A386768 a(n) = Sum_{k=0..n} 5^k * 2^(n-k) * binomial(3*n+k-1,k).

1, 17, 589, 23063, 952421, 40527732, 1758058219, 77293019898, 3431959098741, 153547092814172, 6911193017626324, 312596792782451183, 14196172772254858211, 646897139198653660412, 29563753017571135649154, 1354477988702509748029668, 62191803671962046948722581

Offset

0,2

Links

Table of n, a(n) for n=0..16.

Formula

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

a(n) = [x^n] ( (1+2*x)^4/(1-3*x)^3 )^n.

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-3*x)^3 / (1+2*x)^4 ). See A386774.

a(n) = Sum_{k=0..n} 5^k * (-3)^(n-k) * binomial(4*n,k).

D-finite with recurrence +6561*n*(3*n-1)*(866874441*n -1379250238)*(3*n-2)*a(n) +648*(-3285736631046*n^4 +7965087872184*n^3 -621409760406*n^2 -9688518250831*n +5867806764110)*a(n-1) +1920*(-7264105318332*n^4 +60745334410890*n^3 -195779508237450*n^2 +280383483469585*n -148039402286753)*a(n-2) -51200*(2*n-5)*(4*n-9) *(4581663714*n-6698674013)*(4*n-11)*a(n-3)=0. - R. J. Mathar, Aug 03 2025

Recurrence (of order 2): 81*n*(3*n - 2)*(3*n - 1)*(972*n^3 - 4350*n^2 + 6422*n - 3125)*a(n) = 24*(1382184*n^6 - 8267724*n^5 + 19373616*n^4 - 22463754*n^3 + 13362532*n^2 - 3782080*n + 382725)*a(n-1) + 640*(2*n - 3)*(4*n - 7)*(4*n - 5)*(972*n^3 - 1434*n^2 + 638*n - 81)*a(n-2). - Vaclav Kotesovec, Nov 09 2025

Prog

(PARI) a(n) = sum(k=0, n, 5^k*2^(n-k)*binomial(3*n+k-1, k));

Crossrefs

Cf. A386766, A386767.

Cf. A386774.

Sequence in context: A041547 A041544 A202407 * A009709 A162490 A296790

Adjacent sequences: A386765 A386766 A386767 * A386769 A386770 A386771

Keyword

nonn

Author

Seiichi Manyama, Aug 02 2025

Status

approved