cp's OEIS Frontend

A107675 Column 0 of triangle A107674.

%I A107675 #28 Mar 11 2021 17:43:10
%S A107675 1,24,2268,461056,160977375,85624508376,64363893844726,
%T A107675 64928246784463872,84623205378726331245,138408056280920732755000,
%U A107675 277597038523589348539241112,670011760601512512626484887040
%N A107675 Column 0 of triangle A107674.
%C A107675 The o.g.f. A(x) = Sum_{m >= 0} a(m)*x^m is such that, for each integer n > 0, the coefficient of x^n in the expansion of exp(n^3*x)*(1 - x*A(x)) is equal to 0.
%C A107675 Given the o.g.f. A(x), the o.g.f. of A304323 equals 1/(1 - x*A(x)).
%F A107675 O.g.f. A(x) satisfies: [x^n] exp(n^3*x) * (1 - x*A(x)) = 0 for n > 0. - _Paul D. Hanna_, May 12 2018
%F A107675 a(n) = (n+1)^(3*n+3)/(n+1)! - Sum_{k=1..n} (n+1)^(3*k)/k! * a(n-k) for n > 0 with a(0) = 1. - _Paul D. Hanna_, May 12 2018
%F A107675 a(n) = A342202(3,n+1) = Sum_{r=1..(n+1)} (-1)^(r-1) * Sum_{s_1, ..., s_r} (1/(Product_{j=1..r} s_j!)) * Product_{j=1..r} (Sum_{i=1..j} s_i)^(3*s_j)), where the second sum is over lists (s_1, ..., s_r) of positive integers s_i such that Sum_{i=1..r} s_i = n+1. (Thus the second sum is over all compositions of n+1. See _Michel Marcus_'s PARI program in A342202.) - _Petros Hadjicostas_, Mar 10 2021
%e A107675 O.g.f.: A(x) = 1 + 24*x + 2268*x^2 + 461056*x^3 + 160977375*x^4 + 85624508376*x^5 + 64363893844726*x^6 + 64928246784463872*x^7 + ...
%o A107675 (PARI) {a(n)=local(P=matrix(n+1,n+1,r,c,if(r>=c,(r^3)^(r-c)/(r-c)!)), D=matrix(n+1,n+1,r,c,if(r==c,r)));(P^-1*D^2*P)[n+1,1]}
%o A107675 for(n=0,20, print1(a(n),", "))
%o A107675 (PARI) /* From formula: [x^n] exp( n^3*x ) * (1 - x*A(x)) = 0 */
%o A107675 {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*m^3 +x^2*O(x^m)) * (1 - x*Ser(A)) )[m+1] ); A[n+1]}
%o A107675 for(n=0, 25, print1( a(n), ", ")) \\ _Paul D. Hanna_, May 12 2018
%o A107675 (PARI) /* From Recurrence: */
%o A107675 {a(n) = if(n==0,1, (n+1)^(3*n+3)/(n+1)! - sum(k=1,n, (n+1)^(3*k)/k! * a(n-k) ))}
%o A107675 for(n=0,25, print1( a(n),", ")) \\ _Paul D. Hanna_, May 12 2018
%Y A107675 Cf. A107671, A107674, A107676.
%Y A107675 Cf. A304323, A107668, A304394, A304395, A342202.
%K A107675 nonn
%O A107675 0,2
%A A107675 _Paul D. Hanna_, Jun 07 2005