cp's OEIS Frontend

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

%I A192442 #26 Jan 12 2025 17:36:18
%S A192442 1,0,0,3,4,0,15,42,28,84,360,495,715,2860,6006,8463,23660,61880,
%T A192442 104244,220932,596904,1201560,2313003,5753979,12983707,25477100,
%U A192442 57557500,135227235,280913490,600900300,1395727515,3046800900,6447717900,14540497920,32572229976,69844899432
%N A192442 Coefficient of x^n in the expansion of (1+x^3+x^4)^n.
%H A192442 Vincenzo Librandi, <a href="/A192442/b192442.txt">Table of n, a(n) for n = 0..200</a>
%F A192442 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
%F A192442 a(n) = sum(j=floor(n/4)..floor(n/3), binomial(j,n-3*j)*binomial(n,j)). - _Vladimir Kruchinin_, Jun 14 2014
%F A192442 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
%t A192442 Table[Coefficient[(1+x^3+x^4)^n, x, n],{n,0,20}] (* _Vaclav Kotesovec_, Jun 15 2014 *)
%o A192442 (PARI) a(n)=polcoeff((1+x^3+x^4)^n,n);
%o A192442 (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
%o A192442 for(n=0,30,print1(a(n),", "))
%o A192442 (Magma) P<x>:=PolynomialRing(Integers()); [ Coefficients((1+x^3+x^4)^n)[ n+1 ]: n in [0..40] ]; // _Vincenzo Librandi_, Sep 10 2011
%o A192442 (Maxima) makelist((coeff(expand((1+x^3+x^4)^n), x, n)), n, 0, 40); /* _Vincenzo Librandi_, Sep 10 2011 */
%o A192442 (Maxima)
%o A192442 a(n):=sum(binomial(j,n-3*j)*binomial(n,j),j,floor(n/4),floor(n/3)); /* _Vladimir Kruchinin_, Jun 14 2014 */
%Y A192442 Cf. A002426, A192440, A192441.
%K A192442 nonn
%O A192442 0,4
%A A192442 _Joerg Arndt_, Jul 01 2011