A176155 Regular triangle, T(n, k) = f(n, k) - f(n, 0) + 1, where f(n, k) = Sum_{j=0..k} StirlingS1(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} StirlingS1(n, n-j)*binomial(n, j), read by rows.

1, 1, 1, 1, -1, 1, 1, -8, -8, 1, 1, -23, 67, -23, 1, 1, -49, 181, 181, -49, 1, 1, -89, 1906, -6704, 1906, -89, 1, 1, -146, -1511, 9808, 9808, -1511, -146, 1, 1, -223, 49113, -426551, 782671, -426551, 49113, -223, 1, 1, -323, -343547, 3220453, -3873389, -3873389, 3220453, -343547, -323, 1

Offset: 0

Author

Roger L. Bagula, Apr 10 2010

Keywords

• sign
• tabl

Comments

Row sum are: {1, 2, 1, -14, 23, 266, -3068, 16304, 27351, -1993610, 31213301, ...}.

Examples

			Triangle begins as:
  1;
  1,    1;
  1,   -1,       1;
  1,   -8,      -8,       1;
  1,  -23,      67,     -23,        1;
  1,  -49,     181,     181,      -49,        1;
  1,  -89,    1906,   -6704,     1906,      -89,       1;
  1, -146,   -1511,    9808,     9808,    -1511,    -146,       1;
  1, -223,   49113, -426551,   782671,  -426551,   49113,    -223,    1;
  1, -323, -343547, 3220453, -3873389, -3873389, 3220453, -343547, -323, 1;
		

Links

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

Crossrefs

Cf. A048994, A132393, A176153, A176154, A176156, A176157.

Programs

GAP

f:= function(n,k) return Sum([0..k], j-> (-1)^j*Stirling1(n,n-j)*Binomial(n,j)) + Sum([0..n-k], j-> (-1)^j*Stirling1(n, n-j)*Binomial(n,j)); end;
Flat(List([0..10], n-> List([0..n], k-> f(n, k)-f(n,0)+1 ))); # G. C. Greubel, Nov 26 2019

Magma

f:= func< n,k | (&+[StirlingFirst(n,n-j)*Binomial(n,j): j in [0..k]]) + (&+[StirlingFirst(n,n-j)*Binomial(n,j): j in [0..n-k]]) >;
[f(n,k) - f(n,0) + 1: k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 26 2019

Maple

with(combinat);
f:= proc(n, k) option remember; add(stirling1(n, n-j)*binomial(n, j), j=0..k) + add(stirling1(n, n-j)* binomial(n, j), j=0..n-k); end;
seq(seq(f(n,k) -f(n,0) +1, k=0..n), n=0..10); # G. C. Greubel, Nov 26 2019

Mathematica

f[n_, k_]:= Sum[StirlingS1[n, n-j]*Binomial[n, j], {j,0,k}] + Sum[StirlingS1[n, n-j]*Binomial[n, j], {j,0,n-k}];
Table[f[n, k] - f[n, 0] + 1, {n,0,10}, {k,0,n}]//Flatten

PARI

f(n,k) = sum(j=0,k, stirling(n,n-j,1)*binomial(n,j)) + sum(j=0,n-k, stirling(n, n-j,1)*binomial(n,j));
T(n,k) = f(n,k) - f(n,0) + 1; \\ G. C. Greubel, Nov 26 2019

Sage

def f(n, k): return sum((-1)^j*stirling_number1(n,n-j)*binomial(n,j) for j in (0..k)) + sum((-1)^j*stirling_number1(n, n-j)*binomial(n,j) for j in (0..n-k))
[[f(n, k)-f(n,0)+1 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 26 2019

Formula

With f(n, k) = Sum_{j=0..k} StirlingS1(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} StirlingS1(n, n-j)*binomial(n, j) then T(n, k) = f(n, k) - f(n, 0) + 1.

Extensions

Name edited by G. C. Greubel, Nov 27 2019