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

Author

Omar E. Pol, May 31 2015

Keywords

• nonn

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.

Examples

			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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.                _|_| | |_ _ _      |
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.              _|_|_| | |_ _    |   |
.             |_|_|_| | |_ _|_ _|_  |
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.            _|_|_|_| | |_ _    | | |
.           | | | |_| | |_ _|_  | | |
.          _|_|_|_|_| | |_ _  | | | |
.        _|_|_|_|_|_| | |_  | | | | |
.       |_|_|_|_|_|_| | |_|_|_|_|_|_|
.   z  _ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _  x
.                     |  _ _ _ _ _ _
.                     | |_| | | | | |
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.                     z
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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.
		

Crossrefs

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

Programs

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 *)

Formula

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