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A090317 Row sums of triangle in A090285.

%I A090317 #23 Feb 21 2019 15:01:27
%S A090317 1,2,7,28,118,510,2235,9876,43870,195556,873814,3911168,17527904,
%T A090317 78622982,352911939,1584927828,7120769526,32002212252,143859840114,
%U A090317 646819996008,2908670252676,13081556909292,58839348572574,264674150692488,1190649451348908,5356483791828840,24098774900561500
%N A090317 Row sums of triangle in A090285.
%C A090317 Apply the inverse of the Riordan array (1/(1-x^2),x/(1+x)^2) to 2^n. - _Paul Barry_, Mar 13 2009
%C A090317 Hankel transform is A079935. - _Paul Barry_, Mar 13 2009
%H A090317 Vincenzo Librandi, <a href="/A090317/b090317.txt">Table of n, a(n) for n = 0..200</a>
%F A090317 a(n+1) = A000108(n+1) + Sum_{k=0..n} a(n-k)*A001700(k); a(0) = 1.
%F A090317 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
%F A090317 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
%F A090317 a(n) ~ 9^n/2^(n+2). - _Vaclav Kotesovec_, Oct 14 2012
%F A090317 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
%t A090317 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 *)
%o A090317 (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
%o A090317 (Maxima)
%o A090317 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 */
%Y A090317 Cf. A000108, A001700, A090285.
%K A090317 easy,nonn
%O A090317 0,2
%A A090317 _Philippe Deléham_, Jan 25 2004
%E A090317 Term 15 corrected by _Paul Barry_, Mar 13 2009