A206777 G.f.: 1 + Sum_{n>=1} x^n * exp( Sum_{k>=1} binomial(2*n*k-1, n*k) * x^(n*k)/k ).

1, 1, 2, 3, 9, 15, 75, 133, 676, 1712, 7295, 16797, 100889, 208013, 1083061, 3186268, 14634359, 35357671, 209715154, 477638701, 2679305455, 7638371497, 35224117709, 91482563641, 568102780247, 1311836293464, 6979729709485, 21256008509080, 103440124113965

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..500

Formula

G.f.: Sum_{n>=0} x^n * Product_{k=0..n-1} C(u(n)^k*x), where u(n) = exp(2*Pi*I/n) is an n-th root of unity, and C(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers (A000108).

Example

G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 9*x^4 + 15*x^5 + 75*x^6 + 133*x^7 +...
Let C(x) = (1-sqrt(1-4*x))/(2*x), then the g.f. A(x) equals the series:
A(x) = 1 + x*C(x) + x^2*C(x)*C(-x) + x^3*C(x)*C(u(3)*x)*C(u(3)^2*x) + x^4*C(x)*C(I*x)*C(-x)*C(-I*x) + x^5*C(x)*C(u(5)*x)*C(u(5)^2*x)*C(u(5)^3*x)*C(u(5)^4*x) +...
where u(n) = exp(2*Pi*I/n).
The expansions of P(n) = Product_{k=0..n-1} C(u(n)^k*x) begin:
P(1) = C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
P(2) = C(x)*C(-x) = 1 + 3*x^2 + 22*x^4 + 211*x^6 + 2306*x^8 +...
P(3) = 1 + 10*x^3 + 281*x^6 + 10580*x^9 + 457700*x^12 + 21475122*x^15 +...
P(4) = 1 + 35*x^4 + 3830*x^8 + 570451*x^12 + 98118690*x^16 +...
P(5) = 1 + 126*x^5 + 54127*x^10 + 32006130*x^15 + 21932146139*x^20 +...
P(6) = 1 + 462*x^6 + 782761*x^12 + 1841287756*x^18 +...
P(7) = 1 + 1716*x^7 + 11501478*x^14 + 107761838756*x^21 +...
P(8) = 1 + 6435*x^8 + 170974710*x^16 + 6386000085075*x^24 +...
such that A(x) = 1 + x*C(x) + x^2*P(2) + x^3*P(3) + x^4*P(4) +...

Prog

(PARI) {a(n)=polcoeff(1+sum(m=1, n, x^m*exp(sum(k=1, n\m, binomial(2*m*k-1, m*k)*x^(m*k)/k)+x*O(x^n))), n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n)=local(CATALAN=(1-sqrt(1-4*x+x^2*O(x^n)))/(2*x)); A=1+sum(m=1, n+1, x^m*CATALAN*round(prod(k=1, m-1, subst(CATALAN, x, exp(2*Pi*I*k/m)*x+x*O(x^n))))); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A205504, A000108.

Sequence in context: A054416 A291868 A092638 * A173809 A298355 A023147

Adjacent sequences: A206774 A206775 A206776 * A206778 A206779 A206780

Keyword

nonn

Author

Paul D. Hanna, Feb 12 2012

Status

approved