A276432 Sum of the traces of all plane partitions of n.

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

Offset

1,2

Comments

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

Convolution of A340793 and A091360. - Omar E. Pol, Feb 16 2021

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.

Cf. A091360, A340793.

Sequence in context: A322060 A022812 A192306 * A308817 A000293 A000294

Adjacent sequences: A276429 A276430 A276431 * A276433 A276434 A276435

Keyword

nonn

Author

Emeric Deutsch, Sep 24 2016

Status

approved