A396097 G.f. A(x) satisfies Sum_{k=0..n} [x^k] A(x)^n = Sum_{k=0..2*n} [x^k] C(x)^n for n >= 0, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan function.

1, 3, 24, 287, 4082, 63854, 1060893, 18375481, 328140158, 5998110712, 111679078694, 2110614752092, 40383196151858, 780713537950989, 15227082192168750, 299259411611190032, 5920512681769082585, 117815581575201608792, 2356618411595244754811, 47356204772093822680868, 955566142938585617722546

Offset

0,2

Comments

Compare to: Sum_{k=0..n} [x^k] 1/(1-x)^n = (2*n)!/n!^2 = A000984(n).

Compare to: Sum_{k=0..n} [x^k] 1/(1-x)^(2*n) = Sum_{k=0..2*n} [x^k] 1/(1-x)^n = (3*n)!/(n!*(2*n)!) = A005809(n).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..400

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) Sum_{k=0..n} [x^k] A(x)^n = Sum_{k=0..2*n} [x^k] C(x)^n for n >= 0, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan function.

(2) Sum_{k=0..n} [x^k] A(x)^n = A352275(n) where A352275(n) = Sum_{k=0..2*n} binomial(n + 2*k,k) * n/(n + 2*k) for n >= 0.

(3) A(x) = G(x/A(x)) where G(x) = A(x*G(x)) satisfies (G(x) + x*G'(x)) / (G(x) - x*G(x)^2) = Sum_{n>=0} A352275(n) * x^n.

(4) A(x) = (1-x) * (A(x) - x*A'(x)) * Sum_{n>=0} A352275(n) * x^n / A(x)^n.

From Vaclav Kotesovec, May 29 2026: (Start)

a(n) ~ c * d^n / n^(3/2), where

d = 21.7870041673292333577648845136818182882906239511526041767826595572589580609... and

c = 0.14785584766568055166299618504433412209227083010399493968555118456367388152...

Conjecture: d is the largest real root of the equation -132400 + 2959788*d - 23123461*d^2 + 55640000*d^3 - 43613528*d^4 + 6629032*d^5 - 490000*d^6 + 12500*d^7 == 0. (End)

Example

G.f.: A(x) = 1 + 3*x + 24*x^2 + 287*x^3 + 4082*x^4 + 63854*x^5 + 1060893*x^6 + 18375481*x^7 + 328140158*x^8 + 5998110712*x^9 + 111679078694*x^10 + ...
RELATED SERIES.
The series G(x) = A(x*G(x)) begins
G(x) = 1 + 3*x + 33*x^2 + 530*x^3 + 10055*x^4 + 209357*x^5 + 4625679*x^6 + 106500731*x^7 + 2527389563*x^8 + ...
satisfies (G(x) + x*G'(x)) / (G(x) - x*G(x)^2) = D(x) where
D(x) = 1 + 4*x + 64*x^2 + 1429*x^3 + 35072*x^4 + 898129*x^5 + 23571781*x^6 + 628750217*x^7 + ... + A352275(n)*x^n + ...
Thus, G(x) = 1 + Integral G(x)*(D(x)-1)/x - D(x)*G(x)^2 dx
and A(x) = x/Series_Reversion(x*G(x)).
RELATED TABLES.
We shall compare the terms of the following tables to illustrate the defining property of the g.f. of this sequence.
Given the g.f. A(x), the table of coefficients of x^k in A(x)^n begins
  n = 1: [1,  3,  24,  287,  4082,   63854,  1060893, ...];
  n = 2: [1,  6,  57,  718, 10462,  165976,  2783215, ...];
  n = 3: [1,  9,  99, 1320, 19788,  319299,  5414988, ...];
  n = 4: [1, 12, 150, 2120, 32789,  539348,  9266334, ...];
  n = 5: [1, 15, 210, 3145, 50275,  844483, 14719195, ...];
  n = 6: [1, 18, 279, 4422, 73137, 1256142, 22237782, ...];
  ...
Given the Catalan series C(x) = 1 + x*C(x)^2, the table of coefficients of x^k in C(x)^n begins
  n = 1: [1, 1,  2,  5,  14,  42,  132,  429,  1430, ...];
  n = 2: [1, 2,  5, 14,  42, 132,  429, 1430,  4862, ...];
  n = 3: [1, 3,  9, 28,  90, 297, 1001, 3432, 11934, ...];
  n = 4: [1, 4, 14, 48, 165, 572, 2002, 7072, 25194, ...];
  ...
From these two tables, we see that the following partial sums of the rows in the above tables are equal
  n = 1: (1 + 3) = (1 + 1 + 2);
  n = 2: (1 + 6 + 57) = (1 + 2 + 5 + 14 + 42);
  n = 3: (1 + 9 + 99 + 1320) = (1 + 3 + 9 + 28 + 90 + 297 + 1001);
  n = 4: (1 + 12 + 150 + 2120 + 32789) = (1 + 4 + 14 + 48 + 165 + 572 + 2002 + 7072 + 25194);
  ...

Prog

(PARI) /* By Definition (slow): */
{d(n) = if(n==0, 1, sum(k=0, 2*n, binomial(n + 2*k, k) * n/(n + 2*k) ) )}
{a(n) = if(n==0, 1, (d(n) - sum(k=0, n, polcoef(sum(j=0, min(k, n-1), a(j)*x^j)^n + x*O(x^k), k)))/n)}
for(n=0, 12, print1(a(n), ", "))
(PARI) /* Faster, using series reversion: */
{d(n) = if(n==0, 1, sum(k=0, 2*n, binomial(n + 2*k, k) * n/(n + 2*k) ) )}
{a(n) = my(D = sum(k=0, n, d(k)*x^k) +x^3*O(x^n), G=1+x*O(x^n));
for(i=1, n, G = 1 + intformal( (D-1)*G/x - D*G^2));
polcoef(GF = x/serreverse(x*G +x^2*O(x^n)), n)}
{upto(n) = a(n); Vec(GF)}
upto(30)

Crossrefs

Cf. A352275, A232683, A000108.

Sequence in context: A355426 A064037 A257453 * A128572 A052592 A059381

Adjacent sequences: A396094 A396095 A396096 * A396098 A396099 A396100

Keyword

nonn

Author

Paul D. Hanna, May 28 2026

Status

approved