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A249935 G.f. satisfies: A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} x^k * {[x^k] A(x)^(2*n)}. 4
1, 1, 3, 5, 25, 42, 203, 352, 1863, 3221, 17028, 29700, 160011, 279869, 1515002, 2660203, 14496687, 25519004, 139589213, 246299404, 1351864004, 2389786433, 13150095286, 23284570446, 128400299029, 227675571607, 1257685572691, 2232848363136, 12352579717154, 21954187917378 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Here [x^k] A(x)^(2*n) denotes the coefficient of x^k in A(x)^(2*n).

LINKS

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

FORMULA

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

a(n) ~ c * d^n / sqrt(n), where d = 3.18978759025890966... , c = 0.5263107214182... if n is even and c = 0.2984906301198... if n is odd. - Vaclav Kotesovec, Nov 29 2014

EXAMPLE

G.f.: A(x) = 1 + x + 3*x^2 + 5*x^3 + 25*x^4 + 42*x^5 + 203*x^6 + 352*x^7 +...

Related expansions:

A(x)^2 = 1 + 2*x + 7*x^2 + 16*x^3 + 69*x^4 + 164*x^5 + 665*x^6 +...

A(x)^4 = 1 + 4*x + 18*x^2 + 60*x^3 + 251*x^4 + 828*x^5 + 3208*x^6 +...

A(x)^6 = 1 + 6*x + 33*x^2 + 140*x^3 + 630*x^4 + 2478*x^5 + 10144*x^6 +...

A(x)^8 = 1 + 8*x + 52*x^2 + 264*x^3 + 1306*x^4 + 5824*x^5 + 25676*x^6 +...

GENERATING METHOD.

The initial terms, k=0..n, of the (2*n)-th power of g.f. A(x) begin:

n=0: [1];

n=2: [1, 2];

n=4: [1, 4, 18];

n=6: [1, 6, 33, 140];

n=8: [1, 8, 52, 264, 1306];

n=10:[1, 10, 75, 440, 2395, 11832];

n=12:[1, 12, 102, 676, 4029, 21756, 111204];

n=14:[1, 14, 133, 980, 6356, 37170, 203406, 1049764];

n=16:[1, 16, 168, 1360, 9540, 60000, 350056, 1918816, 10031418];

n=18:[1, 18, 207, 1824, 13761, 92556, 573477, 3325212, 18304947, 96438254]; ...

from which the antidiagonal sums form this sequence:

a(0) = 1;

a(1) = 1;

a(2) = 1 + 2 = 3;

a(3) = 1 + 4 = 5;

a(4) = 1 + 6 + 18 = 25;

a(5) = 1 + 8 + 33 = 42;

a(6) = 1 + 10 + 52 + 140 = 203;

a(7) = 1 + 12 + 75 + 264 = 352; ...

ALTERNATE GENERATING METHOD.

Define G(x) such that G(x) = A(x*G(x)^2) = sqrt( (1/x)*Series_Reversion(x/A(x)^2) ):

G(x) = 1 + x + 5*x^2 + 28*x^3 + 199*x^4 + 1474*x^5 + 11668*x^6 + 95316*x^7 + 802213*x^8 + 6892525*x^9 + 60259964*x^10 +...

then A(x) = (1  + 2*x^2 * G'(x^2)/G(x^2)) / (1 - x*G(x^2)^2).

Note that 1  + 2*x^2 * G'(x^2)/G(x^2) begins:

1 + 2*x^2 + 18*x^4 + 140*x^6 + 1306*x^8 + 11832*x^10 + 111204*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))^(2*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=sqrt(1/x*serreverse(x/A^2 +x^2*O(x^n))); A=(1+2*x^2*subst(G'/G, x, x^2))/(1-x*subst(G^2, x, x^2))); polcoeff(A, n)}

for(n=0, 40, print1(a(n), ", "))

CROSSREFS

Cf. A222658, A249936, A249937.

Sequence in context: A347947 A208800 A356274 * A009002 A119882 A276968

Adjacent sequences:  A249932 A249933 A249934 * A249936 A249937 A249938

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 24 2014

STATUS

approved

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Last modified September 26 15:09 EDT 2022. Contains 357000 sequences. (Running on oeis4.)