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.

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A227551 Number T(n,k) of partitions of n into distinct parts with boundary size k; triangle T(n,k), n>=0, 0<=k<=A227568(n), read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 3, 1, 0, 1, 3, 2, 0, 1, 5, 2, 0, 1, 5, 4, 0, 1, 5, 6, 0, 1, 6, 7, 1, 0, 1, 6, 10, 1, 0, 1, 7, 11, 3, 0, 1, 9, 13, 4, 0, 1, 7, 18, 6, 0, 1, 8, 20, 9, 0, 1, 10, 21, 14, 0, 1, 9, 27, 16, 1, 0, 1, 10, 29, 22, 2
Offset: 0

Views

Author

Alois P. Heinz, Jul 16 2013

Keywords

Comments

The boundary size is the number of parts having fewer than two neighbors.

Examples

			T(12,1) = 1: [12].
T(12,2) = 6: [1,11], [2,10], [3,4,5], [3,9], [4,8], [5,7].
T(12,3) = 7: [1,2,3,6], [1,2,9], [1,3,8], [1,4,7], [1,5,6], [2,3,7], [2,4,6].
T(12,4) = 1: [1,2,4,5].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1;
  0, 1, 1;
  0, 1, 1;
  0, 1, 2;
  0, 1, 3;
  0, 1, 3, 1;
  0, 1, 3, 2;
  0, 1, 5, 2;
  0, 1, 5, 4;
  0, 1, 5, 6;
  0, 1, 6, 7, 1;

Links

Crossrefs

Columns k=0-10 give: A000007, A057427, A227559, A227560, A227561, A227562, A227563, A227564, A227565, A227566, A227567.
Row sums give: A000009.
Last elements of rows give: A227552.
Cf. A227345 (a version with trailing zeros), A053993, A201077, A227568, A224878 (one part of size 0 allowed).

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>1, x, 1),
      expand(`if`(i<1, 0, `if`(t>1, x, 1)*b(n, i-1, iquo(t, 2))+
      `if`(i>n, 0, `if`(t=2, x, 1)*b(n-i, i-1, iquo(t, 2)+2)))))
    end:
T:= n-> (p->seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
seq(T(n), n=0..30);
  • Mathematica
    b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t > 1, x, 1], Expand[If[i < 1, 0, If[t > 1, x, 1]*b[n, i - 1, Quotient[t, 2]] + If[i > n, 0, If[t == 2, x, 1]*b[n - i, i - 1, Quotient[t, 2] + 2]]]]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Dec 12 2016, after Alois P. Heinz *)

A186084 Triangle T(n,k) read by rows: number of 1-dimensional sandpiles (see A186085) with n grains and base length k.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 3, 4, 1, 0, 0, 0, 0, 1, 6, 5, 1, 0, 0, 0, 0, 1, 4, 10, 6, 1, 0, 0, 0, 0, 0, 3, 10, 15, 7, 1, 0, 0, 0, 0, 0, 2, 8, 20, 21, 8, 1, 0, 0, 0, 0, 0, 1, 7, 19, 35, 28, 9, 1, 0, 0, 0, 0, 0, 0, 5, 18, 40, 56, 36, 10, 1, 0, 0, 0, 0, 0, 0, 3, 16, 41, 76, 84, 45, 11, 1, 0, 0, 0, 0, 0, 0, 1, 12, 41, 86, 133, 120, 55, 12, 1
Offset: 0

Views

Author

Joerg Arndt, Feb 13 2011

Keywords

Comments

Compositions of n into k nonzero parts such that the first and last parts are 1 and the absolute difference between consecutive parts is <=1.
Row sums are A186085.
Column sums are the Motzkin numbers (A001006).
First nonzero entry in row n appears in column A055086(n).
From Joerg Arndt, Nov 06 2012: (Start)
The transposed triangle (with zeros omitted) is A129181.
For large k, the columns end in reverse([1, 1, 3, 5, 9, 14, 24, 35, ...]) for k even (cf. A053993) and reverse([1, 2, 3, 6, 10, 16, 26, 40, 60, 90, ...]) for k odd (cf. A201077).
The diagonals below the main diagonal are (apart from leading zeros), n, n*(n+1)/2, n*(n+1)*(n+2)/6, and the e-th diagonal appears to have a g.f. of the form f(x)/(1-x)^e.
(End)

