A160707 -id:A160707 - cp's OEIS frontend

A160708 Convolution triangle by rows, row sums = the Robbins sequence, A005130 starting with offset 1.

1, 1, 1, 3, 1, 3, 18, 3, 3, 18, 192, 18, 9, 18, 192, 3472, 192, 54, 54, 192, 3472, 104964, 3472, 576, 324, 576, 3472, 104964, 5606272, 104964, 10416, 3456, 3456, 10416, 104964, 5306272

Offset: 1

Gary W. Adamson, May 24 2009

The terms are not integral in general, see A160707. - Joerg Arndt, Jan 02 2019
Row sums = the Robbins sequence A005130, starting with offset 1: (1, 2, 7, 42, 429,...).
Right and left borders = A160707, the convolution square root of A005130.

			First few rows of the triangle =
1;
1, 1;
3, 1, 3;
18, 3, 3, 18;
192, 18, 9, 18, 192;
3472, 192, 54, 54, 192, 3472;
104964, 3472, 576, 324, 576, 3472, 104964;
5606272, 104964, 10416, 3456, 3456, 10416, 104964, 5306272;
...
Example: row 5 = (192, 18, 9, 18, 192) = (192, 18, 3, 1, 1) * (1, 1, 3, 18, 192); where A005130(5) = 429 = (192 + 18 + 9 + 18 + 192).

Cf. A005130, A160707

Let M = an infinite lower triangular Toeplitz matrix with A160707 in every column: (1, 1, 3, 18, 192, 5472,...); where A160707 = the convolution square root of the Robbins sequence: (1, 2, 7, 42, 429, 7436,...). Let Q = an infinite lower triangular matrix with (1, 1, 3, 18, 192,...) as the main diagonal and the rest zeros. Triangle A160708 = M * Q.

A173312 Partial sums of A005130.

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Feb 16 2010

nonn

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(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.

Table[Sum[Product[(3 k + 1)!/(j + k)!, {k, 0, j - 1}], {j, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 26 2017 *)
Accumulate[Table[Product[(3k+1)!/(n+k)!,{k,0,n-1}],{n,0,20}]] (* Harvey P. Dale, Feb 06 2019 *)

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]

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

cp's OEIS Frontend

A160708 Convolution triangle by rows, row sums = the Robbins sequence, A005130 starting with offset 1.

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