Search: id:A103923
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%I A103923
%S A103923 1,1,1,2,2,1,3,4,2,1,5,7,5,2,1,7,12,9,5,2,1,11,19,17,10,5,2,1,15,30,28,
%T A103923 19,10,5,2,1,22,45,47,33,20,10,5,2,1,30,67,73,57,35,20,10,5,2,1,42,97,
%U A103923 114,92,62,36,20,10,5,2,1,56,139,170,147,102,64,36,20,10,5,2,1,77,195
%N A103923 Triangle of partitions of n with parts of sizes 1,2,...,m, each of two 
               different kinds, m>=1.
%C A103923 The corresponding Fine-Riordan triangle is A008951.
%C A103923 This is the array p_2(n,m) of Gupta et al. written as a triangle. p_2(n,
               m) is defined on p. x of this reference as the number of partitions 
               of n into parts consisting of two varieties of each of the integers 
               1 to m and one variety of each larger integer. Therefore a(n,m) gives 
               these numbers for the partitions of n-m.
%C A103923 a(n,m)= sum over partitions of n+t(m)-m of binomial(q(partition),m), 
               with t(m):=A000217(m) and q the number of distinct parts of a given 
               partition. m>=0.
%C A103923 a(n,m)= number of partitions of 2*n-m with exactly m odd parts.
%C A103923 a(n,m)= sum over partitions of n+t(m)-m of product(k[j],j=1..m), with 
               t(m):=A000217(m) and k[j]=number of parts of size j (exponent of 
               j in a given partition of n), if m>=1. If m=0 then a(n,0)=p(n):=A000041(n) 
               (number of partitions of n). 0 is counted as a part for n=0 and only 
               for this n.
%D A103923 H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, 
               Vol. 4, Cambridge Univ. Press, 1958 (reprinted 1962), pp. 90-121.
%D A103923 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
%H A103923 W. Lang: <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A103923.text">
               First 16 rows.</a>
%F A103923 a(n, m)= a(n-1, m-1) + a(n-m, m), n>=m>=0, with a(n, 0)= A000041(n) (partition 
               numbers), a(n, m)=0 if n<m.
%F A103923 a(n, m) = sum(a(n-1-j*m, m-1), j=0..floor((n-m)/m)), m>=1, input a(n, 
               0)= A000041(n).
%F A103923 G.f. column m: product(1/(1-x^j), j=1..m)*P(x), with P(x)= product(1/
               (1-x^j), j=1..infty), the o.g.f. for the partition numbers A000041.
%F A103923 G.f. column m>=1: (product(1/(1-x^k), k=1..m)^2)*product(1/(1-x^j), j=(m+1)..infty). 
               For m=0 put the first product equal to 1.
%e A103923 [1];[1,1];[2,2,1];[3,4,2,1];[5,7,5,2,1];...
%e A103923 a(4,2)=5 from the partitions of 4-2=2 with two varieties of parts 1 and 
               of 2, namely (2),(2'),(1^2),(1'^2) and (1,1').
%e A103923 a(4,2)=5 from the partitions of 4+t(2)-2=5 which have products of the 
               exponents of parts 1 and 2: 0*0,1*0,0*1,2*1,1*2,5*0 and sum to 4.
%e A103923 a(4,2)=5 from the partitions of 4+t(2)-2=5 which have number of distinct 
               parts (q values) 1,2,2,2,2,2,1. The corresponding binomial(q,2) values 
               are 0,1,1,1,1,0 and sum to 4.
%e A103923 a(4,2)=5 from the partitions of 2*4-2=6 with exactly two odd parts, namely 
               (1,5), (3^2), (1^2,4), (1,2,3) and (1^2,2^2), which are 5 in number.
%Y A103923 The column sequences (without leading 0's) are, for m=0..10: A000041, 
               A000070, A000097, A000098, A000710, A103924-A103929.
%Y A103923 Cf. A000712 A124577.
%Y A103923 Sequence in context: A071453 A080242 A035317 this_sequence A061987 A105809 
               A091594
%Y A103923 Adjacent sequences: A103920 A103921 A103922 this_sequence A103924 A103925 
               A103926
%K A103923 nonn,easy,tabl
%O A103923 0,4
%A A103923 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 
               Mar 24 2005

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