A247332 G.f. A(x) satisfies: 1-x = Sum_{n>=0} (A(x) + (-x)^n)^n * (-x)^n.

1, 2, 3, 7, 16, 41, 107, 286, 782, 2179, 6172, 17702, 51272, 149727, 440341, 1303130, 3877837, 11596602, 34832573, 105041452, 317900632, 965240836, 2939479066, 8976098663, 27478467863, 84314278171, 259262013763, 798802232031, 2465713674230, 7624219181757, 23612883510015, 73241919575468

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

G.f. A(x) satisfies: 1-x = Sum_{n>=1} x^(n*(n-1)) / (1 - A(x)*(-x)^n)^n.

a(n) ~ c * d^n / n^(3/2), where d = 3.2506054895139..., c = 1.82859376... . - Vaclav Kotesovec, Sep 29 2014

Example

G.f.: A(x) = 1 + 2*x + 3*x^2 + 7*x^3 + 16*x^4 + 41*x^5 + 107*x^6 + 286*x^7 +...
where
1-x = 1 - (A(x) - x)*x + (A(x) + x^2)^2*x^2 - (A(x) - x^3)^3*x^3 + (A(x) + x^4)^4*x^4 - (A(x) - x^5)^5*x^5 + (A(x) + x^6)^6*x^6 - (A(x) - x^7)^7*x^7 +-...
Also, the g.f. satisfies the series identity:
1-x = 1/(1 + A(x)*x) + x^2/(1 - A(x)*x^2)^2 + x^6/(1 + A(x)*x^3)^3 + x^12/(1 - A(x)*x^4)^4 + x^20/(1 + A(x)*x^5)^5 + x^30/(1 - A(x)*x^6)^6 + x^42/(1 + A(x)*x^7)^7 +...

Prog

(PARI) {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=Vec(sum(k=0, #A, (Ser(A) + (-x)^k)^k*(-x)^k))[#A+1]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=Vec(sum(k=1, sqrtint(#A)+1, x^(k^2-k)/(1 - Ser(A)*(-x)^k)^k ))[#A+1]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A317997.

Sequence in context: A352816 A107387 A091487 * A162092 A143884 A227546

Adjacent sequences: A247329 A247330 A247331 * A247333 A247334 A247335

Keyword

nonn

Author

Paul D. Hanna, Sep 26 2014

Status

approved