A143229 a(n) = A000041(n) * A000070(n).

1, 2, 8, 21, 60, 133, 330, 675, 1474, 2910, 5838, 10920, 20944, 37673, 68580, 120384, 211365, 359964, 614845, 1022630, 1701678, 2776752, 4517016, 7232565, 11557350, 18201568, 28579152, 44373420, 68634280, 105109125, 160436916, 242692582, 365853180, 547346709

Offset: 0

Author

Gary W. Adamson, Jul 31 2008

Keywords

• nonn

Comments

Row sums of triangle A143228.

Examples

			a(4) = 60 = A000041(4) * A000070(4) = 5 * 12.
a(4) = 60 = sum of row 4 terms of row 4 in triangle A143228: (5 + 5 + 10 + 15 + 25).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000041, A000070, A143228.

Programs

Magma

A143229:= func< n | NumberOfPartitions(n)*(&+[NumberOfPartitions(k): k in [0..n]]) >;
[A143229(n): n in [0..50]]; // G. C. Greubel, Aug 27 2024

Mathematica

A143229[n_]:= PartitionsP[n]*Sum[PartitionsP[k], {k, 0, n}];
Table[A143229[n], {n,0,50}] (* G. C. Greubel, Aug 27 2024 *)

SageMath

def p(n): return number_of_partitions(n) # A000041
def A143229(n): return p(n)*sum(p(k) for k in range(n+1))
[A143229(n) for n in range(51)] # G. C. Greubel, Aug 27 2024

Formula

a(n) = A000041(n) * A000070(n).

a(n) ~ exp(2*sqrt(2*n/3)*Pi)/(8*sqrt(6)*Pi*n^(3/2)) * (1 + (5*Pi/(12*sqrt(6)) - sqrt(3/2)/Pi)/sqrt(n) + (13*Pi^2/1728 - 19/48)/n). - Vaclav Kotesovec, Nov 04 2016

Extensions

More terms from Vaclav Kotesovec, Nov 04 2016