A243953 E.g.f.: exp( Sum_{n>=1} A000108(n-1)*x^n/n ), where A000108(n) = binomial(2*n,n)/(n+1) forms the Catalan numbers.

1, 1, 2, 8, 56, 592, 8512, 155584, 3456896, 90501632, 2728876544, 93143809024, 3550380249088, 149488545697792, 6890674623094784, 345131685337530368, 18664673706719019008, 1083931601731053223936, 67278418002152175960064, 4444711314548967826259968

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

Formula

E.g.f. A(x) satisfies:

(1) A(x) = exp(1 - sqrt(1-4*x)) * (1 + sqrt(1-4*x))/2.

(2) A(x)^2 - A(x)*A'(x) + x*A'(x)^2 = 0 (differential equation).

(3) [x^n/n!] A(x)^(n+1) = (n+1)^(n-1)*2^n for n>=0.

(4) A(x) = G(x/A(x)) such that A(x*G(x)) = G(x) = Sum_{n>=0} (n+1)^(n-2)*2^n*x^n/n!.

(5) A(x) = x / Series_Reversion(x*G(x)) where G(x) = Sum_{n>=0} (n+1)^(n-2)*2^n*x^n/n!.

(6) x = -LambertW(-2*x/A(x)) * (2 + LambertW(-2*x/A(x)))/4. [From a formula by Vaclav Kotesovec in A127670]

a(n) ~ 2^(2*n-5/2) * n^(n-2) / exp(n-1). - Vaclav Kotesovec, Jun 22 2014

a(n) = sum(i=0..n-1, (n-1)!/(n-i-1)!*binomial(2*i,i)/(i+1)*a(n-i-1)), a(0)=1. - Vladimir Kruchinin, Feb 22 2015

From Peter Bala, Apr 14 2017: (Start)

a(n+2) = 2^(n+1)*A001515(n).

a(n+1) = Sum_{k = 0..n} binomial(n+k-1,2*k)*2^(n-k)*(2*k)!/k!.

D-finite with recurrence a(n) = (4*n - 10)*a(n-1) + 4*a(n-2) with a(0) = a(1) = 1.

The derivative A'(x) of the e.g.f. is equal to exp(2*x*c(x)), that is, A'(x) is the Catalan transform of exp(2*x) as defined in Barry, Section 3. (End)

E.g.f. A(x) satisfies (x/A(x))' = 1/A'(x). - Alexander Burstein, Oct 31 2023

Example

G.f.: A(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 56*x^4/4! + 592*x^5/5! + 8512*x^6/6! +...
such that the logarithmic derivative of the e.g.f. equals the Catalan numbers:
log(A(x)) = x + x^2/2 + 2*x^3/3 + 5*x^4/4 + 14*x^5/5 + 42*x^6/6 + 132*x^7/7 + 429*x^8/8 +...+ A000108(n-1)*x^n/n +...
thus A'(x)/A(x) = C(x) where C(x) = 1 + x*C(x)^2.
Also, e.g.f. A(x) satisfies:
A(x) = 1 + x/A(x) + 4*(x/A(x))^2/2! + 32*(x/A(x))^3/3! + 400*(x/A(x))^4/4! + 6912*(x/A(x))^5/5! +...+ (n+1)^(n-2)*2^n*(x/A(x))^n/n! +...
If we form a table of coefficients of x^k/k! in A(x)^n, like so:
[1, 1,  2,    8,    56,    592,    8512,   155584,    3456896, ...];
[1, 2,  6,   28,   200,   2064,   28768,   511424,   11106432, ...];
[1, 3, 12,   66,   504,   5256,   72288,  1259712,   26822016, ...];
[1, 4, 20,  128,  1064,  11488,  158752,  2740480,   57517184, ...];
[1, 5, 30,  220,  2000,  22680,  319600,  5525600,  115094400, ...];
[1, 6, 42,  348,  3456,  41472,  602352, 10533024,  219321216, ...];
[1, 7, 56,  518,  5600,  71344, 1075648, 19176304,  401916032, ...];
[1, 8, 72,  736,  8624, 116736, 1835008, 33554432,  712166016, ...];
[1, 9, 90, 1008, 12744, 183168, 3009312, 56687040, 1224440064, ...]; ...
then the main diagonal equals (n+1)^(n-1) * 2^n for n>=0:
[1, 2, 12, 128, 2000, 41472, 1075648, 33554432, 1224440064, ...].
Note that Sum_{n>=0} (n+1)^(n-2) * 2^n * x^n/n! is an e.g.f. of A127670.

Mathematica

CoefficientList[Series[E^(1 - Sqrt[1-4*x]) * (1 + Sqrt[1-4*x])/2, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jun 22 2014 *)

Prog

(PARI) /* Explicit formula: */
{a(n)=n!*polcoeff( exp(1-sqrt(1-4*x +x*O(x^n))) * (1 + sqrt(1-4*x +x*O(x^n)))/2, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* Logarithmic derivative of e.g.f. equals Catalan numbers: */
{A000108(n) = binomial(2*n, n)/(n+1)}
{a(n)=n!*polcoeff( exp(sum(m=1, n, A000108(m-1)*x^m/m)+x*O(x^n)), n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* From [x^n/n!] A(x)^(n+1) = (n+1)^(n-1)*2^n */
{a(n)=n!*polcoeff(x/serreverse(x*sum(m=0, n+1, (m+1)^(m-2)*(2*x)^m/m!)+x^2*O(x^n)), n)}
for(n=0, 25, print1(a(n), ", "))
(Maxima)
a(n):=if n=0 then 1 else sum((n-1)!/(n-i-1)!*binomial(2*i, i)/(i+1)*a(n-i-1), i, 0, n-1); /* Vladimir Kruchinin, Feb 22 2015 */

Crossrefs

Cf. A000108, A127670, A243954, A213507.

Cf. A251663, A251664, A251665, A251666, A251667, A251668, A251669, A251670.

Cf. A251573, A251574, A251575, A251576, A251577, A251578, A251579, A251580.

Cf. A001515.

Sequence in context: A325290 A197949 A363589 * A005439 A128814 A108208

Adjacent sequences: A243950 A243951 A243952 * A243954 A243955 A243956

Keyword

nonn,easy

Author

Paul D. Hanna, Jun 21 2014

Status

approved