cp's OEIS Frontend

A307047 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1+x)^k-x^k).

%I A307047 #24 Aug 05 2024 08:41:33
%S A307047 1,1,0,1,-2,0,1,-3,4,0,1,-4,6,-8,0,1,-5,10,-9,16,0,1,-6,15,-20,9,-32,
%T A307047 0,1,-7,21,-35,36,0,64,0,1,-8,28,-56,70,-64,-27,-128,0,1,-9,36,-84,
%U A307047 126,-125,120,81,256,0,1,-10,45,-120,210,-252,200,-240,-162,-512,0
%N A307047 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1+x)^k-x^k).
%H A307047 Seiichi Manyama, <a href="/A307047/b307047.txt">Antidiagonals n = 0..139, flattened</a>
%F A307047 A(n,k) = (-1)^n * Sum_{j=0..floor(n/k)} (-1)^((k mod 2) * j) * binomial(n+k-1,k*j+k-1).
%e A307047 Square array begins:
%e A307047    1,    1,    1,    1,    1,    1,     1,     1, ...
%e A307047    0,   -2,   -3,   -4,   -5,   -6,    -7,    -8, ...
%e A307047    0,    4,    6,   10,   15,   21,    28,    36, ...
%e A307047    0,   -8,   -9,  -20,  -35,  -56,   -84,  -120, ...
%e A307047    0,   16,    9,   36,   70,  126,   210,   330, ...
%e A307047    0,  -32,    0,  -64, -125, -252,  -462,  -792, ...
%e A307047    0,   64,  -27,  120,  200,  463,   924,  1716, ...
%e A307047    0, -128,   81, -240, -275, -804, -1715, -3432, ...
%e A307047    0,  256, -162,  496,  275, 1365,  2989,  6436, ...
%t A307047 T[n_, k_] := (-1)^n * Sum[(-1)^(j * Mod[k, 2]) * Binomial[n + k - 1, k*j + k - 1], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* _Amiram Eldar_, May 20 2021 *)
%Y A307047 Columns 1-7 give A000007, A122803, A000748, (-1)^n * A000749(n+3), A000750, A006090, A049018.
%Y A307047 Cf. A039912 (square array A(n,k), n >= 0, k >= 2), A306913, A306914, A306915.
%K A307047 sign,tabl
%O A307047 0,5
%A A307047 _Seiichi Manyama_, Mar 21 2019