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

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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

a(n) = SUM[Sum_{i=0..n] } A005130(i) = SUM[Sum_{i=0..n]PRODUCT[} Product_{k=0..i-1} (3k+1)!/(i+k)!. [corrected by Vaclav Kotesovec, Oct 26 2017]

a(n) = SUM[i=0..n] A005130(i) = SUM[i=0..n]PRODUCT[k=0..i-1} (3k+1)!/(ni+k)!. [corrected by _Vaclav Kotesovec_, Oct 26 2017]

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

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_Jonathan Vos Post (jvospost3(AT)gmail.com), _, Feb 16 2010

Partial sums of A005130.

1, 2, 4, 11, 53, 482, 7918, 226266, 11076482, 922911942, 130457184642, 31226202037017, 12642538061714517, 8652026056359367017, 10004193381504526849017, 19539080428042781631746217

0,2

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)?

a(n) = SUM[i=0..n] A005130(i) = SUM[i=0..n]PRODUCT[k=0..i-1} (3k+1)!/(n+k)!.

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

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

nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 16 2010

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