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A211978
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Total number of parts in all partitions of n plus the sum of largest parts of all partitions of n.
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22
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0, 2, 6, 12, 24, 40, 70, 108, 172, 256, 384, 550, 798, 1112, 1560, 2136, 2926, 3930, 5288, 6996, 9260, 12104, 15798, 20412, 26348, 33702, 43044, 54588, 69090, 86906, 109126, 136270, 169854, 210732, 260924, 321752, 396028, 485624, 594402, 725174, 883092, 1072208
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OFFSET
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0,2
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COMMENTS
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Also twice A006128, because the total number of parts in all partitions of n equals the sum of largest parts of all partitions of n. For a proof without words see the illustration of initial terms. Note that the sum of the lengths of all horizontal segments equals the sum of largest parts of all partitions of n. On the other hand, the sum of the lengths of all vertical segments equals the total number of parts of all partition of n. Therefore the sum of lengths of all horizontal segments equals the sum of lengths of all vertical segments.
a(n) is also the sum of the semiperimeters of the Ferrers boards of the partitions of n. Example: a(2)=6; indeed, the Ferrers boards of the partitions [2] and [1,1] of 2 are 2x1 rectangles; the sum of their semiperimeters is 3 + 3 = 6. - Emeric Deutsch, Oct 07 2016
a(n) is also the sum of the semiperimeters of the regions of the set of partitions of n. See the first illustration in the Example section. For more information see A278355. - Omar E. Pol, Nov 23 2016
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LINKS
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FORMULA
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The trivariate g.f. G(t,s,x) of the partitions of a nonnegative integer relative to weight (marked by x), number of parts (marked by t), and largest part (marked by s) is G(t,s,x) = Sum_{i>=1} t*s^i*x^i/product_{j=1..i} (1-tx^j). Setting s = t, we obtain the bivariate g.f. of the partitions relative to weight (marked by x) and semiperimeter of the Ferrers board (marked by t). The g.f. of a(n) is g(x) = Sum_{i>=1} ((x^i*(1 + i + Q(x))/R(x)), where Q(x) = sum_{j=1..i} (x^j/(1 - x^j)) and R(x) = product_{j=1..i}(1-x^j). g(x) has been obtained by setting t = 1 in dG(t,t,x))/dt. - Emeric Deutsch, Oct 07 2016
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EXAMPLE
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Illustration of initial terms as a minimalist diagram of regions of the set of partitions of n, for n = 1..6:
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. _ _ _ _ _ _ _|_ | _ _ _|_ | |
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. _ _ _ _ _|_ | _ _|_ | | _ _|_ | | |
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. 2 6 12 24 40 70
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Also using the elements from the diagram we can draw an infinite Dyck path in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. Note that the n-th largest peak between two valleys at height 0 is also the partition number A000041(n) as shown below:
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11...........................................................
. /\
. / \
. / \
7.................................. / \
. /\ / \
5.................... / \ /\/ \
. /\ / \ /\ / \
3.......... / \ / \ / \/ \
2..... /\ / \ /\/ \ / \
1.. /\ / \ /\/ \ / \ /\/ \
0 /\/ \/ \/ \/ \/ \
. 0,2, 6, 12, 24, 40, 70...
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MAPLE
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Q := sum(x^j/(1-x^j), j = 1 .. i): R := product(1-x^j, j = 1 .. i): g := sum(x^i*(1+i+Q)/R, i = 1 .. 100): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 0 .. 41); # Emeric Deutsch, Oct 07 2016
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MATHEMATICA
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Array[2 Sum[DivisorSigma[0, m] PartitionsP[# - m], {m, #}] &, 42, 0] (* Michael De Vlieger, Mar 20 2020 *)
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CROSSREFS
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Cf. A006128, A135010, A141285, A186114, A193870, A187219, A194446, A194447, A206437, A211026, A220517, A225600, A278355.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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