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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A332997 a(n) = A000120(A332995(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 3, 4, 1, 2, 1, 1, 2, 3, 2, 3, 1, 1, 2, 3, 1, 1, 2, 2, 2, 3, 2, 3, 3, 3, 4, 5, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 3, 4, 2, 2, 3, 3, 1, 2, 1, 2, 2, 2, 3, 4, 1, 2, 1, 1, 2, 3, 2, 3, 2, 2, 3, 4, 2, 2, 3, 3, 3, 4, 3, 4, 4, 4, 5, 6, 1, 2, 1, 1, 2, 3, 2, 3, 1, 1
Offset: 0

Views

Author

Antti Karttunen, Mar 05 2020

Keywords

Comments

It seems that a(A055010(n)) = n for all n >= 0, and apart from n=1, A055010 seems to give the first occurrence of each n in this sequence.

Links

Crossrefs

Cf. A000120, A332817, A332897, A332995, A332998.
Cf. A052940, A055010.

Programs

Formula

a(n) = A000120(A332995(n)) = A332897(A332817(n)).
a(n) = A000120(n) - A332998(n).

A124732 Triangle P*M, where P is the Pascal triangle written as an infinite lower triangular matrix and M is the infinite bidiagonal matrix with (1,2,1,2,...) in the main diagonal and (2,1,2,1,...) in the subdiagonal.

Original entry on oeis.org

1, 3, 2, 5, 5, 1, 7, 9, 5, 2, 9, 14, 14, 9, 1, 11, 20, 30, 25, 7, 2, 13, 27, 55, 55, 27, 13, 1, 15, 35, 91, 105, 77, 49, 9, 2, 17, 44, 140, 182, 182, 140, 44, 17, 1, 19, 54, 204, 294, 378, 336, 156, 81, 11, 2, 21, 65, 285, 450, 714, 714, 450, 285, 65, 21, 1, 23, 77, 385, 660
Offset: 1

Views

Author

Gary W. Adamson & Roger L. Bagula, Nov 05 2006

Keywords

Comments

Row sums = A052940: (1, 5, 11, 23, 47, 95, ...).

Examples

			First 3 rows of the triangle are (1; 3,2; 5,5,1) since [1,0,0; 1,1,0; 1,2,1] * [1,0,0; 2,2,0; 0,1,1] = [1,0,0; 3,2,0; 5,5,1].
First few rows of the triangle are:
   1;
   3,   2;
   5,   5,   1;
   7,   9,   5,   2;
   9,  14,  14,   9,   1;
  11,  20,  30,  25,   7,   2;
  13,  27,  55,  55,  27,  13,   1;
  15,  35,  91, 105,  77,  49,   9,   2;
  ...

Crossrefs

Cf. A124730, A052940.

Programs

  • Maple
    T:=(n,k)->binomial(n,k)*(3*n-(-1)^k*(n-2*k))/2/n: for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

Formula

T(n,k) = binomial(n,k)*(3n-(-1)^k*(n-2*k))/(2n) (1 <= k <= n).

Extensions

Edited by N. J. A. Sloane, Nov 24 2006

A140182 Binomial transform of an infinite bidiagonal matrix with (1,3,1,3,1,3,...) in the main diagonal, (1,1,1,...) in the subdiagonal, the rest zeros.

Original entry on oeis.org

1, 2, 3, 3, 7, 1, 4, 12, 4, 3, 5, 18, 10, 13, 1, 6, 25, 20, 35, 6, 3, 7, 33, 35, 75, 21, 19, 1, 8, 42, 56, 140, 56, 70, 8, 3, 9, 52, 84, 238, 126, 196, 36, 25, 1, 10, 63, 120, 378, 252, 462, 120, 117, 10, 3, 11, 75, 165, 570, 462, 966, 330, 405, 55, 31, 1
Offset: 0

Views

Author

Gary W. Adamson, May 11 2008

Keywords

Comments

Row sums = A052940: (1, 5, 11, 23, 47, 95, ...).

