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A276432 Sum of the traces of all plane partitions of n. 2
1, 4, 10, 26, 56, 126, 252, 512, 980, 1866, 3427, 6258, 11121, 19618, 33975, 58328, 98732, 165804, 275246, 453544, 740338, 1200088, 1929897, 3083898, 4893775, 7720826, 12106814, 18883104, 29291740, 45215386, 69451631, 106197524, 161656759, 245050410, 369935066 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Convolution of A000203 and A000219. - Vaclav Kotesovec, Sep 25 2016

REFERENCES

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, pp. 179-201.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

FORMULA

G.f.: g(x) = Sum_{j>=1} (j*x^j/(1-x^j))/Product_{k>=1} (1-x^k)^k.

a(n) = Sum(k*A089353(n,k), k>=1).

EXAMPLE

a(3) = 10 because the 6 (=A000219(3)) planar partitions of 3 are [3], [2,1], [2;1], [1,1,1], [1;1;1], [1,1;1] (; indicates a new row); the sum of their traces is 3+2+2+1+1+1 = 10.

MAPLE

g:= (sum(j*x^j/(1-x^j), j = 1..100))/(product((1-x^k)^k, k = 1..100)): gser := series(g, x = 0, 40): seq(coeff(gser, x, m), m = 1 .. 35);

# second Maple program:

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0, add((p

      ->p+[0, j*p[1]])(b(n-i*j, i-1))*binomial(i+j-1, j), j=0..n/i)))

    end:

a:= n-> b(n$2)[2]:

seq(a(n), n=1..50);  # Alois P. Heinz, Sep 24 2018

MATHEMATICA

nmax = 50; Rest[CoefficientList[Series[Sum[j*x^j/(1-x^j), {j, 1, nmax}]*Product[1/(1-x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 25 2016 *)

CROSSREFS

Cf. A000219, A089353, A277029.

Sequence in context: A322060 A022812 A192306 * A000293 A000294 A133086

Adjacent sequences:  A276429 A276430 A276431 * A276433 A276434 A276435

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Sep 24 2016

STATUS

approved

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Last modified June 16 11:12 EDT 2019. Contains 324152 sequences. (Running on oeis4.)