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.

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, 132, 174, 174, 146, 103, 58, 21, 0, 429, 561, 561, 477, 351, 220, 109, 34, 0, 1430, 1859, 1859, 1595, 1205, 801, 453, 201, 55, 0, 4862, 6292, 6292, 5434, 4180, 2884, 1756, 908, 365, 89

Offset

0,6

Links

Table of n, a(n) for n=0..65.

Example

      1,   0,    0,    0,     0,      0,      0,       0,       0, ...
      1,   1,    2,    5,    14,     42,    132,     429,    1430, ... [A000108]
      2,   3,    7,   19,    56,    174,    561,    1859,    6292, ... [A005807]
      3,   7,   19,   56,   174,    561,   1859,    6292,   21658, ... [A005807]
      5,  15,   46,  146,   477,   1595,   5434,   18798,   65858, ...
      8,  30,  103,  351,  1205,   4180,  14651,   51844,  185028, ...
     13,  58,  220,  801,  2884,  10372,  37401,  135420,  492558, ...
     21, 109,  453, 1756,  6621,  24674,  91532,  339184, 1257762, ...
     34, 201,  908, 3734, 14719,  56796, 216698,  821848, 3107583, ...
     55, 365, 1781, 7746, 31872, 127245, 499164, 1937439, 7470819, ...
A000045,A023610,...
Seen as a triangle a refinement of A000958:
[0]                                1
[1]                              0, 1
[2]                            0, 1, 2
[3]                           0, 2, 3, 3
[4]                         0, 5, 7, 7, 5
[5]                      0, 14, 19, 19, 15, 8
[6]                   0, 42, 56, 56, 46, 30, 13
[7]               0, 132, 174, 174, 146, 103, 58, 21
[8]            0, 429, 561, 561, 477, 351, 220, 109, 34
[9]       0, 1430, 1859, 1859, 1595, 1205, 801, 453, 201, 55

Maple

gf := 1 + x*(2*(x + 1))/(sqrt(1 - 4*y) - 2*x*(x + 1) + 1):
serx := series(gf, x, 20): sery := n -> series(coeff(serx, x, n), y, 20):
row := n -> seq(coeff(sery(n), y, j), j=0..9):
seq(lprint(row(n)), n=0..9);

Mathematica

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 *)

Crossrefs

Antidiagonal sums (or row sums of the triangle) are A000958.

Cf. A000045, A000108, A023610, A005807.

Sequence in context: A074660 A002125 A171731 * A185815 A332448 A321132

Adjacent sequences: A323209 A323210 A323211 * A323213 A323214 A323215

Keyword

nonn,easy,tabl

Author

Peter Luschny, Feb 14 2019

Status

approved