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A062992 Row sums of unsigned triangle A062991.

%I A062992 #42 Sep 28 2024 05:41:11
%S A062992 1,3,13,67,381,2307,14589,95235,636925,4341763,30056445,210731011,
%T A062992 1493303293,10678370307,76957679613,558403682307,4075996839933,
%U A062992 29909606989827,220510631755773,1632599134961667,12133359132082173
%N A062992 Row sums of unsigned triangle A062991.
%C A062992 a(n) = N(2; n,x=-1), with the polynomials N(2; n,x) defined in A062991.
%H A062992 Vincenzo Librandi, <a href="/A062992/b062992.txt">Table of n, a(n) for n = 0..200</a>
%H A062992 Paul Barry, <a href="https://arxiv.org/abs/2001.08799">Characterizations of the Borel triangle and Borel polynomials</a>, arXiv:2001.08799 [math.CO], 2020.
%H A062992 L. Guo and 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 A062992 a(n) = (-1)^(n+1) + 2*Sum_{j = 0..n} (-1)^j*C(n-j)*2^(n-j) with C(n) := A000108(n) (Catalan).
%F A062992 G.f.: A(x) = (2*c(2*x) - 1)/(1 + x) with c(x) the g.f. of A000108.
%F A062992 a(n) = (1/(n+1)) * Sum_{k = 0..n} binomial(2*n+2, n-k)*binomial(n+k, k). - _Paul Barry_, May 11 2005
%F A062992 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
%F A062992 Recurrence: (n+1)*a(n) = (7*n-5)*a(n-1) + 4*(2*n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 13 2012
%F A062992 a(n) ~ 2^(3*n+4)/(9*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 13 2012
%F A062992 a(n) = hypergeometric([-n, n+1], [-n-1], 2). - _Peter Luschny_, Nov 30 2014
%F A062992 G.f.: A(x) = exp( Sum_{n >= 1} A119259(n)*x^n/n ). - _Peter Bala_, Jun 08 2023
%t A062992 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 A062992 (PARI) a(n)=polcoeff((1-2*x-sqrt(1-8*x+x^2*O(x^n)))/(2*x+2*x^2),n)
%o A062992 (PARI) a(n)=if(n<0,0,polcoeff(serreverse((x-x^2)/(1+x)^2+O(x^(n+2))),n+1)) \\ _Ralf Stephan_
%o A062992 (Haskell)
%o A062992 a062992 = sum . a234950_row  -- _Reinhard Zumkeller_, Jan 12 2014
%o A062992 (Sage)
%o A062992 def a(n): return hypergeometric([-n, n+1], [-n-1], 2)
%o A062992 [a(n).hypergeometric_simplify() for n in range(21)] # _Peter Luschny_, Nov 30 2014
%o A062992 (Magma)
%o A062992 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A062992 Coefficients(R!( (1-2*x-Sqrt(1-8*x))/(2*x+2*x^2) )); // _G. C. Greubel_, Sep 27 2024
%Y A062992 Cf. A112707 (c(n, -m) triangle). Here m=2 is used. Row sums of A234950.
%Y A062992 Cf. A000108, A062991, A064062, A119259.
%K A062992 nonn,easy
%O A062992 0,2
%A A062992 _Wolfdieter Lang_, Jul 12 2001