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A036469 Partial sums of A000009 (partitions into distinct parts). 39
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

The subsequence of prime partial sums of A000009 (partitions into distinct parts) begins: 2, 3, 5, 7, 19, 43, 137, 307, 1069, 1483, 11383. - Jonathan Vos Post, Feb 11 2010

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

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

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

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 *)

CROSSREFS

Cf. A000009, A265093.

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

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

Sequence in context: A175842 A008581 A172491 * A238658 A116480 A023026

Adjacent sequences:  A036466 A036467 A036468 * A036470 A036471 A036472

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified June 1 19:32 EDT 2020. Contains 334762 sequences. (Running on oeis4.)