cp's OEIS Frontend

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.

Showing 1-2 of 2 results.

A227682 G.f.: exp( Sum_{n>=1} x^n / (n*(1-x)^n * (1-x^n)) ).

Original entry on oeis.org

1, 1, 3, 7, 16, 35, 76, 162, 342, 715, 1484, 3060, 6278, 12824, 26102, 52969, 107224, 216601, 436798, 879584, 1769117, 3554726, 7136736, 14318524, 28711315, 57544864, 115290624, 230910993, 462362571, 925610398, 1852669016, 3707705019, 7419275371, 14844857959
Offset: 0

Views

Author

Paul D. Hanna, Jul 19 2013

Keywords

Comments

Number of compositions of n with k sorts of parts k where the sorts of parts are nondecreasing through the composition, see example. - Joerg Arndt, May 01 2014

Examples

			From _Joerg Arndt_, May 01 2014: (Start)
The a(5) = 35 compositions as described in the first comment are (here p:s stands for a part p of sort s)
01:  [ 1:0  1:0  1:0  1:0  1:0  ]
02:  [ 1:0  1:0  1:0  2:0  ]
03:  [ 1:0  1:0  1:0  2:1  ]
04:  [ 1:0  1:0  2:0  1:0  ]
05:  [ 1:0  1:0  3:0  ]
06:  [ 1:0  1:0  3:1  ]
07:  [ 1:0  1:0  3:2  ]
08:  [ 1:0  2:0  1:0  1:0  ]
09:  [ 1:0  2:0  2:0  ]
10:  [ 1:0  2:0  2:1  ]
11:  [ 1:0  2:1  2:1  ]
12:  [ 1:0  3:0  1:0  ]
13:  [ 1:0  4:0  ]
14:  [ 1:0  4:1  ]
15:  [ 1:0  4:2  ]
16:  [ 1:0  4:3  ]
17:  [ 2:0  1:0  1:0  1:0  ]
18:  [ 2:0  1:0  2:0  ]
19:  [ 2:0  1:0  2:1  ]
20:  [ 2:0  2:0  1:0  ]
21:  [ 2:0  3:0  ]
22:  [ 2:0  3:1  ]
23:  [ 2:0  3:2  ]
24:  [ 2:1  3:1  ]
25:  [ 2:1  3:2  ]
26:  [ 3:0  1:0  1:0  ]
27:  [ 3:0  2:0  ]
28:  [ 3:0  2:1  ]
29:  [ 3:1  2:1  ]
30:  [ 4:0  1:0  ]
31:  [ 5:0  ]
32:  [ 5:1  ]
33:  [ 5:2  ]
34:  [ 5:3  ]
35:  [ 5:4  ]
(End)

Links

Crossrefs

Cf. A227681, A238349, A253829.

Programs

  • Mathematica
    Flatten[{1,Table[SeriesCoefficient[Exp[Sum[x^k / (k*(1-x)^k * (1-x^k)),{k,1,n}]],{x,0,n}], {n,1,40}]}] (* Vaclav Kotesovec, May 01 2014 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/(m*(1-x)^m*(1-x^m +x*O(x^n))) )), n)}
for(n=0, 50, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n+1, x^m*sumdiv(m, d, 1/(1-x +x*O(x^n))^d/d) )), n)}
for(n=0, 50, print1(a(n), ", "))

Formula

G.f.: exp( Sum_{n>=1} x^n * Sum_{d|n} 1/(d*(1-x)^d) ).
G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 16*x^4 + 35*x^5 + 76*x^6 + 162*x^7 +...
where
log(A(x)) = x/((1-x)*(1-x)) + x^2/(2*(1-x)^2*(1-x^2)) + x^3/(3*(1-x)^3*(1-x^3)) + x^4/(4*(1-x)^4*(1-x^4)) + x^5/(5*(1-x)^5*(1-x^5)) +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 13*x^3/3 + 29*x^4/4 + 56*x^5/5 + 107*x^6/6 + 197*x^7/7 + 365*x^8/8 + 679*x^9/9 + 1280*x^10/10 +...
a(n) = A238350(n*(n+3)/2,n), a(n) is the number of compositions of n*(n+3)/2 with exactly n fixed points. - Alois P. Heinz, Apr 11 2014
a(n) ~ c * 2^n, where c = 1/(2*A048651) = 1.73137330972753180576... - Vaclav Kotesovec, May 01 2014
G.f.: Product {n >= 1} 1/(1 - x^n/(1 - x)). Row sums of A253829. - Peter Bala, Jan 20 2015