Examples

			First few rows of the triangle are:
  1;
  2,  3;
  3,  7,  1;
  4, 12,  4,  3;
  5, 18, 10, 13,  1;
  6, 25, 20, 35,  6,  3;
  7, 33, 35, 75, 21, 19,  1;
  ...

Crossrefs

Cf. A052940.

Programs

  • Maple
    T:=proc(n,k) if `mod`(k,2)=0 then binomial(n+1,k+1) else 2*binomial(n,k)+binomial(n+1,k+1) end if end proc: for n from 0 to 10 do seq(T(n,k),k=0..n) end do; # yields sequence in triangular form - Emeric Deutsch, May 18 2008

Formula

A007318 as an infinite lower triangular matrix * a bidiagonal matrix with (1,3,1,3,1,3,...) in the main diagonal, (1,1,1,...) in the subdiagonal and the rest zeros.
From Emeric Deutsch, May 18 2008: (Start)
T(n, 2k) = binomial(n+1, 2k+1);
T(n, 2k+1) = 2*binomial(n, 2k+1) + binomial(n+1, 2k+2). (End)

A213668 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with k vertices of the graph G(n) consisting of a pair of endvertices joined by n internally disjoint paths of length 2 (the n-ary generalized theta graph THETA_{2,2,...2}; n>=1, 1<=k<=n+2).

Original entry on oeis.org

1, 3, 1, 0, 6, 4, 1, 0, 7, 10, 5, 1, 0, 9, 16, 15, 6, 1, 0, 11, 25, 30, 21, 7, 1, 0, 13, 36, 55, 50, 28, 8, 1, 0, 15, 49, 91, 105, 77, 36, 9, 1, 0, 17, 64, 140, 196, 182, 112, 45, 10, 1, 0, 19, 81, 204, 336, 378, 294, 156, 55, 11, 1
Offset: 1

Views

Author

Emeric Deutsch, Jul 06 2012

Keywords

Comments

Row n>=2 contains n+1 entries.
Sum of entries in row n=3*2^n-1 = A052940(n) = A153893(n) = A055010(n+1) = A083329(n+1).
The graph G(n) is the join of the graph consisting of 2 isolated vertices and the graph consisting of n isolated vertices. Then the expression of the domination polynomial follows from Theorem 12 of the Akbari et al. reference.

Examples

			Row 1 is 1,3,1 because the graph G(1) is the path abc; there are 1 dominating subset of size 1 ({b}), 3 dominating subsets of size 2 ({a,b}, {a,c}, {b,c}), and 1 dominating subset of size 3 ({a,b,c}).
Row 2 is 0,6,4,1 because the graph G(2) is the cycle a-b-c-d-a and has dominating subsets ab, ac, ad, bc, bd, cd, abc, abd, acd, bcd, and abcd (see A212634).
Triangle starts:
1,3,1;
0,6,4,1;
0,7,10,5,1;
0,9,16,15,6,1;

References

  • S. Akbari, S. Alikhani, and Y. H. Peng, Characterization of graphs using domination polynomials, European J. Comb., 31, 2010, 1714-1724.

Links

Crossrefs

Cf. A052940, A153893, A055010, A083329, A212634

Programs

  • Maple
    p := proc (n) options operator, arrow: ((1+x)^n-1)*((1+x)^2-1)+x^n+x^2 end proc: for n to 12 do seq(coeff(p(n), x, k), k = 1 .. n+2) end do; # yields sequence in triangular form
  • Mathematica
    T[n_, k_] := SeriesCoefficient[((1+x)^n-1) ((1+x)^2-1)+x^n+x^2, {x, 0, k}];
Table[T[n, k], {n, 1, 9}, {k, 1, n+2}] // Flatten (* Jean-François Alcover, Dec 06 2017 *)

Formula

The generating polynomial of row n is p(n)=((1+x)^n-1)*((1+x)^2-1)+x^n+x^2; by definition, p(n) is the domination polynomial of the graph G(n).
Bivariate g.f.: x*z/(1-x*z)-2*x*z/(1-z)+x*z*(1+x)*(2+x)/(1-z-x*z).
T(n,3)=n^2 for n!=3.
Previous Showing 11-14 of 14 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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