A349535 G.f. A(x) satisfies A(x) = 1 / ((1 - 4 * x) * (1 - x * A(x)^2)).

1, 5, 31, 232, 2007, 19213, 196912, 2114360, 23467039, 266945359, 3094963919, 36433302124, 434269638196, 5230582604476, 63560612650120, 778281652963504, 9593248679529551, 118938874458415987, 1482248642350235245, 18557299563105608336

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..887

Formula

a(n) = 4^n + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).

a(n) = Sum_{k=0..n} binomial(n+k,2*k) * binomial(3*k,k) * 4^(n-k) / (2*k+1).

a(n) = 4^n*F([1/3, 2/3, -n, 1+n], [1/2, 1, 3/2], -3^3/2^6), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 21 2021

a(n) ~ 273^(1/4) * (59 + 3*sqrt(273))^(n + 1/2) / (9 * sqrt(Pi) * n^(3/2) * 2^(3*n + 5/2)). - Vaclav Kotesovec, Nov 22 2021

Mathematica

nmax = 19; A[_] = 0; Do[A[x_] = 1/((1 - 4 x) (1 - x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 4^n + Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 19}]
Table[Sum[Binomial[n + k, 2 k] Binomial[3 k, k] 4^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 19}]

Crossrefs

Cf. A001764, A104498, A199475, A349533, A349534.

Sequence in context: A052773 A062147 A213048 * A069321 A211179 A177797

Adjacent sequences: A349532 A349533 A349534 * A349536 A349537 A349538

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 21 2021

Status

approved