A259095 Triangle read by rows: T(n,r) = number of arrangements of n pennies in rows, with r contiguous pennies in the bottom row, and each higher row consisting of contiguous pennies, each touching two pennies in the row below (1 <= r <= n).

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

Offset: 1

Author

N. J. A. Sloane, Jun 19 2015

Keywords

• nonn
• tabl

Comments

Computed by R. K. Guy (see link).

Examples

			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,
  ...
(An unusually large number of rows are shown in order to explain the related sequences A005575-A005578.)
		

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)
• R. K. Guy, Letter to N. J. A. Sloane, Apr 08 1988 (annotated scanned copy, included with permission)

Crossrefs

Cf. A001524 (row sums), A001519 (column sums).

Cf. also A005575 (a diagonal), A005576, A005577 (row maxima), A005578.

Programs

Maple

b:= proc(n, i, d) option remember; `if`(i*(i+1)/2n, 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 *)

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]