A371774 a(n) = Sum_{k=0..floor(n/3)} binomial(3*n-k+1,n-3*k).

1, 4, 21, 121, 727, 4473, 27949, 176549, 1124332, 7205511, 46411744, 300183757, 1948255421, 12681654613, 82755728730, 541213820732, 3546268982757, 23276100962571, 153004515241866, 1007131032951572, 6637396253259291, 43791520333601111

Offset

0,2

Links

Table of n, a(n) for n=0..21.

Formula

a(n) = [x^n] 1/(((1-x)^2-x^3) * (1-x)^(2*n)).

a(n) = binomial(1+3*n, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [-1-3*n, 1+n, 3/2+n], 27/4). - Stefano Spezia, Apr 06 2024

From Vaclav Kotesovec, Apr 08 2024: (Start)

Recurrence: 2*n*(2*n - 1)*(671*n^4 - 4757*n^3 + 11743*n^2 - 11533*n + 3516)*a(n) = (44957*n^6 - 350256*n^5 + 997889*n^4 - 1236792*n^3 + 563834*n^2 + 39768*n - 60480)*a(n-1) - 10*(19459*n^6 - 156741*n^5 + 461272*n^4 - 575421*n^3 + 211099*n^2 + 106572*n - 60480)*a(n-2) + (93269*n^6 - 753150*n^5 + 2221631*n^4 - 2772678*n^3 + 999800*n^2 + 543408*n - 302400)*a(n-3) - 3*(3*n - 8)*(3*n - 7)*(671*n^4 - 2073*n^3 + 1498*n^2 + 366*n - 360)*a(n-4).

a(n) ~ 3^(3*n + 5/2) / (11 * sqrt(Pi*n) * 2^(2*n)). (End)

Prog

(PARI) a(n) = sum(k=0, n\3, binomial(3*n-k+1, n-3*k));

Crossrefs

Cf. A371773, A371775, A371776.

Cf. A371754, A371770.

Cf. A045721.

Sequence in context: A046090 A045721 A101810 * A274969 A236525 A277292

Adjacent sequences: A371771 A371772 A371773 * A371775 A371776 A371777

Keyword

nonn

Author

Seiichi Manyama, Apr 05 2024

Status

approved