A306063 O.g.f. A(x) satisfies: Sum_{n>=1} (2^n*x - A(x))^n / n = 0.

2, 2, 68, 9398, 4833428, 9454918068, 72006088426248, 2165455076559211174, 259347215815106405220132, 124310299732163916975832447388, 239094057363977384232311742474570360, 1847535112178186477442381068773529944826172, 57378255829217503847229646446662951215946818659912, 7161580198468591866673993366959923994699839199088716021928

Offset

1,1

Comments

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

Links

Paul D. Hanna, Table of n, a(n) for n = 1..80

Example

O.g.f.: A(x) = 2*x + 2*x^2 + 68*x^3 + 9398*x^4 + 4833428*x^5 + 9454918068*x^6 + 72006088426248*x^7 + 2165455076559211174*x^8 + 259347215815106405220132*x^9 + 124310299732163916975832447388*x^10 + 239094057363977384232311742474570360*x^11 + 1847535112178186477442381068773529944826172*x^12 + ...
such that
0 = (2*x - A(x)) + (2^2*x - A(x))^2/2 + (2^3*x - A(x))^3/3 + (2^4*x - A(x))^4/4 + (2^5*x - A(x))^5/5 + (2^6*x - A(x))^6/6 + (2^7*x - A(x))^7/7 + ...
RELATED SERIES.
exp( Sum_{n>=1} 2^(n^2)*x^n / n ) = 1 + 2*x + 10*x^2 + 188*x^3 + 16774*x^4 + 6745436*x^5 + 11466849412*x^6 + 80444398636280*x^7 + ... + A155200(n)*x^n + ...
exp( Sum_{n>=1} A(x)^n / n ) = 1/(1 - A(x)) = 1 + 2*x + 6*x^2 + 84*x^3 + 9714*x^4 + 4872228*x^5 + 9474410908*x^6 + 72043987279208*x^7 + 2165743253217563938*x^8 + ...
Sum_{n>=1} A(x)^n / n = -log(1 - A(x)) = 2*x + 8*x^2/2 + 224*x^3/3 + 38192*x^4/4 + 24263312*x^5/5 + 56787868688*x^6/6 + 504175196453504*x^7/7 + ...

Prog

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

Crossrefs

Cf. A155200.

Sequence in context: A187024 A274477 A231808 * A028372 A130678 A370736

Adjacent sequences: A306060 A306061 A306062 * A306064 A306065 A306066

Keyword

nonn

Author

Paul D. Hanna, Jun 19 2018

Status

approved