A301387 E.g.f. A(x) satisfies: [x^n] A(x)^n = (2*n - 1) * [x^(n-1)] A(x)^n for n>=1.

1, 1, 5, 73, 2185, 108881, 8012941, 809101945, 106751544593, 17777715999265, 3641538959077141, 899203905438721961, 263360370074829490585, 90266849931865728247153, 35797787561010642901996445, 16266200052343102435435467481, 8396767484613064218299219696161, 4887270793881840799764234723049025, 3185975842406919756361250644217366053

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

E.g.f. A(x) satisfies: A(x) = exp( x * (A(x) + x*A'(x)) / (A(x) - x*A'(x)) ).

a(n) ~ c * 2^n * (n!)^2, where c = 0.321697697353832218399635... (is the same as for A301388). - Vaclav Kotesovec, Mar 21 2018

Example

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 73*x^3/3! + 2185*x^4/4! + 108881*x^5/5! + 8012941*x^6/6! + 809101945*x^7/7! + 106751544593*x^8/8! + 17777715999265*x^9/9! + ...
The table of coefficients in A(x)^n begins:
n=1: [(1), (1), 5/2, 73/6, 2185/24, 108881/120, 8012941/720, ...];
n=2: [1, (2), (6), 88/3, 638/3, 10288/5, 1110424/45, ...];
n=3: [1, 3, (21/2), (105/2), 2979/8, 140241/40, 3288981/80, ...];
n=4: [1, 4, 16, (248/3), (1736/3), 79768/15, 2744776/45, ...];
n=5: [1, 5, 45/2, 725/6, (20185/24), (60555/8), 12237185/144, ...];
n=6: [1, 6, 30, 168, 1170, (51744/5), (569184/5), ...]; ...
in which the coefficients in parenthesis are related by
1 = 1*(1); 6 = 3*(2); 105/2 = 5*(21/2); 1736/3 = 7*(248/3); 60555/8 = 9*(20185/24); 569184/5 = 11*(51744/5); ...
illustrating: [x^n] A(x)^n = (2*n - 1) * [x^(n-1)] A(x)^n.
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is an integer power series in x satisfying
log(A(x)) = x * (1 + x*A'(x)/A(x)) / (1 - x*A'(x)/A(x));
explicitly,
log(A(x)) = x + 2*x^2 + 6*x^3 + 22*x^4 + 94*x^5 + 474*x^6 + 2974*x^7 + 24630*x^8 + 271710*x^9 + 3799570*x^10 + ... + A301388(n)*x^n + ...

Prog

(PARI) {a(n) = my(A=1); for(i=1, n, A = exp( x*(A + x*A')/(A - x*A' +x*O(x^n)) ) ); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A301388.

Sequence in context: A144682 A293146 A321189 * A096987 A096538 A355122

Adjacent sequences: A301384 A301385 A301386 * A301388 A301389 A301390

Keyword

nonn

Author

Paul D. Hanna, Mar 20 2018

Status

approved