A316370 E.g.f.: Sum_{n>=0} x^n/n! * Product_{k=1..n} (n+1-k) + k*x.

1, 1, 4, 21, 152, 1410, 15774, 207984, 3153632, 54074952, 1034749080, 21858562440, 505274905992, 12686390177136, 343815306388176, 10003360314147480, 311003061260534400, 10289575224413883840, 360967225620921712704, 13383588039651073512576, 522943874535097662998400, 21477474848621411837159040, 924978962293503284606947200

Offset

0,3

Comments

More generally, we have the following identity. Given the biexponential series

W(x,y) = Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k)*x + k*y,

then for fixed p and q,

Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k + p)*x + (k + q)*y = W(x,y)^(p+q+1) / ( (1 + x*W(x,y))^q * (1 + y*W(x,y))^p ).

Further, W(x,y) satisfies the biexponential functional equation

( W(x,y)/(1 + x*W(x,y)) )^x = ( W(x,y)/(1 + y*W(x,y)) )^y.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:

(1) A(x) = Sum_{n>=0} x^n/n! * Product_{k=1..n} (n+1-k) + k*x.

(2) Sum_{n>=0} x^n/n! * Product_{k=1..n} (n+1-k + p) + (k + q)*x = A(x)^(p+q+1) / ( (1 + x*A(x))^q * (1 + x^2*A(x))^p ), for fixed p and q.

(3) A(x)/(1 + x*A(x)) = ( A(x)/(1 + x^2*A(x)) )^x.

a(n) ~ 2^(n+1) * n^n / (sqrt(log(2)) * exp(n)). - Vaclav Kotesovec, Jul 13 2018

Example

E.g.f.: A(x) = 1 + x + 4*x^2/2! + 21*x^3/3! + 152*x^4/4! + 1410*x^5/5! + 15774*x^6/6! + 207984*x^7/7! + 3153632*x^8/8! + 54074952*x^9/9! + 1034749080*x^10/10! + ...
such that
A(x) = 1 + (1+x)*x + (2 + x)*(1 + 2*x)*x^2/2! + (3 + x)*(2 + 2*x)*(1 + 3*x)*x^3/3! + (4 + x)*(3 + 2*x)*(2 + 3*x)*(1 + 4*x)*x^4/4! + (5 + x)*(4 + 2*x)*(3 + 3*x)*(2 + 4*x)*(1 + 5*x)*x^5/5! + ...
Also,
A(x)^2/(1 + x*A(x)) = 1 + (1 + 2*x)*x + (2 + 2*x)*(1 + 3*x)*x^2/2! + (3 + 2*x)*(2 + 3*x)*(1 + 4*x)*x^3/3! + (4 + 2*x)*(3 + 3*x)*(2 + 4*x)*(1 + 5*x)*x^4/4! + (5 + 2*x)*(4 + 3*x)*(3 + 4*x)*(2 + 5*x)*(1 + 6*x)*x^5/5! + ...
And,
A(x)^3/((1 + x*A(x))*(1 + x^2*A(x))) = 1 + (2 + 2*x)*x + (3 + 2*x)*(2 + 3*x)*x^2/2! + (4 + 2*x)*(3 + 3*x)*(2 + 4*x)*x^3/3! + (5 + 2*x)*(4 + 3*x)*(3 + 4*x)*(2 + 5*x)*x^4/4! + (6 + 2*x)*(5 + 3*x)*(4 + 4*x)*(3 + 5*x)*(2 + 6*x)*x^5/5! + ...
RELATED SERIES.
A(x)/(1 + x*A(x)) = 1 + 2*x^2/2! + 3*x^3/3! + 32*x^4/4! + 190*x^5/5! + 1974*x^6/6! + 21588*x^7/7! + 289232*x^8/8! + 4387752*x^9/9! + ...
A(x)/(1 + x^2*A(x)) = 1 + x + 2*x^2/2! + 9*x^3/3! + 56*x^4/4! + 450*x^5/5! + 4494*x^6/6! + 53424*x^7/7! + 738464*x^8/8! + 11642184*x^9/9! + ...
where ( A(x)/(1 + x^2*A(x)) )^x = A(x)/(1 + x*A(x)).

Prog

(PARI) /* From Biexponential Series: */
{a(n) = my(A); A = sum(m=0, n, x^m/m! * prod(k=1, m, m+1-k + k*x +x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From Biexponential Functional Equation: */
{a(n) = my(A=1); for(i=0, n, A = (1 + x*A)*( A/(1 + x^2*A +x*O(x^n) ) )^x ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A316700, A316701, A316702.

Sequence in context: A330019 A025164 A335848 * A357424 A166901 A060072

Adjacent sequences: A316367 A316368 A316369 * A316371 A316372 A316373

Keyword

nonn

Author

Paul D. Hanna, Jul 12 2018

Status

approved