# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a307393 Showing 1-1 of 1 %I A307393 #25 May 20 2021 04:44:33 %S A307393 1,1,5,1,4,16,1,4,11,42,1,4,10,26,99,1,4,10,21,57,219,1,4,10,20,42, %T A307393 120,466,1,4,10,20,36,84,247,968,1,4,10,20,35,64,169,502,1981,1,4,10, %U A307393 20,35,57,120,340,1013,4017,1,4,10,20,35,56,93,240,682,2036,8100 %N A307393 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-4))/((1-x)^k-x^k). %H A307393 Seiichi Manyama, Antidiagonals n = 0..139, flattened %F A307393 A(n,k) = Sum_{j=0..floor(n/k)} binomial(n+3,k*j+3). %F A307393 A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} binomial(i+1,k*j+1) * binomial(n-i+1,k*j+1). %e A307393 Square array begins: %e A307393 1, 1, 1, 1, 1, 1, 1, 1, ... %e A307393 5, 4, 4, 4, 4, 4, 4, 4, ... %e A307393 16, 11, 10, 10, 10, 10, 10, 10, ... %e A307393 42, 26, 21, 20, 20, 20, 20, 20, ... %e A307393 99, 57, 42, 36, 35, 35, 35, 35, ... %e A307393 219, 120, 84, 64, 57, 56, 56, 56, ... %e A307393 466, 247, 169, 120, 93, 85, 84, 84, ... %e A307393 968, 502, 340, 240, 165, 130, 121, 120, ... %t A307393 T[n_, k_] := Sum[Binomial[n+3, k*j + 3], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* _Amiram Eldar_, May 20 2021 *) %Y A307393 Columns 1-5 give A002662(n+3), A125128(n+1), A111927(n+3), A000749(n+3), A139748(n+3). %Y A307393 Cf. A306915, A306846, A307078, A307394. %K A307393 nonn,tabl %O A307393 0,3 %A A307393 _Seiichi Manyama_, Apr 07 2019 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE