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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. 2
1, 3, 19, 163, 1626, 17769, 206487, 2508195, 31504240, 406214878, 5349255726, 71672186953, 974311431094, 13408623649893, 186491860191519, 2617716792257955, 37040913147928380, 527875569932002608, 7570657419156212256, 109194783587953243038 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Logarithmic derivative of A183070.

A bisection of A003162.

LINKS

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

Pedro J. Miana, Hideyuki Ohtsuka, 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} (C(2n+1,k) - C(2n+1,k-1))^3/C(2n+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

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Last modified December 6 11:56 EST 2022. Contains 358633 sequences. (Running on oeis4.)