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A080959 Square array of coefficients of binomial polynomials, read by antidiagonals.

%I A080959 #13 May 11 2025 19:18:13
%S A080959 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,
%T A080959 454,444,3828,8,-14,85,-200,974,-636,8868,25584,9,-20,126,-416,2024,
%U A080959 -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
%N A080959 Square array of coefficients of binomial polynomials, read by antidiagonals.
%H A080959 G. C. Greubel, <a href="/A080959/b080959.txt">Antidiagonals n = 1..50, flattened</a>
%F A080959 A(n, k) = k!*Sum_{j=1..k} (-1)^(j+1)*binomial(n+j, j)/j (array).
%F A080959 T(n, k) = A(n-k, k) (antidiagonals).
%e A080959 Array, A(n, k), begin as:
%e A080959    1,   1,   5,    14,    94,     444,    3828,      25584,     270576, ... A024167;
%e A080959    2,   1,  11,    14,   214,     444,    8868,      25584,     633456, ... A080958;
%e A080959    3,   0,  20,   -10,   454,    -636,   21468,     -55056,    1722096, ... ;
%e A080959    4,  -2,  34,   -74,   974,   -4236,   56748,    -377616,    5471856, ... ;
%e A080959    5,  -5,  55,  -200,  2024,  -13056,  146208,   -1325136,   16902576, ... ;
%e A080959    6,  -9,  85,  -416,  3968,  -31632,  348816,   -3695952,   47457072, ... ;
%e A080959    7, -14, 126,  -756,  7308,  -67032,  766296,   -9004752,  120758832, ... ;
%e A080959    8, -20, 180, -1260, 12708, -129672, 1563336,  -19925712,  281929392, ... ;
%e A080959    9, -27, 249, -1974, 21018, -234252, 2993436,  -40917312,  611923392, ... ;
%e A080959   10, -35, 335, -2950, 33298, -400812, 5431116,  -79073472, 1248697152, ... ;
%e A080959   11, -44, 440, -4246, 50842, -655908, 9411204, -145250688, 2417424768, ... ;
%e A080959 Antidiagonals, T(n, k), begin as:
%e A080959    1;
%e A080959    2,   1;
%e A080959    3,   1,   5;
%e A080959    4,   0,  11,    14;
%e A080959    5,  -2,  20,    14,   94;
%e A080959    6,  -5,  34,   -10,  214,    444;
%e A080959    7,  -9,  55,   -74,  454,    444,   3828;
%e A080959    8, -14,  85,  -200,  974,   -636,   8868,   25584;
%e A080959    9, -20, 126,  -416, 2024,  -4236,  21468,   25584,  270576;
%e A080959   10, -27, 180,  -756, 3968, -13056,  56748,  -55056,  633456, 2342880;
%e A080959   11, -35, 249, -1260, 7308, -31632, 146208, -377616, 1722096, 2342880, 29400480;
%t A080959 A[n_, k_]:= k!*Sum[(-1)^(j+1)*Binomial[n+j,j]/j, {j,k}];
%t A080959 A080959[n_, k_]:= A[n-k, k];
%t A080959 Table[A080959[n,k], {n,0,12}, {k,n}]//Flatten (* _G. C. Greubel_, May 11 2025 *)
%o A080959 (Magma)
%o A080959 A:= func< n,k | Factorial(k)*(&+[(-1)^(j+1)*Binomial(n+j,j)/j: j in [1..k]]) >;
%o A080959 A080959:= func< n,k | A(n-k,k) >;
%o A080959 [A080959(n,k): k in [1..n], n in [0..12]]; // _G. C. Greubel_, May 11 2025
%o A080959 (SageMath)
%o A080959 def A(n,k): return factorial(k)*sum((-1)^(j+1)*binomial(n+j,j)/j for j in range(1,k+1))
%o A080959 def A080959(n,k): return A(n-k,k)
%o A080959 print(flatten([[A080959(n,k) for k in range(1,n+1)] for n in range(13)])) # _G. C. Greubel_, May 11 2025
%Y A080959 Cf. A024167, A080958.
%Y A080959 Columns: A000027 (k=1), A080956 (k=2), A080957 (k=3).
%K A080959 easy,sign,tabl
%O A080959 1,2
%A A080959 _Paul Barry_, Mar 01 2003