Examples

			Triangle begins:
1;
0,1;
0,0,1;
0,0,1,1;
0,0,0,2,1;
0,0,0,1,3,1;
0,0,0,0,3,4,1;
0,0,0,0,1,6,5,1;
0,0,0,0,1,4,10,6,1;
0,0,0,0,0,3,10,15,7,1;
0,0,0,0,0,2,8,20,21,8,1;
0,0,0,0,0,1,7,19,35,28,9,1;

Links

Crossrefs

Cf. A186085 (sandpiles with n grains, row sums), A001006 (Motzkin numbers, column sums), A055086.
Cf. A186505 (antidiagonal sums).

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, `if`(i=1, 1, 0),
      `if`(n<0 or i<1, 0, expand(x*add(b(n-i, i+j), j=-1..1)) ))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 1)):
seq(T(n), n=0..20);  # Alois P. Heinz, Jul 24 2013
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, If[i == 1, 1, 0], If[n<0 || i<1, 0, Expand[ x*Sum[b[n-i, i+j], {j, -1, 1}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 1]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)
  • PARI
    {T(n,k)=local(A=1+x*y);for(i=1,n,A=1/(1-x*y-x^3*y^2*subst(A,y,x*y+x*O(x^n))));polcoeff(polcoeff(A,n,x),k,y)}
/* Paul D. Hanna */

Formula

G.f. A(x,y) satisfies: A(x,y) = 1/(1 - x*y - x^3*y^2*A(x, x*y) ). [Paul D. Hanna, Feb 22 2011]
G.f.: (formatting to make the structure apparent)
A(x,y) = 1 /
(1 - x^1*y / (1 - x^2*y / (1 - x^5*y^2 /
(1 - x^3*y / (1 - x^4*y / (1 - x^9*y^2 /
(1 - x^5*y / (1 - x^6*y / (1 - x^13*y^2 /
(1 - x^7*y / (1 - x^8*y / (1 - x^17*y^2 / (1 -...)))))))))))))
(continued fraction). [Paul D. Hanna]
G.f.: A(x,y) = 1/(1-x*y - x^3*y^2/(1-x^2*y - x^5*y^2/(1-x^3*y - x^7*y^2/(1 -...)))) (continued fraction). [Paul D. Hanna]

A227552 Number of partitions of n into distinct parts with maximal boundary size.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 1, 2, 2, 4, 6, 1, 1, 3, 4, 6, 9, 14, 1, 2, 3, 5, 8, 11, 17, 24, 1, 1, 3, 5, 8, 11, 18, 24, 35, 49, 1, 2, 3, 6, 9, 14, 21, 30, 42, 60, 81, 1, 1, 3, 5, 9, 13, 21, 29, 43, 60, 84, 113, 156, 1, 2, 3, 6, 10, 15, 24, 35, 50, 71, 99, 134, 184, 246
Offset: 0

Views

Author

Alois P. Heinz, Jul 16 2013

Keywords

Comments

The boundary size is the number of parts having less than two neighbors.

Links

Crossrefs

Last elements of rows of A227551.
Last nonzero elements of rows of A227345.
Cf. A053993, A201077.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>1, x, 1),
      expand(`if`(i<1, 0, `if`(t>1, x, 1)*b(n, i-1, iquo(t, 2))+
      `if`(i>n, 0, `if`(t=2, x, 1)*b(n-i, i-1, iquo(t, 2)+2)))))
    end:
a:= n-> (p->coeff(p, x, degree(p)))(b(n$2, 0)):
seq(a(n), n=0..100);
  • Mathematica
    b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t>1, x, 1], Expand[If[i<1, 0, If[t>1, x, 1]*b[n, i-1, Quotient[t, 2]] + If[i>n, 0, If[t==2, x, 1] * b[n-i, i-1, Quotient[t, 2]+2]]]]]; a[n_] := Function [p, Coefficient[p, x, Exponent[p, x]]][b[n, n, 0]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 15 2017, translated from Maple *)

Formula

a(n) = A227551(n,A227568(n)).
Showing 1-3 of 3 results.

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