A090317 Row sums of triangle in A090285.

1, 2, 7, 28, 118, 510, 2235, 9876, 43870, 195556, 873814, 3911168, 17527904, 78622982, 352911939, 1584927828, 7120769526, 32002212252, 143859840114, 646819996008, 2908670252676, 13081556909292, 58839348572574, 264674150692488, 1190649451348908, 5356483791828840, 24098774900561500

Offset: 0

Author

Philippe Deléham, Jan 25 2004

Keywords

• easy
• nonn

Comments

Apply the inverse of the Riordan array (1/(1-x^2),x/(1+x)^2) to 2^n. - Paul Barry, Mar 13 2009

Hankel transform is A079935. - Paul Barry, Mar 13 2009

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000108, A001700, A090285.

Programs

Mathematica

Table[SeriesCoefficient[(1-x^2*((1-Sqrt[1-4*x])/(2*x))^4)/(1-2*x*((1-Sqrt[1-4*x])/(2*x))^2),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

Maxima

a(n):=if n=0 then 1 else 4*binomial(2*n-1,n)/(n+1)+3*sum(((k+1)*2^(k)*binomial(2*n-1,n-k-1))/(n+k+1),k,1,n-1); /* Vladimir Kruchinin, Feb 21 2019 */

PARI

x='x+O('x^66); Vec((1-x^2*((1-sqrt(1-4*x))/(2*x))^4)/(1-2*x*((1-sqrt(1-4*x))/(2*x))^2)) \\ Joerg Arndt, May 11 2013

Formula

a(n+1) = A000108(n+1) + Sum_{k=0..n} a(n-k)*A001700(k); a(0) = 1.

G.f.: (1-x^2*c(x)^4)/(1-2x*c(x)^2), where c(x) is the g.f. of the Catalan numbers A000108. - Paul Barry, Mar 13 2009

Recurrence: 2*(n+1)*(n+3)*a(n) = (17*n^2+56*n-21)*a(n-1) - 18*(n+4)*(2*n-3)*a(n-2). - Vaclav Kotesovec, Oct 14 2012

a(n) ~ 9^n/2^(n+2). - Vaclav Kotesovec, Oct 14 2012

a(n) = 4*C(2*n-1,n)/(n+1)+3*Sum_{k=1..n-1}(k+1)*2^k*C(2*n-1,n-k-1)/(n+k+1), n>0, a(0)=1. - Vladimir Kruchinin, Feb 21 2019

Extensions

Term 15 corrected by Paul Barry, Mar 13 2009