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A014140 Apply partial sum operator twice to Catalan numbers. 4
1, 3, 7, 16, 39, 104, 301, 927, 2983, 9901, 33615, 116115, 406627, 1440039, 5147891, 18550588, 67310955, 245716112, 901759969, 3325067016, 12312494483, 45766188970, 170702447097, 638698318874, 2396598337975, 9016444758528, 34003644251233, 128524394659942, 486793096819011 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

From Alexander Adamchuk, Jul 04 2006: (Start)

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.

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

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.

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.

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

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

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

p^3 divides a(3p^3+1) for prime p=2,5.. 2^9 divides a(25). (End)

Equals triangle A106270(unsigned) * [1, 2, 3,...]. [Gary W. Adamson, Apr 02 2009]

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

FORMULA

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

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

G.f.: 1/(1-x)^2*(1-sqrt(1-4*x))/(2*x). - Vladimir Kruchinin, Oct 14 2016

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

a(n) ~ 2^(2*n+4) / (9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2016

MATHEMATICA

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 *)

PROG

(PARI)

sm(v)={my(s=vector(#v)); s[1]=v[1]; for(n=2, #v, s[n]=v[n]+s[n-1]); s; }

C(n)=binomial(2*n, n)/(n+1);

sm(sm(vector(66, n, C(n-1))))

/* Joerg Arndt, May 04 2013 */

CROSSREFS

Cf. A000108, A005043, A014137, A106270.

Sequence in context: A190528 A203611 A176604 * A271788 A103439 A147321

Adjacent sequences:  A014137 A014138 A014139 * A014141 A014142 A014143

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Alexander Adamchuk, Jul 04 2006

STATUS

approved

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Last modified December 13 16:57 EST 2017. Contains 295959 sequences.