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%I #9 May 06 2015 10:26:28
%S 1,1,3,13,66,380,2447,17424,135740,1146202,10409616,101031397,
%T 1042361261,11380543227,130980176993,1583726089859,20058575880505,
%U 265416510500487,3660581511822798,52511905732091815,782044494316086134
%N Recurrence: a(n) = Sum_{k=0..n-1} C(2*n-2,n-k-1)*a(k) with a(0)=1.
%H Vaclav Kotesovec, <a href="/A110530/b110530.txt">Table of n, a(n) for n = 0..500</a>
%e a(1) = 1*1 =1,
%e a(2) = 2*1 + 1*1 = 3,
%e a(3) = 6*1 + 4*1 + 1*3 = 13,
%e a(4) = 20*1 + 15*1 + 6*3 + 1*13 = 66,
%e a(5) = 70*1 + 56*1 + 28*3 + 8*13 + 1*66 = 380.
%e This sequence can be generated by the addition table:
%e (1)_1__1___1___1 ...
%e _1__2_(3)__4___5___6 ...
%e _3__4__6___9_(13)_18__24 ...
%e 13_16_20__26__35__48_(66)_90 ...
%e 66_79_95_115_141_176_224_290_(380) ...
%t nmax = 30; aa = ConstantArray[0,nmax+1]; aa[[1]] = 1; Do[aa[[n+1]]=Sum[Binomial[2*n-2, n-k-1]*aa[[k+1]],{k,0,n-1}],{n,1,nmax}]; aa (* _Vaclav Kotesovec_, May 06 2015 , much faster than PARI *)
%o (PARI) a(n)=if(n==0,1,sum(k=0,n-1,binomial(2*n-2,n-k-1)*a(k)))
%Y Cf. A110531.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 25 2005
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