Search: id:A024786
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%I A024786
%S A024786 0,1,1,3,4,8,11,19,26,41,56,83,112,160,213,295,389,526,686,911,1176,1538,
%T A024786 1968,2540,3223,4115,5181,6551,8191,10269,12756,15873,19598,24222,29741,
%U A024786 36532,44624,54509,66261,80524,97446,117862,142029,171036,205290,246211
%N A024786 Number of 2's in all partitions of n.
%C A024786 Also number of partitions of n-1 with a distinguished part different 
               from all the others. [Comment corrected by Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Aug 13 2008]
%C A024786 In general the number of times that j appears in the partitions of n 
               equals Sum_{k<n, k = n (mod j)} P(k). In particular this gives a 
               formula for a(n), A024787, ..., A024794, for j = 2,...,10; it generalizes 
               the formula given for A000070 for j=1. - Jose Luis Arregui (arregui(AT)posta.unizar.es), 
               Apr 05 2002
%D A024786 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 184.
%D A024786 E. Deutsch et al., Problem 11237, Amer. Math. Monthly, 115 (No. 7, 2008), 
               666-667. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 13 
               2008]
%F A024786 a(n) = sum{k=1 to floor(n/2)} A000041(n-2k).
%F A024786 a(n) = Sum_{k<n, k = n (mod 2)} P(k), P(k) =number of partitions of k 
               as in A000041, P(0) = 1. - Jose Luis Arregui (arregui(AT)posta.unizar.es), 
               Apr 05 2002
%F A024786 G.f.: x/((1-x)*(1-x^2)^2))*product(1/(1-x^j), j=3..infty) from Riordan 
               reference second term, last eq.
%t A024786 << DiscreteMath`Combinatorica`; Table[ Count[ Flatten[ Partitions[n]], 
               2], {n, 1, 50} ]
%Y A024786 Cf. A066633, A024787, A024788, A024789, A024790, A024791, A024792, A024793, 
               A024794.
%Y A024786 Column 2 of A060244.
%Y A024786 First differences of A000097.
%Y A024786 Sequence in context: A099108 A001994 A084421 this_sequence A097497 A006167 
               A137504
%Y A024786 Adjacent sequences: A024783 A024784 A024785 this_sequence A024787 A024788 
               A024789
%K A024786 nonn
%O A024786 1,4
%A A024786 Clark Kimberling (ck6(AT)evansville.edu)
%E A024786 Formula and comment from Christian G. Bower (bowerc(AT)usa.net), Jun 
               22 2000

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