A378783 Triangular array T(n,k) read by rows: T(n, k) = c_k(n+1). The sequence c_k(m) has the ordinary generating function C_k(x) which satisfies C_k(x) = 1/(1+C_k(x)*Sum_{t=0..k} x^(t+1)).

-1, 2, 1, -5, -1, -2, 14, 1, 5, 4, -42, -1, -12, -8, -9, 132, 1, 29, 18, 22, 21, -429, -1, -73, -43, -54, -50, -51, 1430, 1, 190, 105, 135, 124, 128, 127, -4862, -1, -505, -262, -345, -315, -326, -322, -323, 16796, 1, 1363, 666, 896, 813, 843, 832, 836, 835

Offset

0,2

Links

Table of n, a(n) for n=0..54.

Formula

G.f. column k: (2 / (sqrt(1+4*Sum_{t=0..k}x^(t+1)) + 1) - 1)/x.

T(n, 0) = (-1)^(n+1)*Catalan(n+1) = A168491(n+1).

T(n, 2) = (-1)^(n+1)*A152171(n+1).

T(n, n) = (-1)^(n+1)*A001006(n) = -A166587(n+1).

A378816(n) = Limit_{k->oo} (T(k, k-n) - T(k, k-n-1)).

Example

Triangle begins:
  [0]    -1
  [1]     2,  1
  [2]    -5, -1,   -2
  [3]    14,  1,    5,    4
  [4]   -42, -1,  -12,   -8,   -9
  [5]   132,  1,   29,   18,   22,   21
  [6]  -429, -1,  -73,  -43,  -54,  -50,  -51
  [7]  1430,  1,  190,  105,  135,  124,  128,  127
  [8] -4862, -1, -505, -262, -345, -315, -326, -322, -323
.

Mathematica

T[n_, k_]:=SeriesCoefficient[(2 / (Sqrt[1+4*Sum[x^(t+1), {t, 0, k}]] + 1) - 1)/x, {x, 0, n}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}]//Flatten (* Stefano Spezia, Dec 08 2024 *)

Prog

(PARI)
column(n, max_n) = { my(s = 1, x = 'x, cu); for(k = 0, max_n-1, cu = cu+polcoeff(1/s+O(x^(k+1)), k, x); cu = cu-polcoeff(1/s+O(x^(k+1)), k-1-n, x); s = s+cu*x^(k+1)); Vec(1/s+O(x^max_n)) };
T(n, k) = column(k, n+2)[n+2]
T(n, k) = polcoeff(2 / (sqrt(1+4*x*sum(t=0, k, x^t)) + 1) + O(x^(n+2)), n+1, x)

Crossrefs

Cf. A001006, A000108, A152171, A166587, A168491, A378816.

Sequence in context: A345355 A132601 A047818 * A055972 A342414 A374964

Adjacent sequences: A378780 A378781 A378782 * A378784 A378785 A378786

Keyword

sign,tabl

Author

Thomas Scheuerle, Dec 07 2024

Status

approved