OFFSET
1,9
COMMENTS
Computed by R. K. Guy (see link).
LINKS
Joerg Arndt, Table of n, a(n) for n = 1..5050 (rows 1..100, flattened)
Joerg Arndt, C++ program for this sequence, 2016
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs (annotated scanned copy)
Anders Claesson, Svante Linusson, Henning Ulfarsson, and Emil Verkama, Inversion monotonicity in subclasses of the 1324-avoiders, arXiv:2604.01143 [math.CO], 2026. See p. 38 (Prop. 7.3).
Richard K. Guy, Letter to N. J. A. Sloane, Apr 08 1988 (annotated scanned copy, included with permission)
FORMULA
T(n,r) = Sum_{D(n,r)} Product_{k=2..m} abs(p[k]-p[k-1]) where the sum ranges over all partitions of n into distinct parts with maximal part r and the product over the m-1 pairs of successive parts; m is the number of parts in the partition under consideration. [Joerg Arndt, Apr 09 2016]
EXAMPLE
Triangle begins:
1,
0,1,
0,1,1,
0,0,2,1,
0,0,1,3,1,
0,0,1,2,4,1,
0,0,0,3,3,5,1,
0,0,0,2,5,4,6,1,
0,0,0,1,5,7,5,7,1,
0,0,0,1,5,8,9,6,8,1,
0,0,0,0,4,10,11,11,7,9,1,
0,0,0,0,3,11,15,14,13,8,10,1,
0,0,0,0,2,9,19,20,17,15,9,11,1,
0,0,0,0,1,9,20,27,25,20,17,10,12,1,
0,0,0,0,1,7,20,32,35,30,23,19,11,13,1,
...
MAPLE
b:= proc(n, i, d) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, 1, b(n, i-1, d+1)+`if`(i>n, 0, d*b(n-i, i-1, 1))))
end:
T:= (n, r)-> b(n-r, r-1, 1):
seq(seq(T(n, r), r=1..n), n=1..15); # Alois P. Heinz, Jul 08 2016
MATHEMATICA
b[n_, i_, d_] := b[n, i, d] = If[i*(i+1)/2 < n, 0, If[n == 0, 1, b[n, i-1, d+1] + If[i > n, 0, d*b[n-i, i-1, 1]]]];
T[n_, r_] := b[n-r, r-1, 1];
Table[T[n, r], {n, 1, 15}, {r, 1, n}] // Flatten (* Jean-François Alcover, Jul 27 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
AUTHOR
N. J. A. Sloane, Jun 19 2015
STATUS
approved