A014160 Apply partial sum operator thrice to partition numbers.

1, 4, 11, 25, 51, 96, 171, 291, 478, 762, 1185, 1803, 2693, 3956, 5727, 8182, 11552, 16134, 22313, 30579, 41559, 56045, 75039, 99796, 131891, 173282, 226405, 294270, 380595, 489945, 627924, 801374, 1018644

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

A014160 convolved with A010815 = A000217, the triangular numbers. - Gary W. Adamson, Nov 09 2008

Unordered partitions of n into parts where the part 1 comes in 4 colors. - Peter Bala, Dec 23 2013

From Omar E. Pol, Mar 01 2023: (Start)

Partial sums of A014153.

Convolution of A000070 and A000027.

Convolution of A000041 and the positive terms of A000217.

Convolution of A002865 and the positive terms of A000292. (End)

Links

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

Crossrefs

Cf. A000041, A000070, A014153.

Cf. A010815, A000217. - Gary W. Adamson, Nov 09 2008

Column k=4 of A292508.

Cf. A000027, A002865, A000292.

Programs

Mathematica

nmax = 50; CoefficientList[Series[1/((1-x)^3 * Product[1-x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 30 2015 *)

Formula

From Peter Bala, Dec 23 2013: (Start)

O.g.f.: 1/(1 - x)^3 * Product_{k >= 1} 1/(1 - x^k).

a(n-1) + a(n-2) = Sum_{parts k in all partitions of n} J_2(k), where J_2(n) is the Jordan totient function A007434(n). (End)

a(n) ~ 3*sqrt(n) * exp(Pi*sqrt(2*n/3)) / (sqrt(2)*Pi^3). - Vaclav Kotesovec, Oct 30 2015

a(n) = Sum_{k=0..n} A014153(k). - Sean A. Irvine, Oct 14 2018