A098131 -id:A098131 - cp's OEIS frontend

A219282 Number of superdiagonal bargraphs with area n.

1, 1, 1, 2, 3, 4, 6, 9, 13, 18, 25, 35, 49, 68, 93, 126, 170, 229, 308, 413, 551, 731, 965, 1269, 1664, 2177, 2842, 3701, 4806, 6222, 8031, 10337, 13272, 17003, 21740, 27745, 35343, 44936, 57021, 72213, 91274, 115149, 145010, 182309, 228841, 286819, 358964, 448614, 559857, 697694

Offset: 0

Joerg Arndt, Dec 04 2012

nonn

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

			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)

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.

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.

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 *)

N=66; q='q+O('q^N);
gf=sum(n=0,N, q^(n*(n+1)/2) / (1-q)^n );
v=Vec(gf)

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.

A236310 Expansion of Sum_{k>=0} x^((k+1)^2)/(1-x)^k.

Original entry on oeis.org

0, 1, 0, 0, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 13, 17, 22, 28, 35, 43, 52, 62, 74, 89, 108, 132, 162, 199, 244, 298, 362, 437, 524, 625, 743, 882, 1047, 1244, 1480, 1763, 2102, 2507, 2989, 3560, 4233, 5022, 5943, 7015, 8261, 9709, 11393, 13354, 15641, 18312, 21435, 25089, 29365, 34367, 40213, 47036, 54985, 64227, 74950

Offset: 0

Joerg Arndt, Apr 22 2014

nonn

a(n) is the number of compositions of n such that the first part is equal to the number of parts and all parts are greater than or equal to the first part. - John Tyler Rascoe, Feb 10 2024

			From _John Tyler Rascoe_, Feb 10 2024: (Start)
The compositions for n = 9..11 are:
9:  [3,3,3], [2,7];
10: [3,4,3], [3,3,4], [2,8];
11: [3,4,4], [3,3,5], [3,5,3], [2,9].
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A098131 (g.f. Sum_{k>=0} x^(k^2)/(1-x)^k).
Cf. A219282 (g.f. Sum_{k>=0} x^(k*(k+1)/2)/(1-x)^k).
Cf. A063978 (g.f. Sum_{k>=0} x^(k^2)/(1-x)^(k+1)).

N=66; q='q+O('q^N);
gf=sum(n=0, N, q^((n+1)^2) / (1-q)^n );
concat([0],Vec(gf))

G.f.: Sum_{k>=0} x^((k+1)^2)/(1-x)^k.

G.f.: Sum_{k>0} A(x,k) where A(x,k) = (x^k)*(x^k/(1-x))^(k-1) is the g.f. for compositions of this kind with first part k. - John Tyler Rascoe, Feb 10 2024

cp's OEIS Frontend

A219282 Number of superdiagonal bargraphs with area n.

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