This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A156041 #51 Jan 24 2025 18:35:06
%S A156041 1,1,1,1,2,1,1,2,3,1,1,3,4,4,1,1,3,6,7,5,1,1,4,8,11,11,6,1,1,4,11,17,
%T A156041 19,16,7,1,1,5,13,26,32,31,22,8,1,1,5,17,35,54,56,48,29,9,1,1,6,20,48,
%U A156041 82,102,93,71,37,10,1,1,6,24,63,120,172,180,148,101,46,11,1,1,7,28,81,170
%N A156041 Array A(n,k) (n>=1, k>=1) read by antidiagonals, where A(n,k) is the number of compositions (ordered partitions) of n into exactly k parts, some of which may be zero, with the first part greater than or equal to all the rest.
%C A156041 A(n,k) is of course smaller than the number of ordered partitions of n into k parts and at least the number of partitions into k parts in descending order.
%C A156041 The sums of the antidiagonals give A079500 - 1. - _N. J. A. Sloane_, Feb 26 2011
%C A156041 For an alternative definition of essentially the same sequence, as a triangle, and which avoids the use of parts of size zero, see A184957. - _N. J. A. Sloane_, Feb 27 2011
%H A156041 R. H. Hardin, <a href="/A156041/b156041.txt">Table of n, a(n) for n = 1..2278</a>
%H A156041 Michele Graffeo, Sergej Monavari, Riccardo Moschetti, and Andrea T. Ricolfi, <a href="https://arxiv.org/abs/2501.10267">Enumeration of partitions via socle reduction</a>, arXiv:2501.10267 [math.CO], 2025. See p. 32.
%F A156041 A(n,k) = [x^n] Sum_{i=0..n} x^i*((1 - x^(i+1))/(1-x))^(k-1). - _Geoffrey Critzer_, Jul 15 2013
%e A156041 The array A(n,k) begins:
%e A156041 1 1 1 1 1 1 1 1 1 ...
%e A156041 1 2 3 4 5 6 7 8 9 ...
%e A156041 1 2 4 7 11 16 22 29 ...
%e A156041 1 3 6 11 19 31 48 ...
%e A156041 1 3 8 17 32 56 ...
%e A156041 1 4 11 26 54 ...
%e A156041 1 4 13 35 ...
%e A156041 ...
%e A156041 The antidiagonals are:
%e A156041 1,
%e A156041 1, 1,
%e A156041 1, 2, 1,
%e A156041 1, 2, 3, 1,
%e A156041 1, 3, 4, 4, 1,
%e A156041 1, 3, 6, 7, 5, 1,
%e A156041 1, 4, 8, 11, 11, 6, 1,
%e A156041 1, 4, 11, 17, 19, 16, 7, 1,
%e A156041 1, 5, 13, 26, 32, 31, 22, 8, 1,
%e A156041 ...
%e A156041 A(3,5) = 11 and the 11 partition of 3 into 5 parts of this type are: (3,0,0,0,0), (2,1,0,0,0), (2,0,1,0,0), (2,0,0,1,0), (2,0,0,0,1), (1,1,1,0,0), (1,1,0,1,0), (1,1,0,0,1), (1,0,1,1,0), (1,0,1,0,1), (1,0,0,1,1).
%p A156041 b:= proc(n, i, m) option remember;
%p A156041 if n<0 then 0
%p A156041 elif n=0 then 1
%p A156041 elif i=1 then `if`(n<=m, 1, 0)
%p A156041 else add(b(n-k, i-1, m), k=0..m)
%p A156041 fi
%p A156041 end:
%p A156041 A:= (n, k)-> add(b(n-m, k-1, m), m=ceil(n/k)..n):
%p A156041 seq(seq(A(d-k, k), k=1..d-1), d=1..14); # _Alois P. Heinz_, Jun 14 2009
%t A156041 (* Returns rectangular array *) nn=10;Table[Table[Coefficient[Series[Sum[x^i((1-x^(i+1))/(1-x))^(k-1),{i,0,n}],{x,0,nn}],x^n],{k,1,nn}],{n,1,nn}]//Grid (* _Geoffrey Critzer_, Jul 15 2013 *)
%Y A156041 A156039 gives A(n,4) and A156040 gives A(n,3). A156042 is the part on or below the main diagonal. A(n,2) is A008619. A(2,n) is A000027. A(3,n) is A000124.
%Y A156041 Cf. A079500.
%K A156041 nonn,tabl
%O A156041 1,5
%A A156041 _Jack W Grahl_, Feb 02 2009, Feb 11 2009
%E A156041 More terms from _Alois P. Heinz_, Jun 14 2009
%E A156041 Edited by _N. J. A. Sloane_, Feb 26 2011