A036469 Partial sums of A000009 (partitions into distinct parts).

1, 2, 3, 5, 7, 10, 14, 19, 25, 33, 43, 55, 70, 88, 110, 137, 169, 207, 253, 307, 371, 447, 536, 640, 762, 904, 1069, 1261, 1483, 1739, 2035, 2375, 2765, 3213, 3725, 4310, 4978, 5738, 6602, 7584, 8697, 9957, 11383, 12993, 14809, 16857, 19161, 21751, 24661

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Also number of 1's in all partitions of n+1 into odd parts. Example: a(4)=7 because the partitions of 5 into odd parts are [5], [3,1,1], [1,1,1,1,1], having a total number of 7 1's. - Emeric Deutsch, Mar 29 2006

Convolved with A035363 = A000070. - Gary W. Adamson, Jun 09 2009

Equals row sums of triangle A166240. - Gary W. Adamson, Oct 09 2009

a(n) = if n <= 1 then A201377(1,n) else A201377(n,1). - Reinhard Zumkeller, Dec 02 2011

a(n) equals the sum of the parts of the form 2^k (k >= 0) in all partitions of n + 1 into distinct parts. Example: a(6) = 14. The partitions of 7 into distinct parts are [7], [6,1], [5,2], [4,3] and [4,2,1] having sum over parts of the form 2^k equal to 1 + 2 + 4 + 4 + 2 + 1 = 14. - Peter Bala, Dec 01 2013

Number of partitions of the (n+1)-multiset {0,...,0,1} with n 0's into distinct multisets; a(3) = 5: 0|00|1, 00|01, 000|1, 0|001, 0001. Also number of factorizations of 3*2^n into distinct factors; a(3) = 5: 2*3*4, 4*6, 3*8, 2*12, 24. - Alois P. Heinz, Jul 30 2021

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
• A. V. Chekhonadskikh, Some classical number sequences in control system design, Siberian Electronic Mathematical Reports, Volume 14, p. 620-628. See Theorem 2.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 774

Crossrefs

Cf. A000009, A265093.

Cf. A035363, A000070. - Gary W. Adamson, Jun 09 2009

Cf. A166240. - Gary W. Adamson, Oct 09 2009

Column k=1 of A346520.

Programs

Maple

g:=1/(1-x)/product(1-x^(2*j-1),j=1..30): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=0..46); # Emeric Deutsch, Mar 29 2006
# second Maple program:
b:= proc(n, i) b(n, i):= `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1)+`if`(i>n, 0, b(n-i, min(n-i, i-1)))))
    end:
a:= proc(n) option remember; b(n, n) +`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 21 2012

Mathematica

CoefficientList[ Series[Product[(1 + t^i), {i, 1, Infinity}]/(1 - t), {t, 0, 46}], t] (* Geoffrey Critzer, May 16 2010 *)
b[n_, i_] := If[n == 0, 1, If[i<1, 0, b[n, i-1]+If[i>n, 0, b[n-i, Min[n-i, i-1]]]]]; a[n_] := a[n] = b[n, n]+If[n>0, a[n-1], 0]; Table[a[n], {n, 0, 60}] // Flatten (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)
Accumulate[Table[PartitionsQ[n], {n, 0, 50}]] (* Vaclav Kotesovec, Oct 26 2016 *)

Formula

G.f.: 1/[(1-x)*product(1-x^(2j-1), j=1..infinity)]. - Emeric Deutsch, Mar 29 2006

a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (2*Pi*n^(1/4)) * (1 + (18+13*Pi^2) / (48*Pi*sqrt(3*n)) + (2916 - 1404*Pi^2 + 121*Pi^4)/(13824*Pi^2*n)). - Vaclav Kotesovec, Feb 26 2015, updated Oct 26 2016

For n > 0, a(n) = A026906(n) + 1. - Vaclav Kotesovec, Oct 26 2016

Faster converging g.f.: A(x) = (1/(1 - x))*Sum_{n >= 0} x^(n*(2*n-1))/Product_{k = 1..2*n} (1 - x^k). - Peter Bala, Feb 02 2021