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 A014140 Apply partial sum operator twice to Catalan numbers. 4

%I

%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 prime p=7,13,19,31,37,43,61,67..=A002476[n] Primes of form 6n + 1.

%C p divides a((p-5)/2) for prime p=13,37,61,73,97,109..=A068228[n] Primes congruent to 1 (mod 12).

%C p divides a(2p+1) for prime p=2,3,5,7,11,17,23,29,41,47,53,59,71.. All primes except 13,19,31,37,43,61,67..=A002476[n] Primes of form 6n + 1 excluding 7.

%C p divides a(3p+1) for prime p=3,5,7,11,17,23,29,41,47.. All odd primes except 13,19,31,37,43..=A002476[n] Primes of form 6n + 1 excluding 7.

%C p^2 divides a(p^2-1) for prime p>3.

%C p divides a(3p^3+1) for prime p=2,3,5,7,11..

%C p^2 divides a(3p^3+1) for prime p=2,3,5,11..

%C p^3 divides a(3p^3+1) for prime p=2,5.. 2^9 divides a(25). (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 a(n) = Sum[Sum[(2k)!/k!/(k+1)!,{k,0,m}],{m,0,n}]. a(n) = Sum[(n+1-k)*(2k)!/k!/(k+1)!,{k,0,n}]. - _Alexander Adamchuk_, Jul 04 2006

%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

%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 Cf. A000108, A005043, A014137, A106270.

%K nonn

%O 0,2

%A _N. J. A. Sloane_.

%E More terms from _Alexander Adamchuk_, Jul 04 2006

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Last modified November 18 07:01 EST 2018. Contains 317279 sequences. (Running on oeis4.)