A171682 -id:A171682 - cp's OEIS frontend

A224960 Number of compositions [p(1), p(2), ..., p(k)] of n such that p(j) >= p(1) - 1.

1, 1, 2, 4, 7, 14, 26, 52, 101, 200, 396, 787, 1564, 3117, 6214, 12398, 24749, 49427, 98740, 197303, 394323, 788201, 1575695, 3150265, 6298732, 12594595, 25184598, 50361842, 100711888, 201404839, 402779246, 805509560, 1610940381, 3221753990

Offset: 0

Joerg Arndt, Apr 21 2013

nonn

			The a(5) = 14 such compositions of 5 are
01:  [ 1 1 1 1 1 ]
02:  [ 1 1 1 2 ]
03:  [ 1 1 2 1 ]
04:  [ 1 1 3 ]
05:  [ 1 2 1 1 ]
06:  [ 1 2 2 ]
07:  [ 1 3 1 ]
08:  [ 1 4 ]
09:  [ 2 1 1 1 ]
10:  [ 2 1 2 ]
11:  [ 2 2 1 ]
12:  [ 2 3 ]
13:  [ 3 2 ]
14:  [ 5 ]
(the two forbidden compositions are [ 3 1 1 ] and [ 4 1 ]).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A171682 (compositions such that p(j) >= p(1)).
Cf. A079501 (compositions such that p(j) > p(1)).
Cf. A048888 (compositions such that p(j) <= p(1) + 1).
Cf. A007059 (compositions such that p(j) < p(1)).
Cf. A079500 (compositions such that p(j) <= p(1)).

b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j,
      `if`(i=0, max(1, j-1), i)), j=`if`(i=0, 1, i)..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, May 02 2013

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - j, If[i == 0, Max[1, j - 1], i]], {j, If[i == 0, 1, i], n}]];
a[n_] := b[n, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

a(n) ~ 3 * 2^(n-3). - Vaclav Kotesovec, May 01 2014

A238347 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n where the k-th part is the first occurrence of a smallest part, n>=1, 1<=k<=n.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 6, 2, 0, 0, 10, 5, 1, 0, 0, 20, 9, 3, 0, 0, 0, 37, 19, 7, 1, 0, 0, 0, 72, 36, 16, 4, 0, 0, 0, 0, 140, 71, 33, 11, 1, 0, 0, 0, 0, 275, 139, 67, 26, 5, 0, 0, 0, 0, 0, 540, 274, 135, 58, 16, 1, 0, 0, 0, 0, 0, 1069, 539, 269, 123, 42, 6, 0, 0, 0, 0, 0, 0, 2118, 1068, 534, 254, 99, 22, 1, 0

Offset: 1

Joerg Arndt and Alois P. Heinz, Feb 25 2014

Column k=0 is A238351. Column k=1 is A171682.

Row sums are A011782.

			Triangle starts:
01:  1,
02:  2, 0,
03:  3, 1, 0,
04:  6, 2, 0, 0,
05:  10, 5, 1, 0, 0,
06:  20, 9, 3, 0, 0, 0,
07:  37, 19, 7, 1, 0, 0, 0,
08:  72, 36, 16, 4, 0, 0, 0, 0,
09:  140, 71, 33, 11, 1, 0, 0, 0, 0,
10:  275, 139, 67, 26, 5, 0, 0, 0, 0, 0,
11:  540, 274, 135, 58, 16, 1, 0, 0, 0, 0, 0,
12:  1069, 539, 269, 123, 42, 6, 0, 0, 0, 0, 0, 0,
13:  2118, 1068, 534, 254, 99, 22, 1, 0, 0, 0, 0, 0, 0,
14:  4206, 2117, 1062, 516, 220, 64, 7, 0, 0, 0, 0, 0, 0, 0,
15:  8365, 4205, 2111, 1040, 470, 163, 29, 1, 0, 0, 0, 0, 0, 0, 0,
...

Joerg Arndt, Table of n, a(n) for n = 1..465 (rows 1..30, flattened)

cp's OEIS Frontend

A224960 Number of compositions [p(1), p(2), ..., p(k)] of n such that p(j) >= p(1) - 1.

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