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%I #40 Feb 13 2022 10:42:06
%S 1,3,7,16,39,104,301,927,2983,9901,33615,116115,406627,1440039,
%T 5147891,18550588,67310955,245716112,901759969,3325067016,12312494483,
%U 45766188970,170702447097,638698318874,2396598337975,9016444758528,34003644251233,128524394659942,486793096819011
%N Apply partial sum operator twice to Catalan numbers.
%C From _Alexander Adamchuk_, Jul 04 2006: (Start)
%C p divides a(p-1) and a((p-3)/2) for primes in A002476.
%C p divides a((p-5)/2) for primes in A068228.
%C p^2 divides a(p^2-1) for all primes p > 3. (End)
%C Equals triangle A106270(unsigned) * [1, 2, 3, ...]. - _Gary W. Adamson_, Apr 02 2009
%H Vincenzo Librandi, <a href="/A014140/b014140.txt">Table of n, a(n) for n = 0..200</a>
%F 1*C(n) + 2*C(n-1) + 3*C(n-2) + ... + (n+1-k)*C(k) + ... + n*C(1) + (n+1)*C(0), where C(k) = (2k)!/(k!*(k+1)!) is Catalan Number A000108(k). - _Alexander Adamchuk_, Jul 04 2006
%F From _Alexander Adamchuk_, Jul 04 2006: (Start)
%F a(n) = Sum_{m=0..n} Sum_{k=0..m} (2k)!/(k!*(k+1)!).
%F a(n) = Sum_{k=0..n} (n+1-k)*(2k)!/(k!*(k+1)!). (End)
%F G.f.: 1/(1-x)^2*(1-sqrt(1-4*x))/(2*x). - _Vladimir Kruchinin_, Oct 14 2016
%F a(n) = Sum_{k=0..n} binomial(n+2,k+2)*r(k), where r(k) are the Riordan numbers A005043. - _Vladimir Kruchinin_, Oct 14 2016
%F a(n) ~ 2^(2*n+4) / (9*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 14 2016
%p b:= proc(n) option remember; `if`(n<0, [0$2], (q->(f->
%p [f[2]+q, q]+f)(b(n-1)))(binomial(2*n, n)/(n+1)))
%p end:
%p a:= n-> b(n)[1]:
%p seq(a(n), n=0..28); # _Alois P. Heinz_, Feb 13 2022
%t Table[Sum[Sum[(2k)!/k!/(k+1)!,{k,0,m}],{m,0,n}],{n,0,50}] Table[Sum[(n+1-k)*(2k)!/k!/(k+1)!,{k,0,n}],{n,0,50}] (* _Alexander Adamchuk_, Jul 04 2006 *)
%o (PARI)
%o sm(v)={my(s=vector(#v)); s[1]=v[1]; for(n=2, #v, s[n]=v[n]+s[n-1]); s; }
%o C(n)=binomial(2*n, n)/(n+1);
%o sm(sm(vector(66, n, C(n-1))))
%o /* _Joerg Arndt_, May 04 2013 */
%Y Partial sums of A014137.
%Y Cf. A000108, A005043, A106270.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Alexander Adamchuk_, Jul 04 2006
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