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A299426 E.g.f. Sum_{n>=0} Series_Reversion( x*exp(-n*x) )^n. 2
1, 1, 4, 39, 688, 18765, 722016, 37003267, 2426725120, 197569640889, 19498786969600, 2288864397602871, 314642887620065280, 50002239358837749061, 9086161251793519280128, 1870292375864728408312875, 432549223770637009573052416, 111598253780454986901562293489, 31918043775392401233962828169216, 10063176613379061071841096012386911, 3479934178969181147698311417364480000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..400

FORMULA

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

(1) A(x) = Sum_{n>=0} Series_Reversion( x*exp(-n*x) )^n.

(2) A(x) = Sum_{n>=1} x^n * W(x,n)^(n^2), where

(2.a) W(x,n) = exp( -LambertW(-n*x)/n ),

(2.b) W(x,n) = exp( x*W(x,n)^n ),

(2.c) W(x,n)^n = Series_Reversion( x*exp(-n*x) ) / x,

(2.d) W(x,n)^n = Sum_{k>=0} n^k * (k+1)^(k-1) * x^k/k!;

(2.e) W(x,n)^(n^2) = Sum_{k>=0} n^(k+1) * (n+k)^(k-1) * x^k/k!.

FORMULAS FOR TERMS.

a(n) = Sum_{k=0..n} n!/k! * n^(k-1) * (n-k)^(k+1).

EXAMPLE

E.g.f. A(x) = 1 + x + 4*x^2/2! + 39*x^3/3! + 688*x^4/4! + 18765*x^5/5! + 722016*x^6/6! + 37003267*x^7/7! + 2426725120*x^8/8! + ...

such that

A(x) = 1 + x*W(x,1) + x^2*W(x,2)^4 + x^3*W(x,3)^9 + x^4*W(x,4)^16 + x^5*W(x,5)^25 + x^6*W(x,6)^36 + ...

where series W(x,n) = exp( x*W(x,n)^n ) begin:

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

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

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

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

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

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

...

and series W(x,n)^n = Series_Reversion( x*exp(-n*x) ) / x begin:

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

W(x,2)^2 = 1 + 2*x + 12*x^2/2! + 128*x^3/3! + 2000*x^4/4! + ...

W(x,3)^3 = 1 + 3*x + 27*x^2/2! + 432*x^3/3! + 10125*x^4/4! + ...

W(x,4)^4 = 1 + 4*x + 48*x^2/2! + 1024*x^3/3! + 32000*x^4/4! + ...

W(x,5)^5 = 1 + 5*x + 75*x^2/2! + 2000*x^3/3! + 78125*x^4/4! + ...

W(x,6)^6 = 1 + 6*x + 108*x^2/2! + 3456*x^3/3! + 162000*x^4/4! + ...

...

and series W(x,n)^(n^2) begin:

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

W(x,2)^4 = 1 + 4*x + 32*x^2/2! + 400*x^3/3! + 6912*x^4/4! + ...

W(x,3)^9 = 1 + 9*x + 135*x^2/2! + 2916*x^3/3! + 83349*x^4/4! + ...

W(x,4)^16 = 1 + 16*x + 384*x^2/2! + 12544*x^3/3! + 524288*x^4/4! + ...

W(x,5)^25 = 1 + 25*x + 875*x^2/2! + 40000*x^3/3! + 2278125*x^4/4! + ...

W(x,6)^36 = 1 + 36*x + 1728*x^2/2! + 104976*x^3/3! + 7776000*x^4/4! + ...

...

PROG

(PARI) {a(n) = my(A); A = sum(m=0, n+1, serreverse( x*exp(-m*x +x^2*O(x^n) ) )^m ); n!*polcoeff(A, n)}

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

(PARI) {a(n) = if(n==0, 1, sum(k=0, n, n!/k! * n^(k-1) * (n-k)^(k+1) ) )}

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

CROSSREFS

Cf. A244468, A299044.

Sequence in context: A065760 A132612 A129463 * A188418 A136653 A165434

Adjacent sequences:  A299423 A299424 A299425 * A299427 A299428 A299429

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 18 2018

STATUS

approved

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Last modified January 21 01:45 EST 2022. Contains 350473 sequences. (Running on oeis4.)