A110531 - OEIS

%I #9 May 06 2015 10:26:57

%S 1,1,4,19,103,628,4258,31753,257815,2259718,21230800,212579938,

%T 2257364371,25315773751,298758986356,3698444546248,47893544997832,

%U 647174968407262,9105301419381562,133116659482393549

%N Recurrence: a(n) = Sum_{k=0..n-1} C(2*n-1,n-k-1)*a(k) with a(0)=1.

%H Vaclav Kotesovec, <a href="/A110531/b110531.txt">Table of n, a(n) for n = 0..500</a>

%e a(1) = 1*1 = 1,

%e a(2) = 3*1 + 1*1 = 4,

%e a(3) = 10*1 + 5*1 + 1*4 = 19,

%e a(4) = 35*1 + 21*1 + 7*4 + 1*19 = 103,

%e a(5) = 126*1 + 84*1 + 36*4 + 9*19 + 1*103 = 628.

%e This sequence can be generated by the addition table:

%e __1__(1)__1___1___1___1 ...

%e __1___2___3__(4)__5___6___7 ...

%e __4___5___7__10__14_(19)_25___32 ...

%e _19__23__28__35__45__59__78_(103)_135 ...

%e 103_122_145_173_208_253_312__390__493_(628) ...

%t nmax = 30; aa = ConstantArray[0,nmax+1]; aa[[1]] = 1; Do[aa[[n+1]]=Sum[Binomial[2*n-1, 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-1,n-k-1)*a(k)))

%Y Cf. A110530.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 25 2005