A375458 Expansion of g.f. A(x) satisfying A(x) = x + (A(x+x^2) - A(x-x^2))/2.

1, 1, 2, 6, 24, 122, 756, 5532, 46696, 446748, 4778004, 56496384, 731876160, 10308645192, 156862861920, 2564468629464, 44828643667200, 834403045921936, 16476039916372680, 344000551441403460, 7572120904973371152, 175258231266231242316, 4255020239957453122152

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) A(x) = x + (A(x+x^2) - A(x-x^2))/2.

(2) A(x) = 2*C(x) - 2*A(C(x)) + A(2*C(x) - x), where C(x) = x + C(x)^2 is the g.f. of the Catalan numbers (A000108).

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 24*x^5 + 122*x^6 + 756*x^7 + 5532*x^8 + 46696*x^9 + 446748*x^10 + ...
where A(x) = x + (A(x+x^2) - A(x-x^2))/2.
RELATED SERIES.
A(x+x^2) = x + 2*x^2 + 4*x^3 + 13*x^4 + 54*x^5 + 280*x^6 + 1752*x^7 + 12900*x^8 + 109388*x^9 + 1050222*x^10 + ...
A(x-x^2) = x + x^4 + 6*x^5 + 36*x^6 + 240*x^7 + 1836*x^8 + 15996*x^9 + 156726*x^10 + ...

Prog

(PARI) {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A=concat(A, 0); Ax=Ser(A);
A[#A] = polcoeff( x + (subst(Ax, x, x+x^2 ) - subst(Ax, x, x-x^2 ))/2 - Ax, #A-1) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A000108.

Sequence in context: A352437 A144167 A104983 * A144251 A304198 A243806

Adjacent sequences: A375455 A375456 A375457 * A375459 A375460 A375461

Keyword

nonn

Author

Paul D. Hanna, Aug 19 2024

Status

approved