%I #102 Oct 23 2023 02:11:09
%S 1,3,7,14,26,45,75,120,187,284,423,618,890,1263,1771,2455,3370,4582,
%T 6179,8266,10980,14486,18994,24757,32095,41391,53123,67865,86325,
%U 109350,137979,173450,217270,271233,337506,418662,517795,638565,785350,963320,1178628
%N Expansion of 1/((1-x)^2*Product_{k>=1} (1-x^k)).
%C 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
%C Partial sums of the partial sums of the partition numbers A000041. Partial sums of A000070. Euler transform of 3,1,1,1,...
%C 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
%C Equals row sums of triangle A141157. - _Gary W. Adamson_, Jun 12 2008
%C 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
%C From _Omar E. Pol_, May 25 2012: (Start)
%C a(n) is also the sum of all parts of the (n+1)st column of a version of the section model of partitions in which every section has its parts aligned to the right margin (cf. A210953, A210970, A135010).
%C Rows of triangle A210952 converge to this sequence. (End)
%C Using the above result (see Jovovic's comment) of Jovovic and Mertens'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
%C 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
%C Convolution of A000041 and A000027. - _Omar E. Pol_, Jun 17 2021
%C Convolution of A002865 and the positive terms of A000217. Partial sums give A014160. - _Omar E. Pol_, Mar 01 2023
%H Alois P. Heinz and Vaclav Kotesovec, <a href="/A014153/b014153.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%H Mircea Merca and Maxie D. Schmidt, <a href="https://arxiv.org/abs/2310.13658">The partition function p(n) in terms of the classical Möbius function</a>, arXiv:2310.13658 [math.CO], 2023.
%F 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
%F a(n) = (n+1)*A000070(n+1) - A182738(n+1). - _Vaclav Kotesovec_, Nov 04 2016
%F 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
%p with(numtheory):
%p a:= proc(n) option remember;
%p `if`(n=0, 1, add((2+sigma(j)) *a(n-j), j=1..n)/n)
%p end:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Feb 13 2012
%t 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-François Alcover_, Mar 03 2014, after _Alois P. Heinz_ *)
%t Table[Sum[(n-k)*PartitionsP[k],{k,0,n}],{n,1,50}] (* _Vaclav Kotesovec_, Jun 23 2015 *)
%t 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 *)
%o (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
%o (PARI) x='x+O('x^45); Vec(1/((1-x)^2*prod(k=1,50, 1-x^k))) \\ _G. C. Greubel_, Oct 15 2018
%Y Cf. A000041, A000070, A116930, A141157.
%Y Cf. A010815. - _Gary W. Adamson_, Nov 09 2008
%Y Column k=3 of A292508.
%Y Cf. A000217, A002865, A014160.
%K nonn
%O 0,2
%A _N. J. A. Sloane_