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 1-4 of 4 results.

A153861 Triangle read by rows, binomial transform of triangle A153860.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 6, 4, 1, 4, 10, 10, 5, 1, 5, 15, 20, 15, 6, 1, 6, 21, 35, 35, 21, 7, 1, 7, 28, 56, 70, 56, 28, 8, 1, 8, 36, 84, 126, 126, 84, 36, 9, 1, 9, 45, 120, 210, 252, 210, 120, 45, 10, 1, 10, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
Offset: 0

Views

Author

Gary W. Adamson, Jan 03 2009

Keywords

Comments

Row sums = A095121: (1, 2, 6, 14, 30, 62, 126,...).
Triangle T(n,k), 0<=k<=n, read by rows, given by [1,1,-1,1,0,0,0,0,0,0,0,...] DELTA [1,0,-1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 03 2009
A123110*A007318 as infinite lower triangular matrices. - Philippe Deléham, Jan 06 2009
A153861 is the fusion of polynomial sequences p(n,x)=x^n+x^(n-1)+...+x+1 and q(n,x)=(x+1)^n; see A193722 for the definition of fusion. - Clark Kimberling, Aug 06 2011

Examples

			First few rows of the triangle are:
1;
1, 1;
2, 3, 1;
3, 6, 4, 1;
4, 10, 10, 5, 1;
5, 15, 20, 15, 6, 1;
6, 21, 35, 35, 21, 7, 1;
7, 28, 56, 70, 56, 28, 8, 1;
8, 36, 84, 126, 126, 84, 36, 9, 1;
9, 45, 120, 210, 252, 210, 120, 45, 10, 1;
...

Links

Crossrefs

Cf. A153860, A095121, A130405.
This is A137396 without the initial column and without signs.

Programs

  • Mathematica
    z = 10; c = 1; d = 1;
p[0, x_] := 1
p[n_, x_] := x*p[n - 1, x] + 1; p[n_, 0] := p[n, x] /. x -> 0;
q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193815 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]   (* A153861 *)
(* Clark Kimberling, Aug 06 2011 *)

Formula

Triangle read by rows, A007318 * A153860. Remove left two columns of Pascal's triangle and append (1, 1, 2, 3, 4, 5,...).
As a recursive operation by way of example, row 3 = (3, 6, 4, 1) =
[1, 1, 1, 0] * (flipped Pascal's triangle matrix) = [1, 3, 3, 1]
[1, 2, 1, 0]
[1, 1, 0, 0]
[1, 0, 0, 0].
(Cf. analogous operation in A130405, but in A153861 the linear multiplier = [1,1,1,...,0].)
T(n,k) = 2*T(n-1,k)+T(n-1,k-1)-T(n-2,k)-T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0)=2, T(2,1)=3, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Dec 15 2013
G.f.: (1-x+x^2+x^2*y)/((x-1)*(-1+x+x*y)). - R. J. Mathar, Aug 11 2015

A153864 Triangle read by rows, A000012 * A153860 * (A066983 * 0^(n-k)).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 2, 2, 2, 6, 3, 1, 2, 2, 6, 6, 7, 2, 2, 2, 6, 6, 14, 9, 1, 2, 2, 6, 6, 14, 18, 17, 2, 2, 2, 6, 6, 14, 18, 34, 25, 1, 2, 2, 6, 6, 14, 18, 34, 50, 43, 2, 2, 2, 6, 6, 14, 18, 34, 50, 86, 67
Offset: 0

Views

Author

Gary W. Adamson, Jan 03 2009

Keywords

Comments

Row sums = A066629: (1, 2, 5, 8, 15, 24, 41, 66, 109,...).
Right border = A066983: (1, 1, 1, 3, 3, 7, 9, 17,...).

