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A174123 Partial sums of A002893. 0
1, 4, 19, 112, 751, 5404, 40573, 313408, 2471167, 19791004, 160459069, 1313922064, 10847561089, 90174127684, 754009158019, 6336733626112, 53489159252671, 453258909448636, 3854034482891725, 32871004555812112, 281127047928811201 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Partial sums of (2n)-th moment of the distance from the origin of a 3-step random walk in the plane. The subsequence of primes in this partial sum begins: 19, 751, 10847561089, 53489159252671.

LINKS

Table of n, a(n) for n=0..20.

Tewodros Amdeberhan and Roberto Tauraso, Two triple binomial sum supercongruences, arXiv:1607.02483 [math.NT], Jul 08 2016.

FORMULA

a(n) = SUM[i=0..n] A002893(i) = SUM[i=0..n] SUM[p+q+r=i} (i!/(p!q!r!))^2 with p,q,r >=0.

From Sergey Perepechko Feb 16 2011: (Start)

O.g.f.: 2*sqrt(2)/Pi/(1-z)/sqrt(1-6*z-3*z^2+sqrt((1-z)^3*(1-9*z)))* EllipticK(8*z^(3/2)/(1-6*z-3*z^2+sqrt((1-z)^3*(1-9*z)))).

9*(n+2)^2*a(n) - (99+86*n+19*n^2)*a(n+1) + (72+56*n+11*n^2)*a(n+2) - (n+3)^2*a(n+3)=0.

(End)

a(n) ~ 3^(2*n + 7/2) / (32*Pi*n). - Vaclav Kotesovec, Jul 11 2016

MATHEMATICA

Accumulate[Table[Sum[Binomial[n, k]^2 Binomial[2k, k], {k, 0, n}], {n, 0, 20}]] (* Harvey P. Dale, May 05 2013 *)

PROG

(PARI) a(n)=sum(m=0, n, sum(k=0, m, binomial(m, k)^2*binomial(2*k, k)))

CROSSREFS

Cf. A002893, A000172, A002895, A000984.

Sequence in context: A004212 A243241 A060905 * A127548 A122835 A013185

Adjacent sequences:  A174120 A174121 A174122 * A174124 A174125 A174126

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Mar 08 2010

STATUS

approved

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Last modified November 20 13:59 EST 2017. Contains 294972 sequences.