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A062992 Row sums of unsigned triangle A062991. 11

%I

%S 1,3,13,67,381,2307,14589,95235,636925,4341763,30056445,210731011,

%T 1493303293,10678370307,76957679613,558403682307,4075996839933,

%U 29909606989827,220510631755773,1632599134961667,12133359132082173

%N Row sums of unsigned triangle A062991.

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

%H Vincenzo Librandi, <a href="/A062992/b062992.txt">Table of n, a(n) for n = 0..200</a>

%H L. Guo, W. Y. Sit, <a href="http://dx.doi.org/10.1007/s11786-010-0061-2">Enumeration and generating functions of Rota-Baxter Words</a>, Math. Comput. Sci. 4 (2010) 313-337

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

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

%F a(n)=(1/(n+1))*sum{k=0..n, binomial(2n+2, n-k)*binomial(n+k, k)}. - _Paul Barry_, May 11 2005

%F 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

%F Recurrence: (n+1)*a(n) = (7*n-5)*a(n-1) + 4*(2*n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 13 2012

%F a(n) ~ 2^(3*n+4)/(9*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 13 2012

%F a(n) = hypergeometric([-n, n+1], [-n-1], 2). - _Peter Luschny_, Nov 30 2014

%t 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 *)

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

%o (PARI) a(n)=if(n<0,0,polcoeff(serreverse((x-x^2)/(1+x)^2+O(x^(n+2))),n+1)) \\ _Ralf Stephan_

%o (Haskell)

%o a062992 = sum . a234950_row -- _Reinhard Zumkeller_, Jan 12 2014

%o (Sage)

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

%o [a(n).hypergeometric_simplify() for n in range(21)] # _Peter Luschny_, Nov 30 2014

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

%Y Cf. A064062.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Jul 12 2001

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Last modified March 22 12:23 EDT 2019. Contains 321421 sequences. (Running on oeis4.)