cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A306913 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).

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%I A306913 #22 May 26 2021 00:55:12
%S A306913 1,1,-2,1,-2,4,1,-3,2,-8,1,-4,6,0,16,1,-5,10,-11,-4,-32,1,-6,15,-20,
%T A306913 21,8,64,1,-7,21,-35,34,-42,-8,-128,1,-8,28,-56,70,-48,85,0,256,1,-9,
%U A306913 36,-84,126,-127,48,-171,16,-512,1,-10,45,-120,210,-252,220,0,342,-32,1024
%N A306913 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 A306913 Seiichi Manyama, <a href="/A306913/b306913.txt">Antidiagonals n = 0..139, flattened</a>
%F A306913 A(n,k) = (-1)^n * Sum_{j=0..floor(n/k)} (-1)^(((k+1) mod 2) * j) * binomial(n+k-1,k*j+k-1).
%e A306913 Square array begins:
%e A306913       1,  1,    1,    1,    1,    1,     1,     1, ...
%e A306913      -2, -2,   -3,   -4,   -5,   -6,    -7,    -8, ...
%e A306913       4,  2,    6,   10,   15,   21,    28,    36, ...
%e A306913      -8,  0,  -11,  -20,  -35,  -56,   -84,  -120, ...
%e A306913      16, -4,   21,   34,   70,  126,   210,   330, ...
%e A306913     -32,  8,  -42,  -48, -127, -252,  -462,  -792, ...
%e A306913      64, -8,   85,   48,  220,  461,   924,  1716, ...
%e A306913    -128,  0, -171,    0, -385, -780, -1717, -3432, ...
%e A306913     256, 16,  342, -164,  715, 1209,  3017,  6434, ...
%t A306913 A[n_, k_] := (-1)^n * Sum[(-1)^(Mod[k+1, 2] * j) * Binomial[n + k - 1, k*j + k - 1], {j, 0, Floor[n/k]}]; Table[A[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* _Amiram Eldar_, May 25 2021 *)
%Y A306913 Columns 1-2 give A122803, A108520.
%Y A306913 Cf. A039912, A306914, A306915.
%K A306913 sign,tabl
%O A306913 0,3
%A A306913 _Seiichi Manyama_, Mar 16 2019