OFFSET
0,2
COMMENTS
Compare e.g.f. to: exp(x) = Sum_{n>=0} (n+1)^(n-1)/n! * x^n/exp(x)^n.
FORMULA
E.g.f. A(x) satisfies: [x^n] A(x)^(n+1) = (n+1)^(2*n)/n! for n>=0.
EXAMPLE
E.g.f.: A(x) = 1 + 2*x + 19*x^2/2! + 634*x^3/3! + 47649*x^4/4! + 6274106*x^5/5! + 1266115531*x^6/6! + 361443760434*x^7/7! + 138409787933185*x^8/8! + 68440264233480946*x^9/9! + 42456829504270563171*x^10/10! +...
such that
A(x) = 1 + 2*x/A(x) + 3^3*(x/A(x))^2/2! + 4^5*(x/A(x))^3/3! + 5^7*(x/A(x))^4/4! + 6^9*(x/A(x))^5/5! + 7^11*(x/A(x))^6/6! +...+ (n+1)^(2*n-1) * (x/A(x))^n/n! +...
The table of coefficients of x^k/k! in A(x)^(n+1) begins:
[1, 2, 19, 634, 47649, 6274106, 1266115531, 361443760434, ...];
[1, 4, 46, 1496, 107608, 13742112, 2718008656, 765460154944, ...];
[1, 6, 81, 2634, 182613, 22621158, 4382123445, 1216833883674, ...];
[1, 8, 124, 4096, 275784, 33164864, 6288898048, 1720947264768, ...];
[1, 10, 175, 5930, 390625, 45667170, 8473311895, 2283859227970, ...];
[1, 12, 234, 8184, 531024, 60466176, 10975478976, 2912393501184, ...];
[1, 14, 301, 10906, 701253, 77947982, 13841287201, 3614236635114, ...];
[1, 16, 376, 14144, 905968, 98550528, 17123083840, 4398046511104, ...];
[1, 18, 459, 17946, 1150209, 122767434, 20880407043, 5273571977298], ...]; ...
in which the main diagonal begins:
[1, 4, 81, 4096, 390625, 60466176, 13841287201, ..., (n+1)^(2*n), ...].
RELATED SERIES.
log(A(x)) = 2*x + 15*x^2/2! + 536*x^3/3! + 42310*x^4/4! + 5741184*x^5/5! + 1181144664*x^6/6! + 341617297664*x^7/7! + 132031460161584*x^8/8! + 65730875486238720*x^9/9! + 40986501016038645760*x^10/10! + ...
PROG
(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)/m! - V[m+1])/(m+1); ); G=Ser(A); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 06 2018
STATUS
approved