cp's OEIS Frontend

A260894 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 - x^(n+k))/(1 - x^k).

%I A260894 #20 Oct 13 2022 10:58:47
%S A260894 1,1,2,2,4,5,8,10,15,20,28,36,50,64,86,110,145,184,238,300,384,481,
%T A260894 608,756,948,1172,1456,1790,2208,2700,3310,4026,4906,5941,7200,8680,
%U A260894 10470,12570,15092,18050,21583,25718,30634,36376,43174,51102,60446,71324,84102,98948
%N A260894 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 - x^(n+k))/(1 - x^k).
%C A260894 Antidiagonal sums of irregular triangle A063746 yields this sequence at offset 1.
%C A260894 From _Jeremy Lovejoy_, Oct 13 2022: (Start)
%C A260894 a(n) is also the number of partitions of n whose rank is at least -1, the rank of a partition being the largest part minus the number of parts.   For example, the 7 partitions of 5 are (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), and (1,1,1,1,1), having ranks 4,2,1,0,-1,-2, and -4, respectively, and so a(5) = 5.
%C A260894 a(n) is also the number of partitions of n such that the smallest positive multiple of 3 not occurring in the partition is not a multiple of 6.   For example, in the 7 partitions of 5 listed above, the smallest positive multiples of 3 not occurring are 3,3,6,6,3,3, and 3, respectively, and so a(5) = 5.  (End)
%H A260894 George E. Andrews, David Newman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Andrews/andrews5.html">The Minimal Excludant in Integer Partitions</a>, J. Int. Seq., Vol. 23 (2020), Article 20.2.3.
%H A260894 Rupam Barman, Ajit Singh, <a href="https://arxiv.org/abs/2009.11602">On Mex-related partition functions of Andrews and Newman</a>, arXiv:2009.11602 [math.NT], 2020.
%F A260894 G.f.: (1/Product_{n>=1}(1-x^n))*Sum_{n>=0}(-1)^n*x^((3*n^2+3*n)/2). - _Jeremy Lovejoy_, Oct 13 2022.
%e A260894 G.f.: A(x) = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 5*x^5 + 8*x^6 + 10*x^7 +...
%e A260894 where
%e A260894 A(x) = 1 + x*(1-x^2)/(1-x) + x^2*(1-x^3)*(1-x^4)/((1-x)*(1-x^2)) + x^3*(1-x^4)*(1-x^5)*(1-x^6)/((1-x)*(1-x^2)*(1-x^3)) + x^4*(1-x^5)*(1-x^6)*(1-x^7)*(1-x^8)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) +...
%t A260894 Table[SeriesCoefficient[Sum[x^k * Product[(1-x^(k+j))/(1-x^j), {j,1,k}], {k,0,n}], {x,0,n}], {n,0,50}] (* _Vaclav Kotesovec_, Aug 08 2015 *)
%o A260894 (PARI) {a(n) = local(A=1); A = sum(m=0,n,x^m*prod(k=1,m,(1-x^(m+k))/(1-x^k +x*O(x^n)))); polcoeff(A,n)}
%o A260894 for(n=0,60,print1(a(n),", "))
%Y A260894 Cf. A064174, A063746.
%Y A260894 Cf. A000041, A064173, A064174.
%K A260894 nonn
%O A260894 0,3
%A A260894 _Paul D. Hanna_, Aug 03 2015