A387338 -id:A387338 - cp's OEIS frontend

A387340 a(n) = Sum_{k=0..n} 3^k * binomial(n+3,k) * binomial(n+3,k+3).

1, 16, 175, 1640, 14189, 117152, 939036, 7379040, 57188010, 438810592, 3342302821, 25316084248, 190937278805, 1435287936320, 10760879892008, 80509920297792, 601343784616830, 4485466826475360, 33420579148668670, 248788060638391120, 1850652536242372786

Offset: 0

Seiichi Manyama, Aug 27 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A069835, A331792, A387339.

Cf. A002696, A387338.

[&+[3^k * Binomial(n+3,k) * Binomial(n+3,k+3): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025

Table[Sum[3^k * Binomial[n+3,k]*Binomial[n+3, k+3],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 29 2025 *)

a(n) = sum(k=0, n, 3^k*binomial(n+3, k)*binomial(n+3, k+3));

n*(n+6)*a(n) = (n+3) * (4*(2*n+5)*a(n-1) - 4*(n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 3^k * 4^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+4*x+3*x^2)^(n+3).
E.g.f.: exp(4*x) * BesselI(3, 2*sqrt(3)*x) / (3*sqrt(3)), with offset 3.

A387342 Expansion of 1/(1 - 6*x + x^2)^(7/2).

Original entry on oeis.org

1, 21, 280, 3024, 28854, 253638, 2103024, 16689816, 128014887, 955485531, 6974119152, 49965080256, 352366829724, 2451595670748, 16858071545664, 114737706591984, 773866620578205, 5177539121330961, 34391021091689416, 226956883258736400, 1488970185631858930

Offset: 0

Seiichi Manyama, Aug 27 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A387338.

R := PowerSeriesRing(Rationals(), 34); f := 1/(1 - 6*x + x^2)^(7/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025

CoefficientList[Series[1/(1-6*x+x^2)^(7/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)

my(N=30, x='x+O('x^N)); Vec(1/(1-6*x+x^2)^(7/2))

n*a(n) = 3*(2*n+5)*a(n-1) - (n+5)*a(n-2) for n > 1.
a(n) = (binomial(n+6,3)/20) * A387338(n).
a(n) = (-1)^n * Sum_{k=0..n} (1/6)^(n-2*k) * binomial(-7/2,k) * binomial(k,n-k).

cp's OEIS Frontend

A387340 a(n) = Sum_{k=0..n} 3^k * binomial(n+3,k) * binomial(n+3,k+3).

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