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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. 22
1, 1, 2, 8, 56, 592, 8512, 155584, 3456896, 90501632, 2728876544, 93143809024, 3550380249088, 149488545697792, 6890674623094784, 345131685337530368, 18664673706719019008, 1083931601731053223936, 67278418002152175960064, 4444711314548967826259968 (list; graph; refs; listen; history; text; internal format)
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!.

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)

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: A124212 A325290 A197949 * A005439 A128814 A108208

Adjacent sequences:  A243950 A243951 A243952 * A243954 A243955 A243956

KEYWORD

nonn,easy

AUTHOR

Paul D. Hanna, Jun 21 2014

STATUS

approved

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Last modified April 26 04:04 EDT 2019. Contains 322469 sequences. (Running on oeis4.)