A253830 Triangular array with g.f. Product_{n >= 1} (1 + (x*z)^n/(1 - z)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 2, 0, 1, 1, 4, 3, 3, 0, 1, 1, 5, 4, 5, 4, 0, 1, 1, 6, 5, 7, 8, 4, 0, 1, 1, 7, 6, 9, 13, 10, 6, 0, 1, 1, 8, 7, 11, 19, 16, 13, 8, 0, 1, 1, 9, 8, 13, 26, 23, 22, 18, 10, 0, 1, 1, 10, 9, 15, 34, 31, 33, 31, 25, 12, 0, 1, 1, 11, 10, 17, 43, 40, 46, 47, 47, 30, 15
Offset: 0

Views

Author

Peter Bala, Jan 20 2015

Keywords

Comments

A colored composition of n is defined as a composition of n where each part p comes in one of p colors (denoted by an integer from 1 to p) and the color numbers are nondecreasing through the composition.
The color numbers thus form a partition, called the color partition, of some integer. For example, 2(c1) + 1(c1) + 5(c3) + 4(c3) + 6(c4) is a colored composition of 18 (the color number of a part is shown after the part prefaced by the letter c) and has the associated color partition (1,1,3,3,4).
T(n,k) equals the number of colored compositions of n whose associated color partition has distinct parts with sum (called the weight of the color partition) equal to k. An example is given below.

Examples

			Triangle begins
n\k| 0  1  2  3  4  5  6  7
= = = = = = = = = = = = = =
0  | 1
1  | 0  1
2  | 0  1  1
3  | 0  1  1  2
4  | 0  1  1  3  2
5  | 0  1  1  4  3  3
6  | 0  1  1  5  4  5  4
7  | 0  1  1  6  5  7  8  4
...
Row 5 polynomial: x + x^2 + 4*x^3 + 3*x*4 + 3*x^5.
Colored             x^(weight of color partition)
compositions
of 5 with
distinct colored
parts
= = = = = = = = = = = = = = = = = = = = = =
5(c1)                        x
5(c2)                        x^2
1(c1) + 4(c2)                x^3
2(c1) + 3(c2)                x^3
3(c1) + 2(c2)                x^3
5(c3)                        x^3
1(c1) + 4(c3)                x^4
2(c1) + 3(c3)                x^4
5(c4)                        x^4
1(c1) + 4(c4)                x^5
2(c2) + 3(c3)                x^5
5(c5)                        x^5

Links

Crossrefs

Cf. A008289, A126348 (row sums), A253829.

Programs

  • Maple
    G := product(1+(x*z)^j/(1-z), j = 1 .. 12): Gser := simplify(series(G, z = 0, 14)): for n to 12 do P[n] := coeff(Gser, z^n) end do: for n to 12 do seq(coeff(P[n], x^j), j = 1 .. n) end do;

Formula

G.f.: G(x,z) := Product_{n >= 1} (1 + (x*z)^n/(1 - z)) = 1 + x*z + (x + x^2)*z^2 + (x + x^2 + 2*x^3)*z^3 + (x + x^2 + 3*x^3 + 2*x^4)*z^4 + .... Note, G(x*z/(x - 1),(x - 1)/x) is the generating function of A008289.
T(n,k) = Sum_{i = 1..k} binomial(i+n-k-1,i-1)*A008289(k,i).
Row sums are A126348.
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