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A306067 E.g.f. A(x) satisfies:  Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n-k) + (k+1)*A(x)  =  1. 3
1, 4, 21, 178, 2279, 39066, 835132, 21400198, 640239525, 21920851282, 845615003996, 36298192983482, 1716348366690487, 88653661788525666, 4967006270867149524, 300043327305644202366, 19440451816128996788777, 1344909407655243937857826, 98949254253416815493778796, 7714902418308597200477578514, 635444724815621395463510504211, 55134789286331454820101232131938 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..128 (terms 0..100 from Paul D. Hanna)

FORMULA

E.g.f. A(x) satisfies:

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

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

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

a(n)/n! ~ c * d^n / n^(3/2), where d = 4.423034555284689... and c = 3.17922741818... - Vaclav Kotesovec, Jul 12 2018

EXAMPLE

G.f.: A(x) = 1 + 4*x + 21*x^2/2! + 178*x^3/3! + 2279*x^4/4! + 39066*x^5/5! + 835132*x^6/6! + 21400198*x^7/7! + 640239525*x^8/8! + 21920851282*x^9/9! + 845615003996*x^10/10! + ...

such that

1  =  (0 + A(x))  -  (1 + A(x))*(0 + 2*A(x))*x  +  (2 + A(x))*(1 + 2*A(x))*(0 + 3*A(x))*x^2/2!  -  (3 + A(x))*(2 + 2*A(x))*(1 + 3*A(x))*(0 + 4*A(x))*x^3/3!  +  (4 + A(x))*(3 + 2*A(x))*(2 + 3*A(x))*(1 + 4*A(x))*(0 + 5*A(x))*x^4/4!  -  (5 + A(x))*(4 + 2*A(x))*(3 + 3*A(x))*(2 + 4*A(x))*(1 + 5*A(x))*(0 + 6*A(x))*x^5/5!  +  ...

PROG

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = -Vec( sum(n=0, #A, (-x)^n/n!* prod(k=0, n, (n-k) + (k+1)*Ser(A) ) ) )[#A] ); n!*A[n+1]}

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

CROSSREFS

Cf. A306090.

Sequence in context: A317104 A163948 A278993 * A230682 A231220 A231434

Adjacent sequences:  A306064 A306065 A306066 * A306068 A306069 A306070

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 25 2018

STATUS

approved

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Last modified June 13 00:57 EDT 2021. Contains 344980 sequences. (Running on oeis4.)