A183069 L.g.f.: Sum_{n>=1,k>=0} CATALAN(n,k)^2 * x^(n+k)/n = Sum_{n>=1} a(n)*x^n/n, where CATALAN(n,k) = n*C(n+2*k-1,k)/(n+k) is the coefficient of x^k in C(x)^n and C(x) is the g.f. of the Catalan numbers.

1, 3, 19, 163, 1626, 17769, 206487, 2508195, 31504240, 406214878, 5349255726, 71672186953, 974311431094, 13408623649893, 186491860191519, 2617716792257955, 37040913147928380, 527875569932002608, 7570657419156212256, 109194783587953243038

Offset

1,2

Comments

Logarithmic derivative of A183070.

A bisection of A003162.

We conjecture that the sequence satisfies the supercongruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for all primes p >= 5 and positive integers n and k. - Peter Bala, Mar 20 2023

Links

G. C. Greubel, Table of n, a(n) for n = 1..500

Pedro J. Miana, Hideyuki Ohtsuka, and Natalia Romero, Sums of powers of Catalan triangle numbers, arXiv:1602.04347 [math.NT], 2016.

Formula

a(n) = Sum_{k=0..n} (n-k)*C(n+k-1,k)^2/n for n>=1.

a(n) = Sum_{k=0..n-1} (C(2*n-1,k) - C(2*n-1,k-1))^3/C(2*n-1,n). [From a formula given in A003162 by Michael Somos.]

Recurrence: 2*n^2*(2*n-1)*(7*n^2 - 20*n + 14)*a(n) = (455*n^5 - 2427*n^4 + 4850*n^3 - 4406*n^2 + 1728*n - 216)*a(n-1) - 4*(n-2)*(2*n - 3)^2*(7*n^2 - 6*n + 1)*a(n-2). - Vaclav Kotesovec, Mar 06 2014

a(n) ~ 16^n/(9*Pi*n^2). - Vaclav Kotesovec, Mar 06 2014

Example

L.g.f.: L(x) = x + 3*x^2/2 + 19*x^3/3 + 163*x^4/4 + 1626*x^5/5 + ...
L(x) = (1 + x + 2^2*x^2 + 5^2*x^3 + 14^2*x^4 + ...)*x
+ (1 + 2^2*x + 5^2*x^2 + 14^2*x^3 + 42^2*x^4 + ...)*x^2/2
+ (1 + 3^2*x + 9^2*x^2 + 28^2*x^3 + 90^2*x^4 + ...)*x^3/3
+ (1 + 4^2*x + 14^2*x^2 + 48^2*x^3 + 165^2*x^4 + ...)*x^4/4
+ (1 + 5^2*x + 20^2*x^2 + 75^2*x^3 + 275^2*x^4 + ...)*x^5/5
+ (1 + 6^2*x + 27^2*x^2 + 110^2*x^3 + 429^2*x^4 + ...)*x^6/6 + ...
which consists of the squares of coefficients in the powers of C(x),
where C(x) = 1 + x*C(x)^2 is g.f. of the Catalan numbers.
...
Exponentiation yields the g.f. of A183070:
exp(L(x)) = 1 + x + 2*x^2 + 8*x^3 + 49*x^4 + 380*x^5 + 3400*x^6 + ...

Mathematica

Table[Sum[(n-k)*Binomial[n+k-1, k]^2/n, {k, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Mar 06 2014 *)

Prog

(PARI) {a(n)=if(n<1, 0, sum(k=0, n, (n-k)*binomial(n+k-1, k)^2/n))}
(PARI) {a(n)=sum(k=0, n, (binomial(2*n+1, k)-binomial(2*n+1, k-1))^3)/binomial(2*n+1, n)}
(Magma) [&+[(n-k)*Binomial(n+k-1, k)^2/n: k in [0..n]]: n in [1..30]]; // Vincenzo Librandi, Feb 16 2016

Crossrefs

Cf. A183070, A003162, A000108, A009766.

Sequence in context: A199559 A136474 A337818 * A256710 A215093 A201123

Adjacent sequences: A183066 A183067 A183068 * A183070 A183071 A183072

Keyword

nonn

Author

Paul D. Hanna, Dec 23 2010

Extensions

a(19)-a(20) from Vincenzo Librandi, Feb 16 2016

Status

approved