A277643 -id:A277643 - cp's OEIS frontend

A211971 Column 0 of square array A211970 (in which column 1 is A000041).

1, 1, 2, 4, 6, 10, 16, 24, 36, 54, 78, 112, 160, 224, 312, 432, 590, 802, 1084, 1452, 1936, 2568, 3384, 4440, 5800, 7538, 9758, 12584, 16160, 20680, 26376, 33520, 42468, 53644, 67552, 84832, 106246, 132706, 165344, 205512, 254824, 315256, 389168, 479368

Offset: 0

Omar E. Pol, Jun 10 2012

nonn

Partial sums give A015128. - Omar E. Pol, Jan 09 2014

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000041, A006950, A008794, A036820, A057077, A195152, A195848, A195849, A195850, A195851, A195852, A196933, A195825, A210964, A277643.

Flatten[{1, Differences[Table[Sum[PartitionsP[n-k]*PartitionsQ[k], {k, 0, n}], {n, 0, 60}]]}] (* Vaclav Kotesovec, Oct 25 2016 *)
CoefficientList[Series[(1 - x)/EllipticTheta[4, 0, x], {x, 0, 43}], x] (* Robert G. Wilson v, Mar 06 2018 *)

a(n) ~ exp(Pi*sqrt(n))*Pi / (16*n^(3/2)) * (1 - (3/Pi + Pi/4)/sqrt(n) + (3/2 + 3/Pi^2+ Pi^2/24)/n). - Vaclav Kotesovec, Oct 25 2016, extended Nov 04 2016

G.f.: (1 - x)/theta_4(x), where theta_4() is the Jacobi theta function. - Ilya Gutkovskiy, Mar 05 2018

A291552 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^k)^p(k), where p(k) is the number of partitions of k (A000041).

Original entry on oeis.org

1, 2, 5, 11, 25, 52, 110, 221, 444, 868, 1685, 3212, 6082, 11361, 21071, 38693, 70570, 127670, 229557, 409963, 728069, 1285522, 2258318, 3947115, 6867238, 11893648, 20513199, 35235429, 60292928, 102787903, 174620017, 295644893, 498931699, 839367287, 1407864040, 2354559426, 3926878130

Offset: 0

Ilya Gutkovskiy, Aug 30 2017

nonn

Partial sums of A001970.

			Equivalently (Cayley), a(n) = total number of 2-dimensional partitions of all nonnegative integers <= n.
a(3) = 11 because we have:
0...1...2.11.1...3.21.2.111.11.1
.............1........1.....1..1
...............................1
and 1 + 1 + 3 + 6 = 11.

Index entries for related partition-counting sequences

Cf. A000041, A000070, A001970, A026905, A036469, A091360, A277643.

with(numtheory): with(combinat):
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      numbpart(d), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; b(n)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 11 2017

nmax = 36; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^k)^PartitionsP[k], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: (1/(1 - x))*Product_{k>=1} 1/(1 - x^k)^p(k), where p(k) = [x^k] Product_{k>=1} 1/(1 - x^k).

A309266 Expansion of (1 + x) * Product_{k>=1} (1 + x^k)/(1 - x^k).

Original entry on oeis.org

1, 3, 6, 12, 22, 38, 64, 104, 164, 254, 386, 576, 848, 1232, 1768, 2512, 3534, 4926, 6812, 9348, 12736, 17240, 23192, 31016, 41256, 54594, 71890, 94232, 122976, 159816, 206872, 266768, 342756, 438868, 560064, 712448, 903526, 1142478, 1440528, 1811384, 2271720, 2841800, 3546224

Offset: 0

Ilya Gutkovskiy, Jul 20 2019

nonn

Cf. A001650, A015128, A052816, A084376, A207641, A211971, A277643.

nmax = 42; CoefficientList[Series[(1 + x) Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = Sum[PartitionsP[k] PartitionsQ[n - k], {k, 0, n}]; Table[a[n] + a[n - 1], {n, 0, 42}]

G.f.: (1 + x)/theta_4(x), where theta_4() is the Jacobi theta function.
a(n) = A015128(n) + A015128(n-1).
a(n) ~ exp(Pi*sqrt(n)) / (4*n) * (1 - (Pi/4 + 1/Pi)/sqrt(n)). - Vaclav Kotesovec, Jul 20 2019

cp's OEIS Frontend

A211971 Column 0 of square array A211970 (in which column 1 is A000041).

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A291552 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^k)^p(k), where p(k) is the number of partitions of k (A000041).

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A309266 Expansion of (1 + x) * Product_{k>=1} (1 + x^k)/(1 - x^k).

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