cp's OEIS Frontend

A058395 Square array read by antidiagonals. Based on triangular numbers (A000217) with each term being the sum of 2 consecutive terms in the previous row.

%I A058395 #14 Apr 13 2023 14:22:45
%S A058395 1,0,1,3,1,1,0,3,2,1,6,3,4,3,1,0,6,6,6,4,1,10,6,9,10,9,5,1,0,10,12,15,
%T A058395 16,13,6,1,15,10,16,21,25,25,18,7,1,0,15,20,28,36,41,38,24,8,1,21,15,
%U A058395 25,36,49,61,66,56,31,9,1,0,21,30,45,64,85,102,104,80,39,10,1,28,21,36,55,81,113,146,168,160,111,48,11,1
%N A058395 Square array read by antidiagonals. Based on triangular numbers (A000217) with each term being the sum of 2 consecutive terms in the previous row.
%C A058395 Changing the formula by replacing T(2n, 0) = T(n, 3) with T(2n, 0) = T(n, m) for some other value of m would change the generating function to the coefficient of x^n in expansion of (1 + x)^k / (1 - x^2)^m. This would produce A058393, A058394, A057884 (and effectively A007318).
%F A058395 T(n, k) = T(n-1, k-1) + T(n, k-1) with T(0, k) = 1, T(2*n, 0) = T(n, 3) and T(2*n + 1, 0) = 0. Coefficient of x^n in expansion of (1 + x)^k / (1 - x^2)^3.
%e A058395 The array T(n, k) starts:
%e A058395 [0] 1, 0,  3,   0,   6,   0,  10,    0,   15,    0, ...
%e A058395 [1] 1, 1,  3,   3,   6,   6,  10,   10,   15,   15, ...
%e A058395 [2] 1, 2,  4,   6,   9,  12,  16,   20,   25,   30, ...
%e A058395 [3] 1, 3,  6,  10,  15,  21,  28,   36,   45,   55, ...
%e A058395 [4] 1, 4,  9,  16,  25,  36,  49,   64,   81,  100, ...
%e A058395 [5] 1, 5, 13,  25,  41,  61,  85,  113,  145,  181, ...
%e A058395 [6] 1, 6, 18,  38,  66, 102, 146,  198,  258,  326, ...
%e A058395 [7] 1, 7, 24,  56, 104, 168, 248,  344,  456,  584, ...
%e A058395 [8] 1, 8, 31,  80, 160, 272, 416,  592,  800, 1040, ...
%e A058395 [9] 1, 9, 39, 111, 240, 432, 688, 1008, 1392, 1840, ...
%p A058395 gf := n -> (1 + x)^n / (1 - x^2)^3: ser := n -> series(gf(n), x, 20):
%p A058395 seq(lprint([n], seq(coeff(ser(n), x, k), k = 0..9)), n = 0..9); # _Peter Luschny_, Apr 12 2023
%t A058395 T[0, k_] := If[OddQ[k], 0, (k+2)(k+4)/8];
%t A058395 T[n_, k_] := T[n, k] = If[k == 0, 1, T[n-1, k-1] + T[n-1, k]];
%t A058395 Table[T[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Apr 13 2023 *)
%Y A058395 Rows are A000217 with zeros, A008805, A002620, A000217, A000290, A001844, A005899.
%Y A058395 Columns are A000012, A001477, A016028.
%Y A058395 Diagonals include A058396, A049611, A001793, A001788, A055580, A055581, A055582.
%Y A058395 The triangle A055252 also appears in half of the array.
%K A058395 nonn,tabl
%O A058395 0,4
%A A058395 _Henry Bottomley_, Nov 24 2000