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.
%I A326476 #29 May 11 2025 10:31:09 %S A326476 1,0,1,0,1,1,0,1,2,1,0,1,8,3,1,0,1,32,21,4,1,0,1,128,183,40,5,1,0,1, %T A326476 512,1641,544,65,6,1,0,1,2048,14763,8320,1205,96,7,1,0,1,8192,132861, %U A326476 131584,26465,2256,133,8,1,0,1,32768,1195743,2099200,628805,64896,3787,176,9,1 %N A326476 A(n, k) = (m*k)! [x^k] MittagLefflerE(m, x)^n, for m = 2, n >= 0, k >= 0; square array read by descending antidiagonals. %F A326476 A(n,k) = (2*k)! * [x^(2*k)] cosh(x)^n. - _Seiichi Manyama_, May 11 2025 %e A326476 Array starts: %e A326476 [0] 1, 0, 0, 0, 0, 0, 0, 0, ... A000007 %e A326476 [1] 1, 1, 1, 1, 1, 1, 1, 1, ... A000012 %e A326476 [2] 1, 2, 8, 32, 128, 512, 2048, 8192, ... A081294 %e A326476 [3] 1, 3, 21, 183, 1641, 14763, 132861, 1195743, ... A054879 %e A326476 [4] 1, 4, 40, 544, 8320, 131584, 2099200, 33562624, ... A092812 %e A326476 [5] 1, 5, 65, 1205, 26465, 628805, 15424865, 382964405, ... A121822 %e A326476 [6] 1, 6, 96, 2256, 64896, 2086656, 71172096, 2499219456, ... %e A326476 [7] 1, 7, 133, 3787, 134953, 5501167, 243147373, 11266376947, ... %e A326476 [8] 1, 8, 176, 5888, 250496, 12397568, 676591616, 39316226048, ... %e A326476 [9] 1, 9, 225, 8649, 427905, 24943689, 1624354785, 114066126729, ... %e A326476 A000567, %e A326476 Seen as a triangle: %e A326476 1; %e A326476 0, 1; %e A326476 0, 1, 1; %e A326476 0, 1, 2, 1; %e A326476 0, 1, 8, 3, 1; %e A326476 0, 1, 32, 21, 4, 1; %e A326476 0, 1, 128, 183, 40, 5, 1; %e A326476 0, 1, 512, 1641, 544, 65, 6, 1; %e A326476 0, 1, 2048, 14763, 8320, 1205, 96, 7, 1; %e A326476 0, 1, 8192, 132861, 131584, 26465, 2256, 133, 8, 1; %t A326476 (* The function MLPower is defined in A326327. *) %t A326476 For[n = 0, n < 8, n++, Print[MLPower[2, n, 8]]] %o A326476 (Sage) # uses[MLPower from A326327] %o A326476 for n in (0..6): print(MLPower(2, n, 9)) %o A326476 (PARI) a(n, k) = (2*k)!*polcoef(cosh(x+x*O(x^(2*k)))^n, 2*k); \\ _Seiichi Manyama_, May 11 2025 %Y A326476 Rows n=0..5 give A000007, A000012, A081294, A054879, A092812, A121822. %Y A326476 Columns include: A000567. %Y A326476 Main diagonal gives A381459. %Y A326476 Variant: A286899. %Y A326476 Cf. A326474 (m=3, p>=0), A326475 (m=3, p<=0), A326327 (m=2, p<=0), this sequence (m=2, p>=0). %K A326476 nonn,tabl %O A326476 0,9 %A A326476 _Peter Luschny_, Jul 08 2019