0,4

Number of compositions n = p(1) + p(2) + ... + p(m) such that p(k) >= k (superdiagonal compositions), see example. - Joerg Arndt, Dec 19 2012

Number of (n-2)-bit binary strings in which the runs of ones are successively (1, 11, 111, 1111, ...), as in for example 00101100111011110011111000... To turn such a string into a composition, add 'X0 to the start of the empty string and the mark ' to the end, replace the runs 1, 11, 111,... with '01, '011, '0111, ... then consider the distances between the marks. - Andrew Woods, Jan 02 2015

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from Vincenzo Librandi)

Margaret Archibald, Aubrey Blecher, Arnold Knopfmacher, and Stephan Wagner, Subdiagonal and superdiagonal compositions, Art Disc. Appl. Math. (2024). See p. 10.

Emeric Deutsch, Emanuele Munarini, and Simone Rinaldi, Skew Dyck paths, area, and superdiagonal bargraphs, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562.

G.f.: Sum_{n>=0} q^(n*(n+1)/2) / (1-q)^n.

a(n) = Sum_{k=0..floor((sqrt(8*n+1)-3)/2)} C(n-1-C(k+1,2),k), for n >= 1.

From Joerg Arndt, Dec 19 2012: (Start)

The a(9) = 18 compositions 9 = p(1) + p(2) + ... + p(m) such that p(k) >= k are

[ 1] [ 1 2 6 ]

[ 2] [ 1 3 5 ]

[ 3] [ 1 4 4 ]

[ 4] [ 1 5 3 ]

[ 5] [ 1 8 ]

[ 6] [ 2 2 5 ]

[ 7] [ 2 3 4 ]

[ 8] [ 2 4 3 ]

[ 9] [ 2 7 ]

[10] [ 3 2 4 ]

[11] [ 3 3 3 ]

[12] [ 3 6 ]

[13] [ 4 2 3 ]

[14] [ 4 5 ]

[15] [ 5 4 ]

[16] [ 6 3 ]

[17] [ 7 2 ]

[18] [ 9 ]

(End)

b:= proc(n, i) option remember; `if`(n=0, 1,

add(b(n-j, i+1), j=i..n))

end:

a:= n-> b(n, 1):

seq(a(n), n=0..60); # Alois P. Heinz, Mar 28 2014

nmax = 50; CoefficientList[Series[Sum[x^(k*(k+1)/2) / (1-x)^k, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 05 2015 *)

b[n_, i_] := b[n, i] = If[n==0, 1, Sum[b[n-j, i+1], {j, i, n}]]; a[n_] := b[n, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 24 2015, after Alois P. Heinz *)

(PARI)

N=66; q='q+O('q^N);

gf=sum(n=0, N, q^(n*(n+1)/2) / (1-q)^n );

v=Vec(gf)

Cf. A063978 (compositions such that p(k) >= k-1 for k >= 2).

Cf. A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A238874 (strictly superdiagonal compositions), A025147 (strictly superdiagonal partitions into distinct parts).

Cf. A008930 (subdiagonal compositions), A238875 (subdiagonal partitions), A010054 (subdiagonal partitions into distinct parts).

Cf. A098131 (compositions with smallest part >= number of parts; g.f. Sum_{k>=0} x^(k^2)/(1-x)^k).

Cf. A143862 (compositions with every part divisible by number of parts; g.f. Sum_{k>=0} x^(k^2) / (1 - x^k)^k).

Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).

Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).

Row sums of A305556.

Sequence in context: A382523 A381996 A384884 * A098578 A303667 A050811

Adjacent sequences: A219279 A219280 A219281 * A219283 A219284 A219285

Joerg Arndt, Dec 04 2012

approved