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A323694 a(n) = [x^n] G(x)^(n+1) / (n+1)^2 for n >= 0, where G(x) is the g.f. of A323693.

%I A323694 #12 Feb 20 2019 12:36:58
%S A323694 1,1,6,80,1780,58212,2609824,153429696,11457990000,1060569950000,
%T A323694 119272908950624,16028231512305792,2537651853233579264,
%U A323694 467637448273232473600,99254069921234336025600,24041252393859722676492288,6592065384977916153080977152,2031547109837445197745286936320,699189635655553248110260224524800
%N A323694 a(n) = [x^n] G(x)^(n+1) / (n+1)^2 for n >= 0, where G(x) is the g.f. of A323693.
%C A323694 The g.f. of A323693, G(x), satisfies: [x^n] G(x)^(n+1) = (n+1)^2 * [x^(n-1)] G(x)^(n+1) for n >= 1.
%H A323694 Paul D. Hanna, <a href="/A323694/b323694.txt">Table of n, a(n) for n = 0..300</a>
%F A323694 a(n) = [x^(n-1)] G(x)^(n+1) for n >= 1, where G(x) is the g.f. of A323693.
%e A323694 The g.f. of A323693 begins
%e A323694 G(x) = 1 + 2*x + 14*x^2 + 228*x^3 + 6332*x^4 + 255800*x^5 + 13862744*x^6 + 962576816*x^7 + 83146713104*x^8 + 8746885895136*x^9 + ...
%e A323694 The table of coefficients of x^k in G(x)^n starts as
%e A323694 n=1: [1, 2, 14, 228, 6332, 255800, 13862744, ...];
%e A323694 n=2: [1, 4, 32, 512, 13772, 543312, 28977968, ...];
%e A323694 n=3: [1, 6, 54, 860, 22488, 866448, 45462704, ...];
%e A323694 n=4: [1, 8, 80, 1280, 32664, 1229568, 63445984, ...];
%e A323694 n=5: [1, 10, 110, 1780, 44500, 1637512, 83069960, ...];
%e A323694 n=6: [1, 12, 144, 2368, 58212, 2095632, 104491088, ...];
%e A323694 n=7: [1, 14, 182, 3052, 74032, 2609824, 127881376, ...]; ...
%e A323694 RELATED SEQUENCES.
%e A323694 In the above table, the main diagonal begins
%e A323694 [1, 4, 54, 1280, 44500, 2095632, 127881376, 9819500544, ...]
%e A323694 which, when divided by n^2, yields this sequence:
%e A323694 [1, 1, 6, 80, 1780, 58212, 2609824, 153429696, 11457990000, ...]
%e A323694 and also yields the secondary diagonal in the above table.
%o A323694 (PARI) {a(n) = my(A=[1], V); for(m=2, n+1, A=concat(A, 0); V=Vec(Ser(A)^m); A[#A] = V[#A-1]*m - V[#A]/m ); polcoeff( Ser(A)^(n+1),n)/(n+1)^2}
%o A323694 for(n=0, 20, print1(a(n), ", "))
%Y A323694 Cf. A323693.
%K A323694 nonn
%O A323694 0,3
%A A323694 _Paul D. Hanna_, Feb 20 2019