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Partial sums of A005130.
1

%I #9 Feb 06 2019 12:10:12

%S 1,2,4,11,53,482,7918,226266,11076482,922911942,130457184642,

%T 31226202037017,12642538061714517,8652026056359367017,

%U 10004193381504526849017,19539080428042781631746217

%N Partial sums of A005130.

%C Partial sums of Robbins numbers. Partial sums of the number of descending plane partitions whose parts do not exceed n. Partial sums of the number of n X n alternating sign matrices (ASM's). After 2, 11, 53, when is this partial sum again prime, as it is not again prime through a(32)?

%F a(n) = Sum_{i=0..n} A005130(i) = Sum_{i=0..n} Product_{k=0..i-1} (3k+1)!/(i+k)!. [corrected by _Vaclav Kotesovec_, Oct 26 2017]

%F a(n) ~ Pi^(1/3) * exp(1/36) * 3^(3*n^2/2 - 7/36) / (A^(1/3) * Gamma(1/3)^(2/3) * n^(5/36) * 2^(2*n^2 - 5/12)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Oct 26 2017

%e a(17) = 1 + 1 + 2 + 7 + 42 + 429 + 7436 + 218348 + 10850216 + 911835460 + 129534272700 + 31095744852375 + 12611311859677500 + 8639383518297652500 + 9995541355448167482000 + 19529076234661277104897200 + 64427185703425689356896743840 + 358869201916137601447486156417296.

%t Table[Sum[Product[(3 k + 1)!/(j + k)!, {k, 0, j - 1}], {j, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 26 2017 *)

%t Accumulate[Table[Product[(3k+1)!/(n+k)!,{k,0,n-1}],{n,0,20}]] (* _Harvey P. Dale_, Feb 06 2019 *)

%Y Cf. A005130, A006366, A048601, also A003827, A005156, A005158, A005160-A005164, A050204, A049503, A160707, A160708.

%K nonn

%O 0,2

%A _Jonathan Vos Post_, Feb 16 2010