A262006 G.f. satisfies: A(x) = 1+x + x*Sum_{n=-oo..+oo} x^n*A(x)^n * (A(x)^n - x^n)^n.

1, 1, 1, 4, 11, 54, 224, 1221, 6506, 37982, 227853, 1436155, 9370157, 63478402, 444856819, 3224349732, 24142103099, 186635429502, 1488745035667, 12246510599338, 103825176789850, 906575491574389, 8147038046590526, 75291889506602676, 714968504199357010, 6970122084115034290, 69699196634887342192

Offset

0,4

Comments

Compare to the curious identity: Sum_{n=-oo..+oo} x^n/a^(n^2+n) * (a^n - x^n)^n = 0 for all fixed a.

Links

Table of n, a(n) for n=0..26.

Formula

G.f. satisfies: A(x) = 1+x + x*Sum_{n=-oo..+oo} (x*A(x))^(n^2-n) / (x^n - A(x)^n)^n.

Example

G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 11*x^4 + 54*x^5 + 224*x^6 + 1221*x^7 +...
such that A(x) = 1 + 2*x + x*N(x) + x*P(x) where
N(x) = Sum_{n<=-1} x^n*A(x)^n * (A(x)^n - x^n)^n,
P(x) = Sum_{n>=1} x^n*A(x)^n * (A(x)^n - x^n)^n;
Equivalently,
N(x) = Sum_{n>=1} (x*A(x))^(n^2-n) / (x^n - A(x)^n)^n,
P(x) = Sum_{n<=-1} (x*A(x))^(n^2-n) / (x^n - A(x)^n)^n,
Explicitly,
N(x) = -1 + 2*x^2 + 2*x^3 + 13*x^4 + 32*x^5 + 199*x^6 + 926*x^7 + 5518*x^8 + 31303*x^9 + 194051*x^10 + 1226544*x^11 + 8123893*x^12 + 55526652*x^13 + 393267242*x^14 + 2876241417*x^15 + 21725489552*x^16 +...
P(x) = x + 2*x^2 + 9*x^3 + 41*x^4 + 192*x^5 + 1022*x^6 + 5580*x^7 + 32464*x^8 + 196550*x^9 + 1242104*x^10 + 8143613*x^11 + 55354509*x^12 + 389330167*x^13 + 2831082490*x^14 + 21265861682*x^15 + 164909939950*x^16 +...

Prog

(PARI) {a(n) = local(A=1+x+x*O(x^n));
for(i=1, n, A=1+x + x*sum(k=-n-1, n+1, x^k*A^k * (A^k-x^k)^k)); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))
(PARI) {a(n) = local(A=1+x+x*O(x^n));
for(i=1, n, A=1+x + x*sum(k=-n-1, n+1, (x*A)^(k^2-k) / (x^k-A^k)^k)); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

Crossrefs

Sequence in context: A149317 A027042 A051770 * A209110 A363664 A282742

Adjacent sequences: A262003 A262004 A262005 * A262007 A262008 A262009

Keyword

nonn

Author

Paul D. Hanna, Sep 21 2015

Status

approved