0,4

Fine-Riordan array S_n(m)=a(n,m) with extra row for n=0 added.

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,...].

Written as triangle this becomes A103923.

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.

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.

W. Lang: First 20 rows and comments.

Riordan gives formula.

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.

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.

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).

Array begins:

m\n 0 1 2 3 4 .5 .6 .7 .8 ...

0 | 1 1 2 3 5 .7 11 15 22 ... (A000041)

1 | . 1 2 4 7 12 19 ... (A000070)

2 | . . . 1 2 .5 .9 ... (A000097)

3 | . . . . . .. .1 ... (A000098)

[1]; [1,1]; [2,2]; [3,4,1]; [5,7,2]; [7,12,5]; [11,19,9,1]...

a(3,1)=4 because the partitions (3), (1,2) and (1^3) have q values 1,2 and 1 which sum to 4.

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.

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').

The first column (m=0) gives A000041(n). Columns m=1..10 are A000070 (partial sums of partition numbers), A000097, A000098, A000710, A103924-A103929.

Adjacent sequences: A008948 A008949 A008950 this_sequence A008952 A008953 A008954

Sequence in context: A159804 A104567 A087824 this_sequence A119473 A002122 A105689

nonn,tabf,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Robert G Bearden (nem636(AT)myrealbox.com), Apr 27 2004

Correction, comments and Riordan formulae from W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Apr 28 2005