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.
G.f. = 1 + x + 3*x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 11*x^6 + 13*x^7 + 17*x^8 + 20*x^9 + ...
The a(4) = 6 compositions of 4 are: (4 0 0), (3 1 0), (3 0 1), (2 2 0), (2 1 1), (2 0 2).
From _Gus Wiseman_, Oct 05 2020: (Start)
The a(0) = 1 through a(7) = 13 triples of nonnegative integers summing to n where the first is at least as great as each of the other two are:
(000) (100) (101) (111) (202) (212) (222) (313)
(110) (201) (211) (221) (303) (322)
(200) (210) (220) (302) (312) (331)
(300) (301) (311) (321) (403)
(310) (320) (330) (412)
(400) (401) (402) (421)
(410) (411) (430)
(500) (420) (502)
(501) (511)
(510) (520)
(600) (601)
(610)
(700)
(End)
a:= proc(n) local m, r; m := iquo(n, 6, 'r'); (4 +6*m +2*r) *m + [1, 1, 3, 4, 6, 8][r+1] end: seq(a(n), n=0..60); # Alois P. Heinz, Jun 14 2009
nn = 58; CoefficientList[Series[x^3/(1 - x^2)^2/(1 - x^3) + 1/(1 - x^2)^2/(1 - x), {x, 0, nn}], x] (* Geoffrey Critzer, Jul 14 2013 *)
CoefficientList[Series[(1 + x^2)/((1 + x) * (1 + x + x^2) * (1 - x)^3), {x, 0, 58}], x] (* L. Edson Jeffery, Jul 29 2014 *)
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {1, 1, 3, 4, 6, 8}, 60] (* Harvey P. Dale, May 28 2015 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n+3,{3}],#[[1]]>=#[[2]]&&#[[1]]>=#[[3]]&]],{n,0,15}] (* Gus Wiseman, Oct 05 2020*)
{a(n) = n*(n+4)\6 + 1}; /* Michael Somos, Mar 26 2017 */
The array A(n,k) begins: 1 1 1 1 1 1 1 1 1 ... 1 2 3 4 5 6 7 8 9 ... 1 2 4 7 11 16 22 29 ... 1 3 6 11 19 31 48 ... 1 3 8 17 32 56 ... 1 4 11 26 54 ... 1 4 13 35 ... ... The antidiagonals are: 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, 19, 16, 7, 1, 1, 5, 13, 26, 32, 31, 22, 8, 1, ... 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).
b:= proc(n, i, m) option remember;
if n<0 then 0
elif n=0 then 1
elif i=1 then `if`(n<=m, 1, 0)
else add(b(n-k, i-1, m), k=0..m)
fi
end:
A:= (n, k)-> add(b(n-m, k-1, m), m=ceil(n/k)..n):
seq(seq(A(d-k, k), k=1..d-1), d=1..14); # Alois P. Heinz, Jun 14 2009
(* 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 *)
Triangle begins: 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 4, 3, 1, 1, 5, 7, 6, 3, 1, 1, 6, 11, 11, 8, 4, 1, 1, 7, 16, 19, 17, 11, 4, 1, 1, 8, 22, 31, 32, 26, 13, 5, 1, 1, 9, 29, 48, 56, 54, 35, 17, 5, 1, ...
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