Examples

			First few rows of the triangle =
1;
1, 1;
2, 2, 1;
1, 2, 2, 3;
2, 2, 2, 6, 3;
1, 2, 2, 6, 6, 7;
2, 2, 2, 6, 6, 14, 9;
1, 2, 2, 6, 6, 14, 18, 17;
2, 2, 2, 6, 6, 14, 18, 34, 25;
1, 2, 2, 6, 6, 14, 18, 34, 50, 43;
...

Crossrefs

Cf. A153860, A066983, A066629

Formula

Triangle read by rows, A000012 * A153860 * (A066983 * 0^(n-k))
Given triangle A000012 * A153860 = partial sums of A153860 starting from the top.
(A066983 * 0^n-k) = an infinite lower triangular matrix with A066983 as the
main diagonal: (1, 1, 1, 3, 3, 7, 9, 17, 25,...) and the rest zeros.

A153284 a(n) = n + Sum_{j=1..n-1} (-1)^j * a(j) for n >= 2, a(1) = 1.

Original entry on oeis.org

1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1
Offset: 1

Views

Author

Walter Carlini, Dec 23 2008

Keywords

Comments

Row sums of triangle A153860. - Gary W. Adamson, Jan 03 2009
1 followed by interleaving of A000012 and A010701. - Klaus Brockhaus, Jan 04 2009

Examples

			a(1)=1, a(2)=2-a(1)=2-1=1, a(3)=3+a(2)-a(1)=3+1-1=3, a(4)=4-a(3)+a(2)-a(1)=4-3+1-1=1, a(5)=5+1-3+1-1=3, a(6)=6-3+1-3+1-1=1, a(7)=7+1-3+1-3+1-1, etc.

Links

Crossrefs

Equals A010684 with the addition of the leading term of 1
The first sequence of a family that includes A153285 and A153286
Cf. A153860.
Cf. A000012 (all 1's sequence), A010701 (all 3's sequence). - Klaus Brockhaus, Jan 04 2009

Programs

  • Magma
    S:=[ 1 ]; for n in [2..105] do Append(~S, n + &+[ (-1)^j*S[j]: j in [1..n-1] ]); end for; S; // Klaus Brockhaus, Jan 04 2009

Formula

a(n)=1 if n is 1 or even; a(n)=3 if n is odd other than 1.
G.f.: x*(1 + x + 2*x^2)/((1+x)*(1-x)). - Klaus Brockhaus, Jan 04 2009 and Oct 15 2009

A063210 Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 42 ).

Original entry on oeis.org

1, 2, 6, 6, 10, 10, 14, 14, 18, 18, 22, 22, 26, 26, 30, 30, 34, 34, 38, 38, 42, 42, 46, 46, 50, 50, 54, 54, 58, 58, 62, 62, 66, 66, 70, 70, 74, 74, 78, 78, 82, 82, 86, 86, 90, 90, 94, 94, 98, 98
Offset: 1

Views

Author

N. J. A. Sloane, Jul 10 2001

Keywords

Comments

Equals triangle A153860 * [1,2,3,...]. - Gary W. Adamson, Jan 03 2009

Links

Crossrefs

Cf. A153860. - Gary W. Adamson, Jan 03 2009

Programs

  • Mathematica
    With[{c=Range[6,102,4]},Join[{1,2},Riffle[c,c]]] (* or *) LinearRecurrence[ {1,1,-1},{1,2,6,6,10},50] (* Harvey P. Dale, Jun 22 2019 *)
  • PARI
    Vec(-x*(x^3-3*x^2-x-1)/((x-1)^2*(x+1)) + O(x^100)) \\ Colin Barker, Sep 08 2013

Formula

a(n) = (n^2 - n)/floor(n/2) for n >=2. (Excludes leading 1.) - William A. Tedeschi, Mar 20 2008
Except for the first term, a(n) = 4*(n-1) - a(n-1), (with a(2)=2). - Vincenzo Librandi, Dec 07 2010
From Colin Barker, Sep 08 2013: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 4.
G.f.: -x*(x^3-3*x^2-x-1) / ((x-1)^2*(x+1)). (End)
Showing 1-4 of 4 results.

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