A087787 -id:A087787 - cp's OEIS frontend

A143228 Triangle read by rows, T(n,k) = p(n) * p(k), where p(n) = the number of partitions of n, for 0 <= k <= n.

1, 1, 1, 2, 2, 4, 3, 3, 6, 9, 5, 5, 10, 15, 25, 7, 7, 14, 21, 35, 49, 11, 11, 22, 33, 55, 77, 121, 15, 15, 30, 45, 75, 105, 165, 225, 22, 22, 44, 66, 110, 154, 242, 330, 484, 30, 30, 60, 90, 150, 210, 330, 450, 660, 900, 42, 42, 84, 126, 210, 294, 462, 630, 924, 1260, 1764

Offset: 0

Gary W. Adamson, Jul 31 2008

			First few rows of the triangle:
   1;
   1,  1;
   2,  2,  4;
   3,  3,  6,  9;
   5,  5, 10, 15, 25;
   7,  7, 14, 21, 35,  49;
  11, 11, 22, 33, 55,  77, 121;
  15, 15, 30, 45, 75, 105, 165, 225;
  ...
T(7,4) = 75 = p(7) * p(4) = 15 * 5.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000041, A143229 (row sums).

Main diagonal gives: A001255.

A143228:= func< n,k | NumberOfPartitions(n)*NumberOfPartitions(k) >;
[A143228(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 27 2024

Table[PartitionsP[n]*PartitionsP[k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 27 2024 *)

def A143215(n,k): return number_of_partitions(n)*number_of_partitions(k)
flatten([[A143215(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 27 2024

T(n, 0) = A000041(n) (left border).
Sum_{k=0..n} T(n, k) = A143229(n) (row sums).
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A000041(n)*A087787(n). - G. C. Greubel, Aug 27 2024

A182722 a(n) = A005291(n+1)-A182712(n).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 10, 14, 26, 36, 60, 83, 128, 175, 261, 351, 504, 674, 943, 1247, 1711, 2243

Offset: 0

Omar E. Pol, Nov 28 2010, Nov 29 2010

nonn

The difference between two apparently unrelated sequences which happen to have the same initial terms. - N. J. A. Sloane, Dec 01 2010

Cf. A005291, A087787, A100818, A135010, A138121, A182703, A182712.

a(n) = A005291(n+1)-A182712(n)

A277963 G.f.: 1/(1+x) * Product_{k>=1} 1/(1-x^k)^k.

Original entry on oeis.org

1, 0, 3, 3, 10, 14, 34, 52, 108, 174, 326, 533, 946, 1539, 2628, 4251, 7046, 11288, 18313, 29017, 46261, 72533, 113942, 176841, 274353, 421680, 647065, 985593, 1497641, 2261971, 3406992, 5105317, 7628112, 11346861, 16829094, 24861952, 36623009, 53756775

Offset: 0

Vaclav Kotesovec, Nov 06 2016

nonn

Convolution of A000219 and A033999.

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

Cf. A000219, A091360, A087787, A191659.

CoefficientList[Series[1/(1+x)*Product[1/(1-x^k)^k, {k, 1, 50}], {x, 0, 50}], x]

a(n) = Sum_{k=0..n} (-1)^(n-k)*A000219(k).

a(n) ~ Zeta(3)^(7/36) * exp(3 * Zeta(3)^(1/3) * (n/2)^(2/3) + 1/12) / (A * sqrt(3*Pi) * 2^(47/36) * n^(25/36)), where A = A074962 is the Glaisher-Kinkelin constant.

A293464 a(n) = Sum_{k=0..n} (-1)^k * 2^k * p(k), where p(k) is the partition function A000041.

Original entry on oeis.org

1, -1, 7, -17, 63, -161, 543, -1377, 4255, -11105, 31903, -82785, 232607, -594785, 1617055, -4150113, 10988703, -27939681, 72985759, -183915361, 473541791, -1187402593, 3015290015, -7512413025, 18911702175, -46787875681, 116689317023, -287306044257

Offset: 0

Vaclav Kotesovec, Oct 09 2017

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A087787, A259400.

Table[Sum[(-1)^k * 2^k * PartitionsP[k], {k, 0, n}], {n, 0, 30}]

a(n) ~ (-1)^n * 2^(n-1) * exp(Pi*sqrt(2*n/3)) / (3^(3/2)*n).

a(n) ~ (-1)^n * 2/3 * 2^n * A000041(n).

A195308 a(n) = A005291(n) + A005291(n+1).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 43, 58, 81, 109, 150, 200, 271, 359, 481, 633, 838, 1095, 1438, 1867, 2430, 3136, 4053, 5200, 6676, 8519, 10871, 13802, 17514, 22129, 27940, 35141, 44155, 55299, 69179, 86286, 107495, 133562, 165744, 205188, 253691, 312975, 385619

Offset: 1

Omar E. Pol, Feb 03 2012

nonn

This sequence arises from A005291 in the same way as A000041 arises from A182712.

Observation: a(3)..a(13) coincide with a sequence related to Stirling's numbers from the Jordan's book.

Charles Jordan, Calculus of finite differences, Chelsea Publishing Co., 1965, chapter IV, pp. 153-155.

Cf. A000041, A005291, A087787, A100818, A182712, A194439, A195017.

a(n) = A000041(n-2), 2 <= n <= 11. - Omar E. Pol, Feb 24 2013

More terms from Amiram Eldar, May 17 2025

cp's OEIS Frontend

A143228 Triangle read by rows, T(n,k) = p(n) * p(k), where p(n) = the number of partitions of n, for 0 <= k <= n.

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A182722 a(n) = A005291(n+1)-A182712(n).

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A277963 G.f.: 1/(1+x) * Product_{k>=1} 1/(1-x^k)^k.

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A293464 a(n) = Sum_{k=0..n} (-1)^k * 2^k * p(k), where p(k) is the partition function A000041.

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A195308 a(n) = A005291(n) + A005291(n+1).

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