A005576 -id:A005576 - cp's OEIS frontend

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

N. J. A. Sloane, Jun 19 2015

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

			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.)

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)

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

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

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

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 *)

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]

A005575 a(n) = A259095(2n,n).

Original entry on oeis.org

0, 0, 1, 2, 5, 11, 20, 37, 63, 110, 174, 283, 435, 671, 1001, 1492, 2160, 3127, 4442, 6269, 8739, 12109, 16597, 22618, 30576, 41077, 54834, 72788, 96056, 126131, 164829, 214327, 277534, 357810, 459507, 587779, 749220, 951473, 1204501, 1519691, 1911618, 2397247, 2997985, 3738482, 4649981, 5768457, 7138640, 8812704, 10854735, 13339286

Offset: 1

N. J. A. Sloane, R. K. Guy

Computed by R. K. Guy in 1988.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..900
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
R. K. Guy, Letter to N. J. A. Sloane, Apr 08 1988 (annotated scanned copy, included with permission)
E. M. Wright, Stacks, III, Quart. J. Math. Oxford, 23 (1972), 153-158.

Cf. A259095, A005576, A005577.

b:= proc(n, i, d) option remember; `if`(i*(i+1)/2n, 0, d*b(n-i, i-1, 1))))
    end:
a:= n-> b(n, n-1, 1):
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 08 2016

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]]]];
a[n_] := b[n, n-1, 1];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Jul 28 2016, after Alois P. Heinz *)

Edited by N. J. A. Sloane, Jun 20 2015

Terms a(25) and beyond from Joerg Arndt, Apr 09 2016

A005577 Maxima of the rows of the triangle A259095.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 11, 15, 20, 27, 35, 44, 56, 73, 91, 115, 148, 186, 227, 283, 358, 435, 538, 671, 813, 1001, 1233, 1492, 1815, 2223, 2673, 3247, 3933, 4713, 5683, 6850, 8170, 9785, 11725, 13948, 16587, 19783, 23468, 27710, 32942, 38956, 45852, 54133, 63879, 75000, 87909, 103471, 121273, 141629

Offset: 1

N. J. A. Sloane, R. K. Guy

Computed by R. K. Guy in 1988.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 100 terms from Joerg Arndt)
F. C. Auluck, On some new types of partitions associated with generalized ferrers graphs, Math. Proc. Camb. Phil. Soc. 47 (1951) 679-686.
R. K. Guy, Letter to N. J. A. Sloane, Apr 08 1988 (annotated scanned copy, included with permission)
E. M. Wright, Stacks (III), The Quarterly J. of Math. (Oxford Journals), 23 (2) (1972) 153-158. MR0299575

Cf. A259095, A005575, A005576.

b:= proc(n, i, d) option remember; `if`(i*(i+1)/2n, 0, d*b(n-i, i-1, 1))))
    end:
a:= n-> max(seq(b(n-r, r-1, 1), r=1..n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Jul 08 2016

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]]]];
a[n_] := Max[Table[b[n-r, r-1, 1], {r, 1, n}]];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jul 28 2016, after Alois P. Heinz *)

Edited by N. J. A. Sloane, Jun 20 2015

cp's OEIS Frontend

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).

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A005575 a(n) = A259095(2n,n).

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A005577 Maxima of the rows of the triangle A259095.

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