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A047993 Number of balanced partitions of n: the largest part equals the number of parts. 24
1, 0, 1, 1, 1, 1, 3, 2, 4, 4, 6, 7, 11, 11, 16, 19, 25, 29, 40, 45, 60, 70, 89, 105, 134, 156, 196, 232, 285, 336, 414, 485, 591, 696, 839, 987, 1187, 1389, 1661, 1946, 2311, 2702, 3201, 3731, 4400, 5126, 6018, 6997, 8195, 9502, 11093, 12849, 14949, 17281, 20062 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

Useful in the creation of plane partitions with C3 or C3v symmetry.

The function T[m,a,b] used here gives the partitions of m whose Ferrers plot fits within an a X b box.

Central terms of triangle in A063995: a(n) = A063995(n,0). - Reinhard Zumkeller, Jul 24 2013

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Erich Friedman, Illustration of initial terms

FORMULA

a(n) = p(n-1) - p(n-2) - p(n-5) + p(n-7) + ... + (-1)^k*(p(n-(3*k^2-k)/2) - p(n-(3*k^2+k)/2)) + ..., where p() is A000041(). E.g., A047993 a(20) = p(19) - p(18) - p(15) + p(13) + p(8) - p(5) = 490 - 385 - 176 + 101 + 22 - 7 = 45. - Vladeta Jovovic, Aug 04 2004

G.f.: Sum_{k>=1} (-1)^k * ( x^((3*k^2+k)/2) - x^((3*k^2-k)/2) ) ) / Product_{k>=1} (1-x^k). - Vladeta Jovovic, Aug 05 2004

Conjecture: a(n) ~ exp(Pi*sqrt(2*n/3))*Pi / (48*sqrt(2)*n^(3/2)) ~ p(n) * Pi / (4*sqrt(6*n)), where p(n) is the partition function A000041. - Vaclav Kotesovec, Oct 06 2016

EXAMPLE

a(12) = 7 because the partitions of 12 where the largest part equals the number of parts are

2 + 3 + 3 + 4,

2 + 2 + 4 + 4,

1 + 3 + 4 + 4,

1 + 2 + 2 + 2 + 5,

1 + 1 + 2 + 3 + 5,

1 + 1 + 1 + 4 + 5, and

1 + 1 + 1 + 1 + 2 + 6.

[Joerg Arndt, Oct 08 2012]

MAPLE

with(combinat): for n from 1 to 36 do P:=partition(n): c:=0: for j from 1 to nops(P) do if P[j][nops(P[j])]=nops(P[j]) then c:=c+1 else c:=c fi od: a[n]:=c: od: seq(a[n], n=1..36); # Emeric Deutsch, Dec 11 2004

MATHEMATICA

Table[ Count[Partitions[n], par_List/; First[par]===Length[par]], {n, 12}] or recur: Sum[T[n-(2m-1), m-1, m-1], {m, Ceiling[Sqrt[n]], Floor[(n+1)/2]}] with T[m_, a_, b_]/; b < a := T[m, b, a]; T[m_, a_, b_]/; m > a*b := 0; T[m_, a_, b_]/; (2m > a*b) := T[a*b-m, a, b]; T[m_, 1, b_] := If[b < m, 0, 1]; T[0, _, _] := 1; T[m_, a_, b_] := T[m, a, b]=Sum[T[m-a*i, a-1, b-i], {i, 0, Floor[m/a]}];

Table[Sum[ -(-1)^k*(p[n-(3*k^2-k)/2] - p[n-(3*k^2+k)/2]), {k, 1, Floor[(1+Sqrt[1+24*n])/6]}] /. p -> PartitionsP, {n, 1, 64}] (* Wouter Meeussen *)

(* also *)

Table[Count[IntegerPartitions[n], q_ /; Max[q] == Length[q]], {n, 24}]

(* Clark Kimberling, Feb 13 2014 *)

PROG

(PARI)

N=66;  q='q + O('q^N );

S=2+2*ceil(sqrt(N));

gf= sum(k=1, S,  (-1)^k * ( q^((3*k^2+k)/2) - q^((3*k^2-k)/2) ) ) / prod(k=1, N, 1-q^k );

/* Joerg Arndt, Oct 08 2012 */

(Haskell)

a047993 = flip a063995 0  -- Reinhard Zumkeller, Jul 24 2013

CROSSREFS

Cf. A000700, A063995, A064173, A064174.

Sequence in context: A112199 A145815 A059851 * A033177 A175512 A240829

Adjacent sequences:  A047990 A047991 A047992 * A047994 A047995 A047996

KEYWORD

nice,nonn

AUTHOR

Wouter Meeussen

STATUS

approved

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Last modified September 19 16:51 EDT 2017. Contains 292243 sequences.