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A014153 Expansion of 1/((1-x)^2*Product_{k>=1} (1-x^k)). 22
1, 3, 7, 14, 26, 45, 75, 120, 187, 284, 423, 618, 890, 1263, 1771, 2455, 3370, 4582, 6179, 8266, 10980, 14486, 18994, 24757, 32095, 41391, 53123, 67865, 86325, 109350, 137979, 173450, 217270, 271233, 337506, 418662, 517795, 638565, 785350, 963320, 1178628 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of partitions of n with three kinds of 1. E.g. a(2)=7 because we have 2, 1+1, 1+1', 1+1", 1'+1', 1'+1", 1"+1". - Emeric Deutsch, Mar 22 2005

Partial sums of the partial sums of the partition numbers A000041. Partial sums of A000070. Euler transform of 3,1,1,1,...

Also sum of parts, counted without multiplicity, in all partitions of n, offset 1. Also Sum phi(p), where the sum is taken over all parts p of all partitions of n, offset 1. - Vladeta Jovovic, Mar 26 2005

Equals row sums of triangle A141157. - Gary W. Adamson, Jun 12 2008

A014153 convolved with A010815 = (1, 2, 3, ...). n-th partial sum sequence of A000041 convolved with A010815 = (n-1)-th column of Pascal's triangle, starting (1, n, ...). - Gary W. Adamson, Nov 09 2008

a(n) is also the sum of all parts of the (n+1)st column of a version of the shell model of partitions in which each shell has its parts aligned to the right margin (Cf. A210953, A210970, A135010). Also rows of triangle A210952 converge to this sequence. - Omar E. Pol, May 25 2012

Using the above result (see Jovovic's comment) of Jovovic and Merten's theorem on the average order of the phi function, we can obtain the estimate a(n-1) = 6/Pi^2*n*p(n) + O(log(n)*A006128(n)), where p(n) is the partition function A000041(n). It can be shown that A006128(n) = O(sqrt(n)*log(n)*p(n)), so we have the asymptotic result a(n) ~ 6/Pi^2*n*p(n). - Peter Bala, Dec 23 2013

a(n-2) is the number of partitions of 2n or 2n-1 with palindromicity 2; that is, partitions that can be listed in palindromic order except for a central sequence of two distinct parts. - Gregory L. Simay, Nov 01 2015

LINKS

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

FORMULA

Let t(n_, k_) = Sum_{i = 0..k} Sum_{j = 0..n} s(n, j)*C(i,  j)*p(k - n - i), where s(n, j) are Stirling numbers of the first kind, C(i, j) are the number of compositions of i distinct objects into j parts, and p is the integer partition function. Then a(k) = t(2, k+2) (conjectured). The formula for t(n, k) is the same as at A126442 except that there the Stirling numbers are of the second kind. - George Beck, May 21 2016

a(n) = (n+1)*A000070(n+1) - A182738(n+1). - Vaclav Kotesovec, Nov 04 2016

a(n) ~ exp(sqrt(2*n/3)*Pi)*sqrt(3)/(2*Pi^2) * (1 + 23*Pi/(24*sqrt(6*n))). - Vaclav Kotesovec, Nov 04 2016

MAPLE

with(numtheory):

a:= proc(n) option remember;

      `if`(n=0, 1, add((2+sigma(j)) *a(n-j), j=1..n)/n)

    end:

seq(a(n), n=0..40);  # Alois P. Heinz, Feb 13 2012

MATHEMATICA

a[n_] := a[n] = If[n == 0, 1, Sum[(2+DivisorSigma[1, j])*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 40}] (* Jean-Fran├žois Alcover, Mar 03 2014, after Alois P. Heinz *)

Table[Sum[(n-k)*PartitionsP[k], {k, 0, n}], {n, 1, 50}] (* Vaclav Kotesovec, Jun 23 2015 *)

t[n_, k_] := Sum[StirlingS1[n, j]* Binomial[i + j - 1, i]* PartitionsP[k - n - i], {j, 0, n}, {i, 0, k - n}]; Print@ Table[t[n, k], {k, 10}, {n, 0, k - 1}]; Table[t[2, k], {k, 3, 43}] (* George Beck, May 25 2016 *)

PROG

(MAGMA) m:=45; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/((1-x)^2*(&*[1-x^k: k in [1..50]])) )); // G. C. Greubel, Oct 15 2018

(PARI) x='x+O('x^45); Vec(1/((1-x)^2*prod(k=1, 50, 1-x^k))) \\ G. C. Greubel, Oct 15 2018

CROSSREFS

Cf. A000041, A000070, A141157.

Cf. A010815. - Gary W. Adamson, Nov 09 2008

Column k=3 of A292508.

Sequence in context: A207381 A008646 A036830 * A001924 A079921 A293767

Adjacent sequences:  A014150 A014151 A014152 * A014154 A014155 A014156

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 20 15:20 EST 2019. Contains 320336 sequences. (Running on oeis4.)