).
T(n, n) = A317274(n). - G. C. Greubel, Aug 03 2021
EXAMPLE
Triangle begins as:
1;
1, 1;
1, -1, -1;
1, -8, -2, -2;
1, -23, 43, 19, 19;
1, -49, 301, -199, -79, -79;
1, -89, 1186, -3314, 796, 76, 76;
1, -146, 3529, -22196, 34644, -2400, 2640, 2640;
1, -223, 8793, -100967, 372863, -362529, 3375, -36945, -36945;
MAPLE
seq(seq( add(combinat[stirling1](n, n-j)*binomial(n, j), j=0..k), k=0..n), n=0..10); # G. C. Greubel, Nov 26 2019
MATHEMATICA
T[n_, k_]:= Sum[StirlingS1[n, n-j]*Binomial[n, j], {j, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten
PROG
(PARI) T(n, k) = sum(j=0, k, stirling(n, n-j, 1)*binomial(n, j)); \\ G. C. Greubel, Nov 26 2019
(Magma) [(&+[StirlingFirst(n, n-j)*Binomial(n, j): j in [0..k]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 26 2019
(Sage) [[sum((-1)^j*stirling_number1(n, n-j)*binomial(n, j) for j in (0..k)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 26 2019
(GAP) Flat(List([0..10], n-> List([0..n], k-> Sum([0..k], j-> (-1)^j*Stirling1(n, n-j)* Binomial(n, j)) ))); # G. C. Greubel, Nov 26 2019
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Apr 10 2010
STATUS
approved