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
Keywords
Examples
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).
Links
- Index entries for linear recurrences with constant coefficients, signature (2, -1).
Crossrefs
Cf. A199257.
Programs
-
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)}
Formula
G.f.: (1+x^2)^2*(1+x^3)/((1-x)*(1-x^2)).