A222658 G.f. satisfies: A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} x^k * {[x^k] A(x)^n}.

1, 1, 2, 3, 9, 14, 42, 67, 221, 352, 1154, 1855, 6222, 10024, 33698, 54520, 184823, 299668, 1019099, 1656234, 5654308, 9205166, 31501343, 51366338, 176178460, 287662788, 988329204, 1615679329, 5559353908, 9097789494, 31343274367, 51341385362, 177069879751, 290293269560

Offset

0,3

Comments

Here [x^k] A(x)^n denotes the coefficient of x^k in A(x)^n.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

G.f. satisfies: A(x) = (1 + x^2*G'(x^2)/G(x^2)) / (1 - x*G(x^2)), where A(x) = G(x/A(x)) and G(x) = A(x*G(x)) = (1/x)*Series_Reversion(x/A(x)).

a(n) ~ c * d^n / sqrt(n), where d = 2.413348608405787... , c = 0.59082988060... if n is even and c = 0.40808981489... if n is odd. - Vaclav Kotesovec, Nov 29 2014

Example

G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 9*x^4 + 14*x^5 + 42*x^6 + 67*x^7 +...
Related expansions:
A(x)^2 = 1 + 2*x + 5*x^2 + 10*x^3 + 28*x^4 + 58*x^5 + 157*x^6 +...
A(x)^3 = 1 + 3*x + 9*x^2 + 22*x^3 + 63*x^4 + 153*x^5 + 416*x^6 +...
A(x)^4 = 1 + 4*x + 14*x^2 + 40*x^3 + 121*x^4 + 328*x^5 + 926*x^6 +...
A(x)^5 = 1 + 5*x + 20*x^2 + 65*x^3 + 210*x^4 + 621*x^5 + 1840*x^6 +...
A(x)^6 = 1 + 6*x + 27*x^2 + 98*x^3 + 339*x^4 + 1080*x^5 + 3368*x^6 +...
GENERATING METHOD.
The initial terms, k=0..n, of the n-th power of g.f. A(x) begin:
n=0: [1];
n=1: [1, 1];
n=2: [1, 2, 5];
n=3: [1, 3, 9, 22];
n=4: [1, 4, 14, 40, 121];
n=5: [1, 5, 20, 65, 210, 621];
n=6: [1, 6, 27, 98, 339, 1080, 3368];
n=7: [1, 7, 35, 140, 518, 1764, 5789, 18138];
n=8: [1, 8, 44, 192, 758, 2744, 9464, 31120, 99489];
n=9: [1, 9, 54, 255, 1071, 4104, 14850, 51093, 169884, 547321];
n=10:[1, 10, 65, 330, 1470, 5942, 22515, 80860, 279290, 932540, 3033585]; ...
from which the antidiagonal sums form this sequence:
a(0) = 1;
a(1) = 1;
a(2) = 1 + 1 = 2;
a(3) = 1 + 2 = 3;
a(4) = 1 + 3 + 5 = 9;
a(5) = 1 + 4 + 9 = 14;
a(6) = 1 + 5 + 14 + 22 = 42;
a(7) = 1 + 6 + 20 + 40 = 67; ...
ALTERNATE GENERATING METHOD.
Define G(x) such that G(x) = A(x*G(x)) = (1/x)*Series_Reversion(x/A(x)):
G(x) = 1 + x + 3*x^2 + 10*x^3 + 42*x^4 + 180*x^5 + 827*x^6 + 3890*x^7 + 18876*x^8 + 93254*x^9 + 468727*x^10 +...
then A(x) = (1  + x^2*G'(x^2)/G(x^2)) / (1 - x*G(x^2)).
Note that 1  + x^2*G'(x^2)/G(x^2) begins:
1 + x^2 + 5*x^4 + 22*x^6 + 121*x^8 + 621*x^10 + 3368*x^12 +...
where the coefficients form the main diagonal of the above triangle.

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*sum(k=0, m, x^k*polcoeff((A+x*O(x^m))^m, k))+x*O(x^n))); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) /* ALTERNATE GENERATING METHOD (faster) */
{a(n)=local(A=1+x, G=1); for(i=0, #binary(n)+1, G=1/x*serreverse(x/A+x^2*O(x^n)); A=(1+x^2*subst(G'/G, x, x^2))/(1-x*subst(G, x, x^2))); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A178852, A249935, A249936, A249937.

Sequence in context: A295857 A047171 A094557 * A227212 A237254 A026307

Adjacent sequences: A222655 A222656 A222657 * A222659 A222660 A222661

Keyword

nonn

Author

Paul D. Hanna, Jun 29 2013

Status

approved