A176154 Triangle T(n,k) = Sum_{j=0..k} Stirling1(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} Stirling1(n, n-j)*binomial(n,j), read by rows.
2, 2, 2, 0, -2, 0, -1, -10, -10, -1, 20, -4, 86, -4, 20, -78, -128, 102, 102, -128, -78, 77, -13, 1982, -6628, 1982, -13, 77, 2641, 2494, 1129, 12448, 12448, 1129, 2494, 2641, -36944, -37168, 12168, -463496, 745726, -463496, 12168, -37168, -36944
Offset: 0
Examples
Triangle begins as:
2;
2, 2;
0, -2, 0;
-1, -10, -10, -1;
20, -4, 86, -4, 20;
-78, -128, 102, 102, -128, -78;
77, -13, 1982, -6628, 1982, -13, 77;
2641, 2494, 1129, 12448, 12448, 1129, 2494, 2641;
Links
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
Programs
-
GAP
T:= 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-> T(n, k) ))); # G. C. Greubel, Nov 26 2019
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Magma
T:= 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]]) >; [T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 26 2019
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Maple
with(combinat); T:= 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(T(n,k), k=0..n), n=0..10); # G. C. Greubel, Nov 26 2019
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Mathematica
T[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[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)) + sum(j=0,n-k, stirling(n, n-j,1)*binomial(n,j)); \\ G. C. Greubel, Nov 26 2019
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Sage
def T(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)) [[T(n, 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) + Sum_{j=0..n-k} Stirling1(n, n-j)*binomial(n,j).
Extensions
Name edited by G. C. Greubel, Nov 27 2019