A349289 G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(x)^3)).

1, 2, 9, 61, 493, 4371, 41065, 401563, 4044097, 41658044, 436862457, 4648331765, 50057856881, 544557984498, 5975422922413, 66059269445451, 735064865871889, 8226310738656892, 92531697191189777, 1045551973586825023, 11862334695799444993

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..500

Formula

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

a(n) = F([1/4, 1/2, 3/4, (1+n)/2, (2+n)/2, -n], [1/3, 2/3, 2/3, 1, 4/3], -2^10/3^6) where F is the generalized hypergeometric function. - Stefano Spezia, Nov 13 2021

a(n) ~ sqrt(1 + 2*r) / (4 * 2^(1/6) * sqrt(3*Pi*(1-r)) * n^(3/2) * r^(n + 1/3)), where r = 0.0816785448577670972635343365300887975661075663022821172271... is the root of the equation 4^4 * r = 3^3 * (1-r)^3. - Vaclav Kotesovec, Nov 14 2021

D-finite with recurrence 81*n*(3*n-1)*(3*n+1)*a(n) +3*(243*n^3-7101*n^2+9986*n-3560)*a(n-1) +(-115027*n^3+514908*n^2-699869*n+269580)*a(n-2) +(-85543*n^3+1604715*n^2-6291692*n+6995280)*a(n-3) +(580211*n^3-6643158*n^2+23063299*n-23830944)*a(n-4) +(-33473*n^3-2231073*n^2+26352470*n-70945392)*a(n-5) +(-872129*n^3+17812344*n^2-119542699*n+264170868)*a(n-6) +(667171*n^3-14196243*n^2+100393472*n-236010000)*a(n-7) -6*(3*n-23)*(9948*n^2-147805*n+548868)*a(n-8) +4044*(3*n-26)*(n-8)*(3*n-22)*a(n-9)=0. - R. J. Mathar, Feb 10 2024

Maple

A349289 := proc(n)
    add( binomial(n+2*k, 3*k)*binomial(4*k, k)/(3*k+1), k=0..n)  ;
end proc:
seq(A349289(n), n=0..50) ; # R. J. Mathar, Feb 10 2024

Mathematica

nmax = 20; A[_] = 0; Do[A[x_] = 1/((1 - x) (1 - x A[x]^3)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 2 k, 3 k] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 20}]

Crossrefs

Cf. A002293, A007317, A199475, A346646, A349290, A349291, A349292, A349293.

Sequence in context: A207649 A289713 A341956 * A053983 A370161 A192939

Adjacent sequences: A349286 A349287 A349288 * A349290 A349291 A349292

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 13 2021

Status

approved