A080959 Square array of coefficients of binomial polynomials, read by antidiagonals.
1, 2, 1, 3, 1, 5, 4, 0, 11, 14, 5, -2, 20, 14, 94, 6, -5, 34, -10, 214, 444, 7, -9, 55, -74, 454, 444, 3828, 8, -14, 85, -200, 974, -636, 8868, 25584, 9, -20, 126, -416, 2024, -4236, 21468, 25584, 270576, 10, -27, 180, -756, 3968, -13056, 56748, -55056, 633456, 2342880, 11, -35, 249, -1260, 7308, -31632, 146208, -377616, 1722096, 2342880, 29400480
Offset: 1
Examples
Array, A(n, k), begin as: 1, 1, 5, 14, 94, 444, 3828, 25584, 270576, ... A024167; 2, 1, 11, 14, 214, 444, 8868, 25584, 633456, ... A080958; 3, 0, 20, -10, 454, -636, 21468, -55056, 1722096, ... ; 4, -2, 34, -74, 974, -4236, 56748, -377616, 5471856, ... ; 5, -5, 55, -200, 2024, -13056, 146208, -1325136, 16902576, ... ; 6, -9, 85, -416, 3968, -31632, 348816, -3695952, 47457072, ... ; 7, -14, 126, -756, 7308, -67032, 766296, -9004752, 120758832, ... ; 8, -20, 180, -1260, 12708, -129672, 1563336, -19925712, 281929392, ... ; 9, -27, 249, -1974, 21018, -234252, 2993436, -40917312, 611923392, ... ; 10, -35, 335, -2950, 33298, -400812, 5431116, -79073472, 1248697152, ... ; 11, -44, 440, -4246, 50842, -655908, 9411204, -145250688, 2417424768, ... ; Antidiagonals, T(n, k), begin as: 1; 2, 1; 3, 1, 5; 4, 0, 11, 14; 5, -2, 20, 14, 94; 6, -5, 34, -10, 214, 444; 7, -9, 55, -74, 454, 444, 3828; 8, -14, 85, -200, 974, -636, 8868, 25584; 9, -20, 126, -416, 2024, -4236, 21468, 25584, 270576; 10, -27, 180, -756, 3968, -13056, 56748, -55056, 633456, 2342880; 11, -35, 249, -1260, 7308, -31632, 146208, -377616, 1722096, 2342880, 29400480;
Links
- G. C. Greubel, Antidiagonals n = 1..50, flattened
Programs
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Magma
A:= func< n,k | Factorial(k)*(&+[(-1)^(j+1)*Binomial(n+j,j)/j: j in [1..k]]) >; A080959:= func< n,k | A(n-k,k) >; [A080959(n,k): k in [1..n], n in [0..12]]; // G. C. Greubel, May 11 2025
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Mathematica
A[n_, k_]:= k!*Sum[(-1)^(j+1)*Binomial[n+j,j]/j, {j,k}]; A080959[n_, k_]:= A[n-k, k]; Table[A080959[n,k], {n,0,12}, {k,n}]//Flatten (* G. C. Greubel, May 11 2025 *) -
SageMath
def A(n,k): return factorial(k)*sum((-1)^(j+1)*binomial(n+j,j)/j for j in range(1,k+1)) def A080959(n,k): return A(n-k,k) print(flatten([[A080959(n,k) for k in range(1,n+1)] for n in range(13)])) # G. C. Greubel, May 11 2025
Formula
A(n, k) = k!*Sum_{j=1..k} (-1)^(j+1)*binomial(n+j, j)/j (array).
T(n, k) = A(n-k, k) (antidiagonals).