A091149 -id:A091149 - cp's OEIS frontend

A217275 Expansion of 2/(1-x+sqrt(1-2x-27x^2)).

1, 1, 8, 22, 141, 561, 3291, 15583, 88691, 459187, 2599570, 14136200, 80391235, 450046143, 2579291352, 14710321998, 85002979083, 491050703739, 2859262171872, 16674374605722, 97747766045679, 574231140306699, 3385974360904227, 20009363692187115, 118582649963026677

Offset: 0

Vaclav Kotesovec, Sep 29 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001006 (z=1), A025235 (z=2), A025237 (z=3), A091147 (z=4), A091148 (z=5), A091149 (z=6).

Table[SeriesCoefficient[2/(1-x+Sqrt[1-2*x-27*x^2]),{x,0,n}],{n,0,25}]
Table[Sum[Binomial[n,2k]*Binomial[2k,k]*7^k/(k+1),{k,0,n}],{n,0,25}]

Generally for G.f. = 2/(1-x+sqrt(1-2x-(4*z-1)*x^2)) is asymptotic
a(n) ~ (1+2*sqrt(z))^(n+3/2)/(2*sqrt(Pi)*z^(3/4)*n^(3/2)); here we have the case z=7.
D-finite with recurrence: (n+2)*a(n)=(2*n+1)*a(n-1)+(4*z-1)*(n-1)*a(n-2);; here with z=7.
G.f.: 1/(1 - x - 7*x^2/(1 - x - 7*x^2/(1 - x - 7*x^2/(1 - x - 7*x^2/(1 - ....))))), a continued fraction. - Ilya Gutkovskiy, May 26 2017

A306684 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2x + (1-4k)*x^2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 7, 9, 1, 1, 1, 5, 10, 21, 21, 1, 1, 1, 6, 13, 37, 61, 51, 1, 1, 1, 7, 16, 57, 121, 191, 127, 1, 1, 1, 8, 19, 81, 201, 451, 603, 323, 1, 1, 1, 9, 22, 109, 301, 861, 1639, 1961, 835, 1

Offset: 0

Seiichi Manyama, May 06 2019

			Square array begins:
   1,   1,   1,    1,    1,    1,     1,     1, ...
   1,   1,   1,    1,    1,    1,     1,     1, ...
   1,   2,   3,    4,    5,    6,     7,     8, ...
   1,   4,   7,   10,   13,   16,    19,    22, ...
   1,   9,  21,   37,   57,   81,   109,   141, ...
   1,  21,  61,  121,  201,  301,   421,   561, ...
   1,  51, 191,  451,  861, 1451,  2251,  3291, ...
   1, 127, 603, 1639, 3445, 6231, 10207, 15583, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..7 give A000012, A001006, A025235, A025237, A091147, A091148, A091149, A217275.
Main diagonal gives A307906.
Cf. A000108, A107267, A247495, A307855.

T[n_, k_] := Sum[If[k == j == 0, 1, k^j] * Binomial[n, 2*j] * CatalanNumber[j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 12 2021 *)

A(n,k) is the coefficient of x^n in the expansion of 1/(n+1) * (1 + x + k*x^2)^(n+1).
A(n,k) = Sum_{j=0..floor(n/2)} k^j * binomial(n,j) * binomial(n-j,j)/(j+1) = Sum_{j=0..floor(n/2)} k^j * binomial(n,2*j) * A000108(j).
(n+2) * A(n,k) = (2*n+1) * A(n-1,k) - (1-4*k) * (n-1) * A(n-2,k).

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A217275 Expansion of 2/(1-x+sqrt(1-2x-27x^2)).

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A306684 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2x + (1-4k)*x^2)).

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cp's OEIS Frontend

A217275 Expansion of 2/(1-x+sqrt(1-2*x-27*x^2)).

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A306684 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2*x + (1-4*k)*x^2)).

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A217275 Expansion of 2/(1-x+sqrt(1-2x-27x^2)).

A306684 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2x + (1-4k)*x^2)).