OFFSET
0,3
COMMENTS
a(0)=1, otherwise row sums of A179748.
For n>=1 cumulative sums of A008930.
a(n) is proportional to A048651*A000079. The error (a(n)-A048651*A000079) divided by sequence A186425 tends to the golden ratio A001622. This can be seen when using about 1000 decimals of the constant A048651 = 0.2887880950866024212... - [Mats Granvik, Jan 01 2015]
From Gus Wiseman, Mar 31 2022: (Start)
Also the number of integer compositions of n with exactly one part on or above the diagonal. For example, the a(1) = 1 through a(5) = 8 compositions are:
(1) (2) (3) (4) (5)
(11) (21) (31) (41)
(111) (112) (212)
(211) (311)
(1111) (1112)
(1121)
(2111)
(11111)
(End)
FORMULA
G.f.: 1 + q/(1-q) * sum(n>=0, q^n * prod(k=1..n, (1-q^k)/(1-q) ) ). [Joerg Arndt, Mar 24 2014]
EXAMPLE
From Joerg Arndt, Mar 24 2014: (Start)
The a(7) = 25 such compositions are:
01: [ 1 1 1 1 1 1 1 ]
02: [ 1 1 1 1 1 2 ]
03: [ 1 1 1 1 2 1 ]
04: [ 1 1 1 1 3 ]
05: [ 1 1 1 2 1 1 ]
06: [ 1 1 1 2 2 ]
07: [ 1 1 1 3 1 ]
08: [ 1 1 1 4 ]
09: [ 1 1 2 1 1 1 ]
10: [ 1 1 2 1 2 ]
11: [ 1 1 2 2 1 ]
12: [ 1 1 2 3 ]
13: [ 1 1 3 1 1 ]
14: [ 1 1 3 2 ]
15: [ 1 2 1 1 1 1 ]
16: [ 1 2 1 1 2 ]
17: [ 1 2 1 2 1 ]
18: [ 1 2 1 3 ]
19: [ 1 2 2 1 1 ]
20: [ 1 2 2 2 ]
21: [ 1 2 3 1 ]
22: [ 2 2 3 ]
23: [ 2 3 2 ]
24: [ 3 4 ]
25: [ 7 ]
(End)
MAPLE
A179748 := proc(n, k) option remember; if k= 1 then 1; elif k> n then 0 ; else add( procname(n-i, k-1), i=1..k-1) ; end if; end proc:
seq(A177510(n), n=1..20) ; # R. J. Mathar, Dec 14 2010
MATHEMATICA
Clear[t, nn]; nn = 39; t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}], 0]; Table[Sum[t[n, k], {k, 1, n}], {n, 1, nn}] (* Mats Granvik, Jan 01 2015 *)
pdw[y_]:=Length[Select[Range[Length[y]], #<=y[[#]]&]]; Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], pdw[#]==1&]], {n, 0, 10}] (* Gus Wiseman, Mar 31 2022 *)
PROG
(Sage)
@CachedFunction
def T(n, k): # A179748
if n == 0: return int(k==0);
if k == 1: return int(n>=1);
return sum( T(n-i, k-1) for i in [1..k-1] );
# to display triangle A179748 including column zero = [1, 0, 0, 0, ...]:
#for n in [0..10]: print([ T(n, k) for k in [0..n] ])
def a(n): return sum( T(n, k) for k in [0..n] )
print([a(n) for n in [0..66]])
# Joerg Arndt, Mar 24 2014
(PARI) N=66; q='q+O('q^N); Vec( 1 + q/(1-q) * sum(n=0, N, q^n * prod(k=1, n, (1-q^k)/(1-q) ) ) ) \\ Joerg Arndt, Mar 24 2014
CROSSREFS
Cf. A238859 (compositions with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
This is column k = 1 of A352525.
A352517 counts weak excedances of standard compositions.
KEYWORD
nonn
AUTHOR
Mats Granvik, Dec 11 2010
EXTENSIONS
New name and a(0) = 1 prepended, Joerg Arndt, Mar 24 2014
STATUS
approved