A176156 Regular triangle, T(n, k) = f(n, k) - f(n, 0) + 1, where f(n, k) = Sum_{j=0..k} (-1)^j*StirlingS2(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} (-1)^j*StirlingS2(n, n-j)*binomial(n, j), read by rows.

1, 1, 1, 1, 3, 1, 1, 10, 10, 1, 1, 25, 67, 25, 1, 1, 51, 281, 281, 51, 1, 1, 91, 646, 1036, 646, 91, 1, 1, 148, -1217, -12536, -12536, -1217, 148, 1, 1, 225, -31079, -287223, -548785, -287223, -31079, 225, 1, 1, 325, -342899, -3906899, -11000741, -11000741, -3906899, -342899, 325, 1

Offset: 0

Author

Roger L. Bagula, Apr 10 2010

Keywords

• sign
• tabl

Comments

Row sum are: {1, 2, 5, 22, 119, 666, 2512, -27208, -1184937, -30500426, -716845999, ...}.

Examples

			Triangle begins as:
  1;
  1,   1;
  1,   3,      1;
  1,  10,     10,       1;
  1,  25,     67,      25,       1;
  1,  51,    281,     281,      51,       1;
  1,  91,    646,    1036,     646,      91,      1;
  1, 148,  -1217,  -12536,  -12536,   -1217,    148,   1;
  1, 225, -31079, -287223, -548785, -287223, -31079, 225, 1;
		

Links

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

Crossrefs

Cf. A048994, A132393, A176153, A176154, A176155, A176157.

Programs

GAP

f:= function(n,k) return Sum([0..k], j-> Stirling1(n,n-j)*Binomial(n,j)) + Sum([0..n-k], 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 | (&+[(-1)^j*StirlingFirst(n,n-j)*Binomial(n,j): j in [0..k]]) + (&+[(-1)^j*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((-1)^j*stirling1(n, n-j)*binomial(n, j), j=0..k) + add((-1)^j*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[(-1)^j*StirlingS1[n, n-j]*Binomial[n, j], {j,0,k}] + Sum[(-1)^j*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, (-1)^j*stirling(n,n-j,1)*binomial(n,j)) + sum(j=0,n-k, (-1)^j*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(stirling_number1(n,n-j)*binomial(n,j) for j in (0..k)) + sum(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} (-1)^j*StirlingS1(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} (-1)^j*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 26 2019