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A389323

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

1, 1, 11, 76, 531, 3851, 28382, 211548, 1590899, 12047170, 91734911, 701710109, 5387995098, 41504066813, 320589683754, 2482245945851, 19259593166083, 149709668838670, 1165639250712398, 9088980758041868, 70964191523639231, 554730210004015821, 4341059335650606723

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OFFSET

0,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = [x^n] (1 + x / (1 - x)^5)^n.

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x / (1 + x / (1 - x)^5) ). See A321799.

MATHEMATICA

Table[Sum[Binomial[n, k]Binomial[n+4*k-1, n-k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Oct 09 2025 *)

PROG

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(n+4*k-1, n-k));