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

Author

Paul Barry, Mar 01 2003

Keywords

• easy
• sign
• tabl

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

Crossrefs

Cf. A024167, A080958.

Columns: A000027 (k=1), A080956 (k=2), A080957 (k=3).

Programs

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

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).