A200535 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} C(2*n,k)^2 * x^k] / A(x)^n * x^n/n ).

1, 1, 4, 5, 9, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236, 240, 244, 248, 252, 256, 260

Offset

0,3

Links

Table of n, a(n) for n=0..67.

Index entries for linear recurrences with constant coefficients, signature (2, -1).

Formula

G.f.: (1+x^2)^2*(1+x^3)/((1-x)*(1-x^2)).

Example

G.f.: A(x) = 1 + x + 4*x^2 + 5*x^3 + 9*x^4 + 12*x^5 + 16*x^6 + 20*x^7 +...
where A(x) = G(x/A(x)) and G(x) = A(x*G(x)) is the g.f. of A199257:
G(x) = 1 + x + 5*x^2 + 18*x^3 + 86*x^4 + 408*x^5 + 2075*x^6 +...
...
The logarithm of the g.f. A(x) equals the series:
log(A(x)) = (1 + 2^2*x + x^2)/A(x) * x +
(1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)/A(x)^2 * x^2/2 +
(1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)/A(x)^3 * x^3/3 +
(1 + 8^2*x + 28^2*x^2 + 56^2*x^3 + 70^2*x^4 + 56^2*x^5 + 28^2*x^6 + 8^2*x^7 + x^8)/A(x)^4 * x^4/4 +
(1 + 10^2*x + 45^2*x^2 + 120^2*x^3 + 210^2*x^4 + 252^2*x^5 + 210^2*x^6 + 120^2*x^7 + 45^2*x^8 + 10^2*x^9 + x^10)/A(x)^5 * x^5/5 +...
which involves the squares of binomial coefficients C(2*n,k).

Prog

(PARI) {a(n)=polcoeff((1+x^2)^2*(1+x^3)/((1-x)*(1-x^2) +x*O(x^n)), n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, n, binomial(2*m, k)^2 *x^k)/(A+x*O(x^n))^m *x^m/m))); polcoeff(A, n)}

Crossrefs

Cf. A199257.

Sequence in context: A332020 A184801 A228914 * A010405 A125603 A239231

Adjacent sequences: A200532 A200533 A200534 * A200536 A200537 A200538

Keyword

nonn

Author

Paul D. Hanna, Nov 18 2011

Status

approved