A062992 Row sums of unsigned triangle A062991.

1, 3, 13, 67, 381, 2307, 14589, 95235, 636925, 4341763, 30056445, 210731011, 1493303293, 10678370307, 76957679613, 558403682307, 4075996839933, 29909606989827, 220510631755773, 1632599134961667, 12133359132082173

Offset: 0

Author

Wolfdieter Lang, Jul 12 2001

Keywords

• nonn
• easy

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
• Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
• L. Guo and W. Y. Sit, Enumeration and generating functions of Rota-Baxter Words, Math. Comput. Sci. 4 (2010) 313-337.

Crossrefs

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

Cf. A000108, A062991, A064062, A119259.

Programs

Haskell

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

Magma

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (1-2*x-Sqrt(1-8*x))/(2*x+2*x^2) )); // G. C. Greubel, Sep 27 2024

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 *)

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

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

Formula

a(n) = (-1)^(n+1) + 2*Sum_{j = 0..n} (-1)^j*C(n-j)*2^(n-j) with C(n) := A000108(n) (Catalan).

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

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

Rewritten: a(n) = (1 - 2*c(n, -2))*(-1)^(n+1), n >= , with c(n, x) := Sum_{k = 0..n} C(k)*x^k 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

G.f.: A(x) = exp( Sum_{n >= 1} A119259(n)*x^n/n ). - Peter Bala, Jun 08 2023