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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
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 (list; graph; refs; listen; history; text; internal format)
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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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

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Last modified April 7 09:31 EDT 2020. Contains 333300 sequences. (Running on oeis4.)