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A116365 Sum of the sizes of the tails below the Durfee squares of all partitions of n. 5
0, 1, 3, 6, 11, 20, 33, 56, 86, 136, 200, 301, 429, 621, 868, 1219, 1669, 2297, 3091, 4171, 5542, 7357, 9648, 12652, 16402, 21250, 27298, 35003, 44556, 56637, 71515, 90160, 113046, 141464, 176189, 219053, 271149, 335044, 412447, 506787, 620597 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

a(n) = Sum(k*A114087(n,k), k=0..n-1).

REFERENCES

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).

G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

FORMULA

G.f.: [(d/dt){sum(q^(k^2)/product((1-q^j)(1-(tq)^j), j=1..k), k=1..infty)}]_{t=1}.

a(n) = (n*A000041(n)-A116503(n))/2. - Vladeta Jovovic, Feb 18 2006

a(n) ~ (1/(8*sqrt(3)) - sqrt(3) * (log(2))^2 / (4*Pi^2)) * exp(Pi*sqrt(2*n/3)). - Vaclav Kotesovec, Jan 03 2019

EXAMPLE

a(4) = 6 because the bottom tails of the five partitions of 4, namely [4], [3,1], [2,2], [2,1,1] and [1,1,1,1], are { }, [1], { }, [1,1] and [1,1,1], respectively, having total size 0+1+0+2+3=6.

MAPLE

g:=sum(z^(k^2)/product((1-z^j)*(1-(t*z)^j), j=1..k), k=1..10): dgdt1:=simplify(subs(t=1, diff(g, t))): dgdt1ser:=series(dgdt1, z=0, 55): seq(coeff(dgdt1ser, z, n), n=1..48);

# 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*add(b(k, d) *b(n-d^2-k, d),

            d=0..floor(sqrt(n))), k=0..n-1):

seq(a(n), n=1..40);  # Alois P. Heinz, Apr 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*Sum[b[k, d]*b[n-d^2-k, d], {d, 0, Floor[Sqrt[n]]}], {k, 0, n-1}]; Table[a[n], {n, 1, 40}] (* Jean-Fran├žois Alcover, Mar 31 2015, after Alois P. Heinz *) *)

CROSSREFS

Cf. A115994, A115995, A114087, A114088, A114089.

Sequence in context: A252479 A320850 A180086 * A297443 A185083 A208851

Adjacent sequences:  A116362 A116363 A116364 * A116366 A116367 A116368

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Feb 12 2006

STATUS

approved

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Last modified July 8 19:20 EDT 2020. Contains 335524 sequences. (Running on oeis4.)