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A036469
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Partial sums of A000009 (partitions into distinct parts).
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45
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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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
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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