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%I #23 Aug 04 2018 06:39:39
%S 1,2,4,6,8,12,16,20,26,33,40,50,61,72,87,104,121,143,167,192,224,259,
%T 295,339,387,437,497,563,631,712,801,893,1000,1117,1238,1379,1532,
%U 1691,1872,2069,2274,2507,2759,3021,3316,3636,3968,4340,4741,5158,5623,6124,6644,7219,7838,8483,9193,9956
%N Number of multiset repetition class defining partitions of N with 1<=N<=n.
%C 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.
%C See A176723 for the characteristic array for these partitions in Abramowitz-Stegun order.
%C 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.
%C 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, ...].
%C a(n) is the total number of such nonincreasing exponent sequences for N from 1 to n.
%C a(n) is also the total number of partitions of N, with 1<=N<=n, into nonzero triangular numbers A000217. See A007294.
%C a(n) is the partial sum of A007294 without the leading 1.
%H Robert Israel, <a href="/A185976/b185976.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = Sum_{k=1..n}A007294(k), n>=1.
%p b:= proc(n, i) option remember;
%p if n<0 then 0
%p elif n=0 then 1
%p elif i=0 then 0
%p else b(n, i-1) +b(n-i*(i+1)/2, i)
%p fi
%p end:
%p a007294:= n-> b(n, floor(sqrt(2*n))): # Alois P. Heinz code for A007294
%p A007294:= [seq(a007294(n),n=1..100)]:
%p ListTools:-PartialSums(A007294); # _Robert Israel_, Apr 15 2016
%t 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_ *)
%Y Cf. A000217, A007294, A176723, A185977.
%K nonn,easy
%O 1,2
%A _Wolfdieter Lang_, Mar 07 2011
%E Changed by the author in response to comments by _Franklin T. Adams-Watters_, Apr 02 2011
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