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Apply partial sum operator thrice to partition numbers.
6

%I #34 Mar 20 2023 17:37:47

%S 1,4,11,25,51,96,171,291,478,762,1185,1803,2693,3956,5727,8182,11552,

%T 16134,22313,30579,41559,56045,75039,99796,131891,173282,226405,

%U 294270,380595,489945,627924,801374,1018644

%N Apply partial sum operator thrice to partition numbers.

%C A014160 convolved with A010815 = A000217, the triangular numbers. - _Gary W. Adamson_, Nov 09 2008

%C Unordered partitions of n into parts where the part 1 comes in 4 colors. - _Peter Bala_, Dec 23 2013

%C From _Omar E. Pol_, Mar 01 2023: (Start)

%C Partial sums of A014153.

%C Convolution of A000070 and A000027.

%C Convolution of A000041 and the positive terms of A000217.

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

%H Vaclav Kotesovec, <a href="/A014160/b014160.txt">Table of n, a(n) for n = 0..10000</a>

%F From _Peter Bala_, Dec 23 2013: (Start)

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

%F 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)

%F a(n) ~ 3*sqrt(n) * exp(Pi*sqrt(2*n/3)) / (sqrt(2)*Pi^3). - _Vaclav Kotesovec_, Oct 30 2015

%F a(n) = Sum_{k=0..n} A014153(k). - _Sean A. Irvine_, Oct 14 2018

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

%Y Cf. A000041, A000070, A014153.

%Y Cf. A010815, A000217. - _Gary W. Adamson_, Nov 09 2008

%Y Column k=4 of A292508.

%Y Cf. A000027, A002865, A000292.

%K nonn

%O 0,2

%A _N. J. A. Sloane_