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A116503 Sum of the areas of the Durfee squares of all partitions of n. 3
1, 2, 3, 8, 13, 26, 39, 64, 98, 148, 216, 322, 455, 648, 904, 1258, 1711, 2336, 3128, 4198, 5548, 7330, 9569, 12496, 16146, 20836, 26674, 34098, 43273, 54846, 69072, 86848, 108627, 135612, 168527, 209066, 258271, 318482, 391321, 479946, 586709 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) = sum(k^2*A115994(n,k), k=1..floor(sqrt(n))).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

FORMULA

G.f.: sum(k^2*z^(k^2)/product((1-z^j)^2, j=1..k), k=1..infinity).

EXAMPLE

a(4) = 8 because the partitions of 4, namely [4], [3,1], [2,2], [2,1,1] and [1,1,1,1], have Durfee squares of sizes 1,1,2,1 and 1, respectively and 1^2+1^2+2^2+1^2+1^2=8.

MAPLE

g:=sum(k^2*z^(k^2)/product((1-z^j)^2, j=1..k), k=1..10): gser:=series(g, z=0, 52): seq(coeff(gser, z^n), n=1..45);

# second Maple program:

b:= proc(n, i) option remember;

      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))

    end:

a:= n-> add(k^2*add(b(m, k)*b(n-k^2-m, k),

            m=0..n-k^2), k=1..floor(sqrt(n))):

seq(a(n), n=1..40);  # Alois P. Heinz, Apr 09 2012

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := Sum [k^2*Sum[b[m, k]*b[n - k^2 - m, k], {m, 0, n - k^2}], {k, 1, Sqrt[n]}]; Table[a[n], {n, 1, 50}] (* Jean-Fran├žois Alcover, Jan 24 2014, after Alois P. Heinz *)

CROSSREFS

Cf. A115994, A115995.

Sequence in context: A213046 A262021 A221181 * A105204 A045692 A103196

Adjacent sequences:  A116500 A116501 A116502 * A116504 A116505 A116506

KEYWORD

easy,nonn

AUTHOR

Emeric Deutsch, Vladeta Jovovic, Feb 18 2006

STATUS

approved

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Last modified December 4 11:13 EST 2016. Contains 278750 sequences.