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%I #12 Jan 23 2018 13:12:22
%S 1,1,7,187,11517,1269821,218962723,54377141463,18394396344313,
%T 8139652855993369,4568235711128252991,3172985215271887844819,
%U 2673595890763195251451381,2687987100922385348506502421,3179641291050281648556689350747,4372634003181176563537837740241711,6918619855455296919138016216697597937,12481750393031643279550408556774581268657
%N E.g.f. satisfies: A(x) = Sum_{n>=0} (n+1)^(2*n-2)/n! * x^n/A(x)^n.
%C Compare e.g.f. to: exp(x) = Sum_{n>=0} (n+1)^(n-1)/n! * x^n/exp(x)^n.
%H Paul D. Hanna, <a href="/A296234/b296234.txt">Table of n, a(n) for n = 0..200</a>
%F E.g.f. A(x) satisfies:
%F (1) [x^n] A(x)^(n+1) = (n+1)^(2*n-1)/n! for n>=0.
%F (2) A(x) = x/Series_Reversion( x * Sum_{n>=0} (n+1)^(2*n-2) * x^n/n! ).
%F (3) A(x) = Sum_{n>=0} (n+1)^(2*n-2) * (x/A(x))^n / n!.
%e E.g.f.: A(x) = 1 + x + 7*x^2/2! + 187*x^3/3! + 11517*x^4/4! + 1269821*x^5/5! + 218962723*x^6/6! + 54377141463*x^7/7! + 18394396344313*x^8/8! + 8139652855993369*x^9/9! + 4568235711128252991*x^10/10! +...
%e such that
%e A(x) = 1 + x/A(x) + 3^2*(x/A(x))^2/2! + 4^4*(x/A(x))^3/3! + 5^6*(x/A(x))^4/4! + 6^8*(x/A(x))^5/5! + 7^10*(x/A(x))^6/6! +...+ (n+1)^(2*n-2) * (x/A(x))^n/n! +...
%e The table of coefficients of x^k/k! in A(x)^(n+1) begins:
%e [1, 1, 7, 187, 11517, 1269821, 218962723, 54377141463, ...];
%e [1, 2, 16, 416, 24824, 2680992, 456281248, 112343845952, ...];
%e [1, 3, 27, 693, 40173, 4249143, 713494215, 174131360553, ...];
%e [1, 4, 40, 1024, 57840, 5991584, 992282944, 239988068352, ...];
%e [1, 5, 55, 1415, 78125, 7927425, 1294484035, 310180891235, ...];
%e [1, 6, 72, 1872, 101352, 10077696, 1622102688, 384996798528, ...];
%e [1, 7, 91, 2401, 127869, 12465467, 1977326743, 464744426517, ...];
%e [1, 8, 112, 3008, 158048, 15115968, 2362541440, 549755813888, ...]; ...
%e in which the main diagonal begins:
%e [1, 2, 27, 1024, 78125, 10077696, 1977326743, ..., (n+1)^(2*n-1), ...].
%e RELATED SERIES.
%e log(A(x)) = x + 6*x^2/2! + 168*x^3/3! + 10700*x^4/4! + 1203960*x^5/5! + 210264432*x^6/6! + 52655421952*x^7/7! + 17914652980128*x^8/8! + 7960047283278720*x^9/9! + 4481097300680675840*x^10/10! +...
%t terms = 18; A[_] = 1; Do[A[x_] = Sum[(n + 1)^(2*n - 2)/n!*x^n/A[x]^n, {n, 0, terms}] + O[x]^terms // Normal, terms];
%t CoefficientList[A[x], x]*Range[0, terms-1]! (* _Jean-François Alcover_, Jan 14 2018 *)
%o (PARI) {a(n) = my(A=[1]); for(m=1,n, A = concat(A,0); V = Vec( Ser(A)^(m+1) ); A[m+1] = ((m+1)^(2*m-1)/m! - V[m+1])/(m+1);); n!*A[n+1]}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A296235.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 06 2018
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