cp's OEIS Frontend

A323212 The Fibonacci-Catalan Hybrid. Expansion of 1 + x*(2*(x + 1))/(sqrt(1 - 4*y) - 2*x*(x + 1) + 1). Square array read by descending antidiagonals, A(n,k) for n,k >= 0.

%I A323212 #13 Mar 20 2019 11:58:10
%S A323212 1,0,1,0,1,2,0,2,3,3,0,5,7,7,5,0,14,19,19,15,8,0,42,56,56,46,30,13,0,
%T A323212 132,174,174,146,103,58,21,0,429,561,561,477,351,220,109,34,0,1430,
%U A323212 1859,1859,1595,1205,801,453,201,55,0,4862,6292,6292,5434,4180,2884,1756,908,365,89
%N A323212 The Fibonacci-Catalan Hybrid. Expansion of 1 + x*(2*(x + 1))/(sqrt(1 - 4*y) - 2*x*(x + 1) + 1). Square array read by descending antidiagonals, A(n,k) for n,k >= 0.
%e A323212       1,   0,    0,    0,     0,      0,      0,       0,       0, ...
%e A323212       1,   1,    2,    5,    14,     42,    132,     429,    1430, ... [A000108]
%e A323212       2,   3,    7,   19,    56,    174,    561,    1859,    6292, ... [A005807]
%e A323212       3,   7,   19,   56,   174,    561,   1859,    6292,   21658, ... [A005807]
%e A323212       5,  15,   46,  146,   477,   1595,   5434,   18798,   65858, ...
%e A323212       8,  30,  103,  351,  1205,   4180,  14651,   51844,  185028, ...
%e A323212      13,  58,  220,  801,  2884,  10372,  37401,  135420,  492558, ...
%e A323212      21, 109,  453, 1756,  6621,  24674,  91532,  339184, 1257762, ...
%e A323212      34, 201,  908, 3734, 14719,  56796, 216698,  821848, 3107583, ...
%e A323212      55, 365, 1781, 7746, 31872, 127245, 499164, 1937439, 7470819, ...
%e A323212 A000045,A023610,...
%e A323212 Seen as a triangle a refinement of A000958:
%e A323212 [0]                                1
%e A323212 [1]                              0, 1
%e A323212 [2]                            0, 1, 2
%e A323212 [3]                           0, 2, 3, 3
%e A323212 [4]                         0, 5, 7, 7, 5
%e A323212 [5]                      0, 14, 19, 19, 15, 8
%e A323212 [6]                   0, 42, 56, 56, 46, 30, 13
%e A323212 [7]               0, 132, 174, 174, 146, 103, 58, 21
%e A323212 [8]            0, 429, 561, 561, 477, 351, 220, 109, 34
%e A323212 [9]       0, 1430, 1859, 1859, 1595, 1205, 801, 453, 201, 55
%p A323212 gf := 1 + x*(2*(x + 1))/(sqrt(1 - 4*y) - 2*x*(x + 1) + 1):
%p A323212 serx := series(gf, x, 20): sery := n -> series(coeff(serx, x, n), y, 20):
%p A323212 row := n -> seq(coeff(sery(n), y, j), j=0..9):
%p A323212 seq(lprint(row(n)), n=0..9);
%t A323212 m = 11; T = PadRight[CoefficientList[#+O[y]^m, y], m]& /@ CoefficientList[1 + 2x(x+1)/(Sqrt[1-4y] - 2x(x+1) + 1) + O[x]^m, x]; Table[T[[n-k+1, k]], {n, 1, m}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Mar 20 2019 *)
%Y A323212 Antidiagonal sums (or row sums of the triangle) are A000958.
%Y A323212 Cf. A000045, A000108, A023610, A005807.
%K A323212 nonn,easy,tabl
%O A323212 0,6
%A A323212 _Peter Luschny_, Feb 14 2019