login
Coefficient of x^n in the expansion of ( (1+x)^3 + x^2 )^n.
1

%I #6 Feb 14 2024 10:47:55

%S 1,3,17,102,645,4193,27764,186231,1261213,8604759,59053167,407217396,

%T 2819252544,19583729766,136426565999,952743556907,6667916884701,

%U 46755146944959,328398159653117,2310073990369926,16271915501598595,114757849228310355

%N Coefficient of x^n in the expansion of ( (1+x)^3 + x^2 )^n.

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(3*n-3*k,n-2*k).

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

%o (PARI) a(n) = sum(k=0, n\2, binomial(n, k)*binomial(3*n-3*k, n-2*k));

%Y Cf. A002426, A006139.

%Y Cf. A188686, A370287.

%Y Cf. A065065.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Feb 14 2024