A176153 Triangle T(n,k) = Sum_{j=0..k} Stirling1(n, n-j)*binomial(n,j), read by rows.

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, 1, -323, 19333, -361691, 2466883, -6010901, 3911515, -33509, 329371, 329371

Offset: 0

Author

Roger L. Bagula, Apr 10 2010

Keywords

• sign
• tabl

Examples

			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;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A008277, A176154, A176155, A176156, A176157, A317274.

Programs

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

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

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

PARI

T(n,k) = sum(j=0,k, stirling(n,n-j,1)*binomial(n,j)); \\ 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

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