A185976 Number of multiset repetition class defining partitions of N with 1<=N<=n.

1, 2, 4, 6, 8, 12, 16, 20, 26, 33, 40, 50, 61, 72, 87, 104, 121, 143, 167, 192, 224, 259, 295, 339, 387, 437, 497, 563, 631, 712, 801, 893, 1000, 1117, 1238, 1379, 1532, 1691, 1872, 2069, 2274, 2507, 2759, 3021, 3316, 3636, 3968, 4340, 4741, 5158, 5623, 6124, 6644, 7219, 7838, 8483, 9193, 9956

Offset

1,2

Comments

A partition of N>=1, given by its positive exponents e[1], e[2], ..., e[M], with largest part M and sum(j*e[j],j=1..M)=N, defines an m-multiset repetition class if it has m:=Sum_{j=1..M}e[j] parts and the exponents are nonincreasing: e[1 >= e[2] >= ... >= e[M] >= 1.

See A176723 for the characteristic array for these partitions in Abramowitz-Stegun order.

The largest part M can be 1,2,...,Mmax(N), where Mmax(N)is the index of the largest triangular number smaller or equal to N. E.g., Mmax(7)= 3.

The minimal number of parts of these partitions of N is given by A185977 = [1, 2, 2, 3, 4, 3, 4, 5, 5, 4, 5, 6, 6, 7, 5, 6, 7, 7, 8, 8, 6, 7, 8, 8, 9, ...].

a(n) is the total number of such nonincreasing exponent sequences for N from 1 to n.

a(n) is also the total number of partitions of N, with 1<=N<=n, into nonzero triangular numbers A000217. See A007294.

a(n) is the partial sum of A007294 without the leading 1.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{k=1..n}A007294(k), n>=1.

Maple

b:= proc(n, i) option remember;
    if n<0 then 0
    elif n=0 then 1
    elif i=0 then 0
    else b(n, i-1) +b(n-i*(i+1)/2, i)
    fi
    end:
a007294:= n-> b(n, floor(sqrt(2*n))): # Alois P. Heinz code for A007294
A007294:= [seq(a007294(n), n=1..100)]:
ListTools:-PartialSums(A007294); # Robert Israel, Apr 15 2016

Mathematica

b[n_, i_] := b[n, i] = Which[n<0, 0, n==0, 1, i==0, 0, True, b[n, i-1] + b[n-i*(i+1)/2, i]]; Accumulate[Table[b[n, Floor[Sqrt[2n]]], {n, 1, 60}]] (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)

Crossrefs

Cf. A000217, A007294, A176723, A185977.

Sequence in context: A346729 A346311 A097921 * A117146 A061553 A138934

Adjacent sequences: A185973 A185974 A185975 * A185977 A185978 A185979

Keyword

nonn,easy

Author

Wolfdieter Lang, Mar 07 2011

Extensions

Changed by the author in response to comments by Franklin T. Adams-Watters, Apr 02 2011

Status

approved