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A219282 Number of superdiagonal bargraphs with area n. 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

LINKS

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

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

FORMULA

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.

EXAMPLE

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)

MAPLE

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

MATHEMATICA

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-Fran├žois Alcover, Mar 24 2015, after Alois P. Heinz *)

PROG

(PARI)

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

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

v=Vec(gf)

CROSSREFS

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. 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).

Sequence in context: A129632 A016028 A239551 * A098578 A050811 A076968

Adjacent sequences:  A219279 A219280 A219281 * A219283 A219284 A219285

KEYWORD

nonn

AUTHOR

Joerg Arndt, Dec 04 2012

STATUS

approved

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Last modified March 27 22:02 EDT 2017. Contains 284182 sequences.