%I A008951
%S A008951 1,1,1,2,2,3,4,1,5,7,2,7,12,5,11,19,9,1,15,30,17,2,22,45,28,5,30,67,
%T A008951 47,10,42,97,73,19,1,56,139,114,33,2,77,195,170,57,5,101,272,253,92,
%U A008951 10,135,373,365,147,20,176,508,525,227,35,1,231,684,738,345,62,2,297
%N A008951 Array read by columns: number of partitions of n into parts of 2 kinds.
%C A008951 Fine-Riordan array S_n(m)=a(n,m) with extra row for n=0 added.
%C A008951 Row n of this triangle has length floor(1/2 + sqrt(2*(n+1))), n>=0. This 
               is sequence A002024(n+1)=[1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,
               6,...].
%C A008951 Written as triangle this becomes A103923.
%C A008951 a(n,m) gives also the number of partitions of n-t(m), where t(m):=A000217(m) 
               (triangular numbers), with two kinds of parts 1,2,..m. See the column 
               o.g.f.'s in table A103923.
%D A008951 H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, 
               Vol. 4, Cambridge Univ. Press, 1958, p. 90.
%D A008951 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
%H A008951 W. Lang: <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A103922.text">
               First 20 rows and comments.</a>
%F A008951 Riordan gives formula.
%F A008951 a(n, m)=sum over partitions of n of product(k[j], j=1..m), with 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). O is counted as a part for n=0 and only for this n.
%F A008951 a(n, m)=sum over partitions of n of binomial(q(partition), m), with q 
               the number of distinct parts of a given partition. m>=0.
%F A008951 a(n, m)=a(n-m, m-1) + a(n-m, m), n>=t(m):=m*(m+1)/2=A000217(m) (triangular 
               numbers), else 0, with input a(n, 0)=p(n):=A000041(n).
%e A008951 Array begins:
%e A008951 m\n 0 1 2 3 4 .5 .6 .7 .8 ...
%e A008951 0 | 1 1 2 3 5 .7 11 15 22 ... (A000041)
%e A008951 1 | . 1 2 4 7 12 19 ... (A000070)
%e A008951 2 | . . . 1 2 .5 .9 ... (A000097)
%e A008951 3 | . . . . . .. .1 ... (A000098)
%e A008951 [1]; [1,1]; [2,2]; [3,4,1]; [5,7,2]; [7,12,5]; [11,19,9,1]...
%e A008951 a(3,1)=4 because the partitions (3), (1,2) and (1^3) have q values 1,
               2 and 1 which sum to 4.
%e A008951 a(3,1)=4 because the exponents of part 1 in the above given partitions 
               of 3 are 0,1,3 and they sum to 4.
%e A008951 a(3,1)=4 because the partitions of 3-t(1)=2 with two kinds of part 1, 
               say 1 and 1' and one kind of part 2 are (2),(1^2), (1'^2) and (11').
%Y A008951 The first column (m=0) gives A000041(n). Columns m=1..10 are A000070 
               (partial sums of partition numbers), A000097, A000098, A000710, A103924-A103929.
%Y A008951 Sequence in context: A159804 A104567 A087824 this_sequence A119473 A002122 
               A105689
%Y A008951 Adjacent sequences: A008948 A008949 A008950 this_sequence A008952 A008953 
               A008954
%K A008951 nonn,tabf,nice
%O A008951 0,4
%A A008951 N. J. A. Sloane (njas(AT)research.att.com).
%E A008951 More terms from Robert G Bearden (nem636(AT)myrealbox.com), Apr 27 2004
%E A008951 Correction, comments and Riordan formulae from W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 
               Apr 28 2005