cp's OEIS Frontend

A085811 Number of partitions of n including 3, but not 1.

%I A085811 #39 Jul 02 2025 16:02:02
%S A085811 0,0,1,0,1,1,2,2,4,4,7,8,12,14,21,24,34,41,55,66,88,105,137,165,210,
%T A085811 253,320,383,478,574,708,847,1039,1238,1507,1794,2167,2573,3094,3660,
%U A085811 4378,5170,6153,7245,8591,10087,11914,13959,16424,19196,22519,26252,30701
%N A085811 Number of partitions of n including 3, but not 1.
%C A085811 Related to the 'number of sums containing k' phenomena reported at link. Define P_k(n,j) to be the number of partitions of n with minimum part j and containing k, P_k(n) as the number of partitions of n that contain k as a part and P(n,j) as the number of partitions of k that have minimum part k, then: P_k(n)=sum{i=1,k-1,P_k(n-i,i)}+P(n-k,k) which (unproved) gives P(n-k). This sequence gives P_3(n,2). E.g. assume P_3(9)=11. P_3(10)=P_3(9,1)+P_3(8,2)+P(7,3)=11+2+2=15, where P(7,3) is given by A008483(7).
%H A085811 Andrew van den Hoeven, <a href="/A085811/b085811.txt">Table of n, a(n) for n = 1..10000</a>
%F A085811 A002865(n) = a(n+3). - _James Sellers_, Dec 06 2005.
%F A085811 G.f.: x^3*(1-x)/prod(n>=1, 1-x^n). [_Joerg Arndt_, Feb 03 2012]
%F A085811 G.f.: x^2 + x^3*(1 - G(0))/(1-x) where G(k) =  1 - x/(1-x^(k+1))/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 23 2013
%e A085811 a(3): 3
%e A085811 a(5): 2+3
%e A085811 a(6): 3+3
%e A085811 a(7): 2+2+3, 3+4
%e A085811 a(8): 2+3+3, 3+5
%e A085811 a(9): 2+3+4, 2+2+2+3, 3+3+3, 3+6
%e A085811 a(10): 2+3+5, 2+2+3+3, 3+7, 3+3+4
%e A085811 a(11): 2+2+3+4, 2+3+6, 2+2+2+2+3, 2+3+3+3, 3+4+4, 3+8, 3+3+5,
%e A085811 a(12): 2+2+2+3+3, 2+3+3+4, 2+3+7, 2+2+3+5, 3+9, 3+3+6, 3+4+5, 3+3+3+3
%e A085811 a(13): 2+2+2+2+2+3, 2+2+2+3+4, 2+2+3+6, 2+2+3+3+3, 2+3+4+4, 2+3+3+5,
%e A085811 2+3+8, 3+10, 3+3+7, 3+4+6, 3+5+5, 3+3+3+4
%t A085811 f[n_] := Block[{c = 0, k = 1, m = PartitionsP[n], p = IntegerPartitions[n] }, While[k < m, If[ Count[ p[[k]], 3] > 0 && Count[ p[[k]], 1] == 0, c++ ]; k++ ]; c]; Table[ f[n], {n, 1, 53}]
%t A085811 (* second program: *)
%t A085811 CoefficientList[x^2*(1-x)/QPochhammer[x] + O[x]^60, x] (* _Jean-François Alcover_, Jan 22 2016, after _Joerg Arndt_ *)
%o A085811 (PARI)  x='x+O('x^66); /* about that many terms */
%o A085811 v=Vec((x^3*(1-x)/eta(x)))  /* _Joerg Arndt_, Feb 03 2012 */
%Y A085811 Cf. A008483, A002865. Essentially the same as A002865.
%K A085811 nonn
%O A085811 1,7
%A A085811 _Jon Perry_, Jul 25 2003
%E A085811 Edited, corrected and extended by _Robert G. Wilson v_
%E A085811 Typo in formula corrected by _Andrew van den Hoeven_, Nov 20 2014