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A062992 Row sums of unsigned triangle A062991. 11
1, 3, 13, 67, 381, 2307, 14589, 95235, 636925, 4341763, 30056445, 210731011, 1493303293, 10678370307, 76957679613, 558403682307, 4075996839933, 29909606989827, 220510631755773, 1632599134961667, 12133359132082173 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n)=N(2; n,x=-1), with the polynomials N(2; n,x) defined in A062991.

LINKS

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

L. Guo, W. Y. Sit, Enumeration and generating functions of Rota-Baxter Words, Math. Comput. Sci. 4 (2010) 313-337

FORMULA

a(n)=2*sum(((-1)^j)*C(n-j)*2^(n-j), j=0..n)-(-1)^n with C(n) := A000108(n) (Catalan).

G.f.: (2*c(2*x)-1)/(1+x) with c(x) g.f. of A000108.

a(n)=(1/(n+1))*sum{k=0..n, binomial(2n+2, n-k)*binomial(n+k, k)}. - Paul Barry, May 11 2005

Rewritten: a(n)= (1-2*c(n, -2))*(-1)^(n+1), n>=0, with c(n, x):=sum(C(k)*x^k, k=0..n) and C(k):=A000108(k) (Catalan). - Wolfdieter Lang, Oct 31 2005

Recurrence: (n+1)*a(n) = (7*n-5)*a(n-1) + 4*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 13 2012

a(n) ~ 2^(3*n+4)/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 13 2012

a(n) = hypergeometric([-n, n+1], [-n-1], 2). - Peter Luschny, Nov 30 2014

MATHEMATICA

Table[2*Sum[(-1)^j*Binomial[2*n-2*j, n-j]/(n-j+1)*2^(n-j), {j, 0, n}]-(-1)^n, {n, 0, 20}] (* Vaclav Kotesovec, Oct 13 2012 *)

PROG

(PARI) a(n)=polcoeff((1-2*x-sqrt(1-8*x+x^2*O(x^n)))/(2*x+2*x^2), n)

(PARI) a(n)=if(n<0, 0, polcoeff(serreverse((x-x^2)/(1+x)^2+O(x^(n+2))), n+1)) \\ Ralf Stephan

(Haskell)

a062992 = sum . a234950_row  -- Reinhard Zumkeller, Jan 12 2014

(Sage)

def a(n): return hypergeometric([-n, n+1], [-n-1], 2)

[a(n).hypergeometric_simplify() for n in range(21)] # Peter Luschny, Nov 30 2014

CROSSREFS

Cf. A112707 (c(n, -m) triangle). Here m=2 is used. Row sums of A234950.

Cf. A064062.

Sequence in context: A239198 A234282 A200754 * A064062 A114191 A107592

Adjacent sequences:  A062989 A062990 A062991 * A062993 A062994 A062995

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Jul 12 2001

STATUS

approved

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Last modified February 17 15:19 EST 2019. Contains 320220 sequences. (Running on oeis4.)