A192442 Coefficient of x^n in the expansion of (1+x^3+x^4)^n.

1, 0, 0, 3, 4, 0, 15, 42, 28, 84, 360, 495, 715, 2860, 6006, 8463, 23660, 61880, 104244, 220932, 596904, 1201560, 2313003, 5753979, 12983707, 25477100, 57557500, 135227235, 280913490, 600900300, 1395727515, 3046800900, 6447717900, 14540497920, 32572229976, 69844899432

Offset: 0

Author

Joerg Arndt, Jul 01 2011

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A002426, A192440, A192441.

Programs

Magma

P:=PolynomialRing(Integers()); [ Coefficients((1+x^3+x^4)^n)[ n+1 ]: n in [0..40] ]; // Vincenzo Librandi, Sep 10 2011

Mathematica

Table[Coefficient[(1+x^3+x^4)^n, x, n],{n,0,20}] (* Vaclav Kotesovec, Jun 15 2014 *)

Maxima

makelist((coeff(expand((1+x^3+x^4)^n), x, n)), n, 0, 40); /* Vincenzo Librandi, Sep 10 2011 */

Maxima

a(n):=sum(binomial(j,n-3*j)*binomial(n,j),j,floor(n/4),floor(n/3)); /* Vladimir Kruchinin, Jun 14 2014 */

PARI

a(n)=polcoeff((1+x^3+x^4)^n,n);

PARI

{a(n)=local(G=(1/x)*serreverse(x/(1+x^3+x^4 +x^2*O(x^n))));polcoeff(1+x*G'/G, n)} \\ Paul D. Hanna, Jun 14 2014
for(n=0,30,print1(a(n),", "))

Formula

Recurrence: 3*n*(3*n-2)*(3*n-1)*(115668*n^3 - 650916*n^2 + 1167723*n - 673723)*a(n) = 2*(n-1)*(231336*n^5 - 1417500*n^4 + 3231306*n^3 - 3349145*n^2 + 1574119*n - 264960)*a(n-1) - 6*(n-2)*(n-1)*(115668*n^4 - 535248*n^3 + 861921*n^2 - 529129*n + 122640)*a(n-2) + 24*(n-2)*(n-1)*(925344*n^4 - 4744656*n^3 + 7608276*n^2 - 4369418*n + 755115)*a(n-3) + 229*(n-3)*(n-2)*(n-1)*(115668*n^3 - 303912*n^2 + 212895*n - 41248)*a(n-4). - Vaclav Kotesovec, Apr 21 2014

a(n) = sum(j=floor(n/4)..floor(n/3), binomial(j,n-3*j)*binomial(n,j)). - Vladimir Kruchinin, Jun 14 2014

G.f.: 1 + x*G'(x)/G(x) where G(x) = 1 + x^3*G(x)^3 + x^4*G(x)^4 = (1/x)*Series_Reversion(x/(1+x^3+x^4)). - Paul D. Hanna, Jun 14 2014