A091355 -id:A091355 - cp's OEIS frontend

A091360 Partial sums of A000219.

1, 2, 5, 11, 24, 48, 96, 182, 342, 624, 1124, 1983, 3462, 5947, 10114, 16993, 28290, 46624, 76225, 123555, 198833, 317627, 504102, 794885, 1246079, 1942112, 3010857, 4643515, 7126749, 10886361, 16555324, 25067633, 37801062, 56776035, 84951990, 126643036, 188127997, 278507781, 410949776, 604437277, 886284200, 1295668181

Offset: 0

Christian G. Bower, Jan 02 2004

nonn

Convergent of columns of A091355.

Joerg Arndt and Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (first 500 terms from Joerg Arndt)
N. J. A. Sloane, Transforms

Cf. A000219, A002117, A074962.

CoefficientList[Series[1/(1-x)*Product[1/(1-x^k)^k,{k,1,50}],{x,0,50}],x] (* Vaclav Kotesovec, Aug 16 2015 *)

N=66; x='x+O('x^N); Vec( 1/((1-x)*prod(n=1,N, (1-x^n)^n )) ) \\ Joerg Arndt, Mar 15 2014

Euler transform of 2, 2, 3, 4, 5, 6, 7, 8, 9, ...
G.f.: 1/( (1-x) * prod(n>=1, (1-x^n)^n ) ). [Joerg Arndt, Mar 15 2014]
From Vaclav Kotesovec, Aug 16 2015: (Start)
a(n) = Sum_{k=0..n} A000219(k).
a(n) ~ (n/(2*Zeta(3)))^(1/3) * A000219(n).
a(n) ~ exp(1/12 + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A * 2^(23/36) * sqrt(3*Pi) * Zeta(3)^(5/36) * n^(13/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.
(End)
G.f.: exp(Sum_{k>=1} (sigma_2(k) + 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018

A091356 Number of planar partitions of n with exactly 2 rows.

Original entry on oeis.org

1, 2, 5, 9, 18, 30, 53, 85, 139, 215, 336, 504, 760, 1115, 1635, 2351, 3375, 4770, 6725, 9368, 13006, 17885, 24510, 33319, 45139, 60743, 81457, 108610, 144334, 190844, 251542, 330082, 431825, 562710, 731154, 946644, 1222305, 1573155, 2019471

Offset: 2

Christian G. Bower, Jan 02 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Column 2 of A091355.

Cf. A000219, A091357, A091358, A091359, A091360.

b:= proc(n, k) option remember; `if`(n=0, 1, add(add(min(d, k)
     *d, d=numtheory[divisors](j))*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n, 2)-b(n, 1):
seq(a(n), n=2..50);  # Alois P. Heinz, Oct 02 2018

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[Sum[Min[d, k] d, {d, Divisors[j]}] b[n - j, k], {j, 1, n}]/n];
a[n_] :=  b[n, 2] - b[n, 1];
a /@ Range[2, 50] (* Jean-François Alcover, Oct 28 2020, after Alois P. Heinz *)

a(n) = A000990(n) - A000041(n).

A091357 Number of planar partitions of n with exactly 3 rows.

Original entry on oeis.org

1, 2, 5, 11, 22, 42, 78, 138, 239, 405, 669, 1088, 1741, 2744, 4267, 6564, 9975, 15019, 22394, 33111, 48549, 70678, 102127, 146636, 209186, 296697, 418401, 586985, 819218, 1137962, 1573336, 2165888, 2968914, 4053563, 5512820, 7469989

Offset: 3

Christian G. Bower, Jan 02 2004

nonn

Column 3 of A091355.

Cf. A000219, A089924, A091356-A091360.

a(n) = A000991(n)-A000990(n).

A091358 Number of planar partitions of n with exactly 4 rows.

Original entry on oeis.org

1, 2, 5, 11, 24, 46, 90, 164, 298, 520, 899, 1512, 2521, 4116, 6659, 10609, 16753, 26126, 40419, 61889, 94067, 141736, 212123, 315087, 465162, 682188, 994857, 1442340, 2080332, 2984724, 4262018, 6056849, 8569913, 12072770, 16938556

Offset: 4

Christian G. Bower, Jan 02 2004

nonn

Column 4 of A091355. Cf. A000219, A091356-A091360.

a(n) = A002799(n)-A000991(n).

A091359 Number of planar partitions of n with exactly 5 rows.

Original entry on oeis.org

1, 2, 5, 11, 24, 48, 94, 176, 324, 580, 1020, 1757, 2985, 4990, 8237, 13428, 21651, 34540, 54583, 85473, 132730, 204484, 312695, 474814, 716217, 1073558, 1599568, 2369781, 3491812, 5118490, 7465789, 10837964, 15661666, 22533586, 32284480

Offset: 5

Christian G. Bower, Jan 02 2004

nonn

Column 5 of A091355. Cf. A000219, A091356-A091360.

a(n) = A001452(n)-A002799(n).

cp's OEIS Frontend

A091360 Partial sums of A000219.

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A091356 Number of planar partitions of n with exactly 2 rows.

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A091357 Number of planar partitions of n with exactly 3 rows.

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A091358 Number of planar partitions of n with exactly 4 rows.

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A091359 Number of planar partitions of n with exactly 5 rows.

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