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A107675
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Column 0 of triangle A107674.
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6
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1, 24, 2268, 461056, 160977375, 85624508376, 64363893844726, 64928246784463872, 84623205378726331245, 138408056280920732755000, 277597038523589348539241112, 670011760601512512626484887040
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OFFSET
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0,2
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COMMENTS
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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.
Given the o.g.f. A(x), the o.g.f. of A304323 equals 1/(1 - x*A(x)).
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LINKS
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FORMULA
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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
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
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
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EXAMPLE
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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 + ...
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PROG
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(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]}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* From formula: [x^n] exp( n^3*x ) * (1 - x*A(x)) = 0 */
{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]}
(PARI) /* From Recurrence: */
{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) ))}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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