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A230320 E.g.f.: Sum_{n>=0} x^n/n! * Product_{k=1..n} ( LambertW(-k*x)/(-k*x) )^(1/k). 3
1, 1, 3, 16, 137, 1746, 31627, 785149, 25715377, 1070214364, 54862242971, 3385895548839, 247409460018217, 21118696317592498, 2080845352648353215, 234093630772343822281, 29777361783749418754593, 4247066958924682143019576, 674393753569770072828136819 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..150

FORMULA

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

EXAMPLE

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 137*x^4/4! + 1746*x^5/5! +...

Let W(x) = LambertW(-x)/(-x), then

W(k*x)^(1/k) = Sum_{j>=0} (k*j+1)^(j-1)*x^j/j!

where

A(x) = 1 + x*W(x) + x^2*W(x)*W(2*x)^(1/2)/2! + x^3*W(x)*W(2*x)^(1/2)*W(3*x)^(1/3)/3! + x^4*W(x)*W(2*x)^(1/2)*W(3*x)^(1/3)*W(4*x)^(1/4)/4! +...

RELATED EXPANSIONS:

W(1*x)^(1/1) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! +...

W(2*x)^(1/2) = 1 + x + 5*x^2/2! + 49*x^3/3! + 729*x^4/4! + 14641*x^5/5! +...

W(3*x)^(1/3) = 1 + x + 7*x^2/2! + 100*x^3/3! + 2197*x^4/4! + 65536*x^5/5! +...

W(4*x)^(1/4) = 1 + x + 9*x^2/2! + 169*x^3/3! + 4913*x^4/4! + 194481*x^5/5! +...

W(5*x)^(1/5) = 1 + x + 11*x^2/2! + 256*x^3/3! + 9261*x^4/4! + 456976*x^5/5! +...

W(6*x)^(1/6) = 1 + x + 13*x^2/2! + 361*x^3/3! + 15625*x^4/4! + 923521*x^5/5! +...

W(7*x)^(1/7) = 1 + x + 15*x^2/2! + 484*x^3/3! + 24389*x^4/4! + 1679616*x^5/5! +...

W(8*x)^(1/8) = 1 + x + 17*x^2/2! + 625*x^3/3! + 35937*x^4/4! + 2825761*x^5/5! +...

W(9*x)^(1/9) = 1 + x + 19*x^2/2! + 784*x^3/3! + 50653*x^4/4! + 4477456*x^5/5! +...

W(10*x)^(1/10) = 1 + x + 21*x^2/2! + 961*x^3/3! + 68921*x^4/4! + 6765201*x^5/5! +...

PROG

(PARI) {a(n)=local(LambertW=serreverse(x*exp(x+x^2*O(x^n))), A=1);

A=sum(m=0, n, x^m/m!*prod(k=1, m, (subst(LambertW, x, -k*x)/(-k*x))^(1/k)));

n!*polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

(PARI) {a(n)=local(W=sum(m=0, n, (m+1)^(m-1)*x^m/m!)+x*O(x^n), A=1);

A=sum(m=0, n, x^m/m!*prod(k=1, m, subst(W, x, k*x)^(1/k)));

n!*polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

(PARI) {a(n)=local(A=1);

A=sum(m=0, n, x^m/m!*prod(k=1, m, sum(j=0, n, (k*j+1)^(j-1)*x^j/j!)+x*O(x^n) ));

n!*polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A230321, A230318.

Sequence in context: A277673 A335356 A135746 * A230318 A006057 A305472

Adjacent sequences:  A230317 A230318 A230319 * A230321 A230322 A230323

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 15 2013

STATUS

approved

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Last modified June 13 14:23 EDT 2021. Contains 345003 sequences. (Running on oeis4.)