A388728 - OEIS

%I #15 Sep 22 2025 14:45:52

%S 1,8,112,1778,29776,513833,9039286,161164599,2902000592,52652740496,

%T 961069474297,17628176117318,324649112983798,5999256527934392,

%U 111183502998164219,2065716744515378213,38463565347521894032,717567269332838675232,13409586559291172964976

%N a(n) = Sum_{k=0..n} binomial(n,k) * binomial(3*n+4*k,k).

%H Vincenzo Librandi, <a href="/A388728/b388728.txt">Table of n, a(n) for n = 0..750</a>

%F a(n) = [x^n] ((1+x)^3 * (x+(1+x)^4))^n.

%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+(1+x)^4)) ).

%t Table[Sum[Binomial[ n,k]*Binomial[3*n+4*k,k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Sep 22 2025 *)

%o (PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(3*n+4*k, k));

%o (Magma) [&+[Binomial(n, k)*Binomial(3*n+4*k,k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Sep 22 2025

%Y Cf. A388535, A388727, A388729.

%Y Cf. A156886. A388726.

%Y Cf. A378685.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Sep 20 2025