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%I #24 Mar 11 2021 17:42:36
%S 1,4,45,816,20225,632700,23836540,1048592640,52696514169,
%T 2976295383100,186548057815801,12845016620629488,963644465255618276,
%U 78224633235142116240,6830914919397129328500,638477522900795994967040
%N Column 0 of triangle A107667.
%C Shift right of column 1 of triangle A107670, which is the matrix square of triangle A107667.
%C 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^2*x)*(1 - x*A(x)) is equal to 0.
%C Given the o.g.f. A(x), the o.g.f. of A304322 equals 1/(1 - x*A(x)).
%F O.g.f. A(x) satisfies: [x^n] exp( n^2*x ) * (1 - x*A(x)) = 0 for n > 0. - _Paul D. Hanna_, May 12 2018
%F a(n) = (n+1)^2 * A107669(n).
%F a(n) = (n+1)^(2*n+2)/(n+1)! - Sum_{k=1..n} (n+1)^(2*k)/k! * a(n-k) for n > 0 with a(0) = 1. - _Paul D. Hanna_, May 12 2018
%F a(n) = A342202(2,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)^(2*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 ordered partitions (i.e., compositions) of n+1. See _Michel Marcus_'s PARI program in A342202.) - _Petros Hadjicostas_, Mar 10 2021
%e O.g.f.: A(x) = 1 + 4*x + 45*x^2 + 816*x^3 + 20225*x^4 + 632700*x^5 + 23836540*x^6 + 1048592640*x^7 + 52696514169*x^8 + 2976295383100*x^9 + ...
%e From _Petros Hadjicostas_, Mar 10 2021: (Start)
%e We illustrate the above formula for a(n) with the compositions of n + 1 for n = 2.
%e The compositions of n + 1 = 3 are 3, 1 + 2, 2 + 1, and 1 + 1 + 1. Thus the above sum has four terms with (r = 1, s_1 = 3), (r = 2, s_1 = 1, s_2 = 2), (r = 2, s_1 = 2, s_2 = 1), and (r = 3, s_1 = s_2 = s_3 = 1).
%e The value of the denominator Product_{j=1..r} s_j! for these four terms is 6, 2, 2, and 1, respectively.
%e The value of the numerator Product_{j=1..r} (Sum_{i=1..j} s_i)^(2*s_j) for these four terms is 729, 81, 144, and 36.
%e Thus a(2) = 729/6 - 81/2 - 144/2 + 36/1 = 45. (End)
%o (PARI) {a(n)=local(A);if(n==0,n+1,A=(n+1)*x+x*O(x^n); for(k=0,n,A+=polcoeff(A,k)*x^k*(n+1-prod(i=0,k,1+(i-n-1)*x))); polcoeff(A,n))}
%o for(n=0,30, print1(a(n),", "))
%o (PARI) /* From formula: [x^n] exp( n^2*x ) * (1 - x*A(x)) = 0 */
%o {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*m^2 +x^2*O(x^m)) * (1 - x*Ser(A)) )[m+1] ); A[n+1]}
%o for(n=0,25, print1( a(n),", ")) \\ _Paul D. Hanna_, May 12 2018
%o (PARI) /* From Recurrence: */
%o {a(n) = if(n==0,1, (n+1)^(2*n+2)/(n+1)! - sum(k=1,n, (n+1)^(2*k)/k! * a(n-k) ))}
%o for(n=0,25, print1( a(n),", ")) \\ _Paul D. Hanna_, May 12 2018
%Y Cf. A107667, A107669, A107670.
%Y Cf. A304322, A107675, A304394, A304395, A342202.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 07 2005
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