A256010 Product of n and the total number of parts in all partitions of n. Also, product of n and the sum of largest parts of all partitions of n.

0, 1, 6, 18, 48, 100, 210, 378, 688, 1152, 1920, 3025, 4788, 7228, 10920, 16020, 23408, 33405, 47592, 66462, 92600, 127092, 173778, 234738, 316176, 421275, 559572, 736938, 967260, 1260137, 1636890, 2112185, 2717664, 3477078, 4435708, 5630660, 7128504, 8984044, 11293638, 14140893, 17661840, 21980264, 27291222

Offset

0,3

Comments

a(n) is also the volume of a three-dimensional model of partitions which is a polycube puzzle that contains n sections and A000041(n) pieces related to the A000041(n) regions of the set of partitions of n. The volume is equivalent to a(n) unit cubes.

Links

Table of n, a(n) for n=0..42.

Formula

a(n) = n * A006128(n).

Example

For n = 6 the total number of parts in all partitions of 6 is equal to 35 so a(n) = 6 * 35 = 210. On the other hand, the sum of largest parts of all partitions of 6 is 1 + 2 + 3 + 2 + 4 + 3 + 5 + 2 + 4 + 3 + 6 = 35, so a(6) is also 6 * 35 = 210.
Illustration of three views of a three-dimensional model of partitions after 6th stage:
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.   z  _ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _  x
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For n = 6 the areas of the views are A006128(6) = 35, A066186(6) = 6 * 11 = 66 and A000290(6) = 6^2 = 36. The structure contains A000041(6) = 11 pieces and the volume is equal to a(6) = 6 * 35 = 210.

Mathematica

lim = 42; CoefficientList[Series[Sum[n x^n Product[1/(1 - x^k), {k, n}], {n, lim}], {x, 0, lim}], x] Range[0, lim] (* Michael De Vlieger, Jul 14 2015, after N. J. A. Sloane at A006128 *)

Crossrefs

Cf. A000041, A000290, A006128, A066186, A135010, A141285, A194446, A206437.

Sequence in context: A188379 A299268 A248462 * A128543 A003202 A003198

Adjacent sequences: A256007 A256008 A256009 * A256011 A256012 A256013

Keyword

nonn

Author

Omar E. Pol, May 31 2015

Status

approved