A305108 O.g.f. A(x) satisfies: [x^n] exp( n^2 * x*A(x)^2 ) * (2 - A(x)) = 0 for n > 0.

1, 1, 12, 342, 16080, 1090400, 99884736, 11885278104, 1786708056832, 331931252093472, 74805826012157600, 20127855750577630968, 6377560491906482613120, 2351353212746078192866032, 998307775668524681354287776, 483648245019895895020555792200, 265226237170176555086800587134976, 163463812810277012465203148994919744, 112505648337664454361768261713783693856

Offset

0,3

Comments

Note: given F(x) = 1 + x * d/dx x*F(x)^2, where x*F(x) is a g.f. of A000699, then

(1) [x^n] exp( x*F(x)^2 ) * (n + 1 - F(x)) = 0 for n > 0,

(2) [x^n] exp( n * x*F(x)^2 ) * (2 - F(x)) = 0 for n > 0.

It is remarkable that this sequence should consist entirely of integers.

Links

Table of n, a(n) for n=0..18.

Example

O.g.f.: A(x) = 1 + x + 12*x^2 + 342*x^3 + 16080*x^4 + 1090400*x^5 + 99884736*x^6 + 11885278104*x^7 + 1786708056832*x^8 + 331931252093472*x^9 + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 25*x^2 + 708*x^3 + 32988*x^4 + 2221168*x^5 + 202453156*x^6 + 24007494000*x^7 + 3600588303536*x^8 + 667824555398528*x^9 + ...
exp(x*A(x)^2) = 1 + x + 5*x^2/2! + 163*x^3/3! + 17665*x^4/4! + 4051301*x^5/5! + 1624532461*x^6/6! + 1032073147855*x^7/7! + 976416036692993*x^8/8! + ...

Prog

(PARI) {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)^2*x*(Ser(A)^2) ) * (2 - Ser(A)) )[m] ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A305107.

Sequence in context: A009680 A218820 A231520 * A027015 A024297 A077296

Adjacent sequences: A305105 A305106 A305107 * A305109 A305110 A305111

Keyword

nonn

Author

Paul D. Hanna, May 27 2018

Status

approved