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 1-3 of 3 results.

A128255 A114219(signed) * A007318.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 2, 6, 5, 1, 3, 10, 13, 7, 1, 3, 15, 27, 23, 9, 1, 4, 21, 48, 57, 36, 1, 4, 28, 78, 118, 104, 52, 13, 1, 5, 36, 118, 218, 246, 172, 71, 15, 1, 5, 45, 170, 370, 510, 458, 265, 93, 17, 1
Offset: 1

Views

Author

Gary W. Adamson, Feb 20 2007

Keywords

Comments

Row sums = A059570: (1, 2, 6, 14, 34, 78, 178,...).

Examples

			First few rows of the triangle are:
1;
1, 1;
2, 3, 1;
2, 6, 5, 1;
3, 10, 13, 7, 1;
3, 15, 27, 23, 9, 1;
4, 21, 48, 57, 36, 11, 1;
...

Crossrefs

Cf. A103450, A114219, A007318, A059570, A097807, A128229.

Formula

Let the signed version of A114219 {1; 0,1; 0,1,1; 0,-1,2,1; 0,1,-2,-3,4;...} = M; and P = Pascal's triangle, A007318. Then A128255 = A114219(signed) * A007318.

A114220 a(n) = Sum_{k=0..floor(n/2)} (k - (k-1)*0^(n-2*k)).

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 4, 6, 7, 10, 11, 15, 16, 21, 22, 28, 29, 36, 37, 45, 46, 55, 56, 66, 67, 78, 79, 91, 92, 105, 106, 120, 121, 136, 137, 153, 154, 171, 172, 190, 191, 210, 211, 231, 232, 253, 254, 276, 277, 300, 301, 325, 326, 351, 352, 378, 379, 406, 407, 435, 436
Offset: 0

Views

Author

Paul Barry, Nov 18 2005

Keywords

Comments

Diagonal sums of A114219.
It appears that the sequence terms from a(4) = 2 onwards are the exponents in the expansion of Sum_{n >= 0} q^(3*n-3) * Product_{k >= n} 1 - q^(2*k) = 1 - q^2 + q^3 - q^4 + q^6 - q^7 + q^10 - q^11 + - .... - Peter Bala, Jan 17 2025

Links

Crossrefs

Cf. A055802, A114219.
Column k=2 of A309049 (for n>0).

Programs

  • Magma
    [(2*n^2-2*n+7 + (9-2*n)*(-1)^n)/16: n in [0..80]]; // G. C. Greubel, Oct 21 2024
  • Mathematica
    CoefficientList[Series[(1-x-x^2+2x^3)/((1-x)(1-x^2)^2), {x,0,80}],x] (* Harvey P. Dale, Mar 24 2011 *)
  • SageMath
    def A114220(n): return (2*n^2-2*n+7 + (9-2*n)*(-1)^n)//16
[A114220(n) for n in range(81)] # G. C. Greubel, Oct 21 2024

Formula

G.f.: (1-x-x^2+2x^3)/((1-x)*(1-x^2)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (2*n^2-2*n+7 + (9-2*n)*(-1)^n)/16.
a(n) = A055802(n+1), n > 1. - R. J. Mathar, Aug 11 2008
E.g.f.: (1/16)*((9 + 2*x)*exp(-x) + (7 + 2*x^2)*exp(x)). - G. C. Greubel, Oct 21 2024

A103450 A figurate number triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 12, 7, 1, 1, 9, 22, 22, 9, 1, 1, 11, 35, 50, 35, 11, 1, 1, 13, 51, 95, 95, 51, 13, 1, 1, 15, 70, 161, 210, 161, 70, 15, 1, 1, 17, 92, 252, 406, 406, 252, 92, 17, 1, 1, 19, 117, 372, 714, 882, 714, 372, 117, 19, 1, 1, 21, 145, 525, 1170, 1722, 1722, 1170, 525, 145, 21, 1
Offset: 0

Views

Author

Paul Barry, Feb 06 2005

Keywords

Comments

Row coefficients are the absolute values of the coefficients of the characteristic polynomials of the n X n matrices A(n) with A(n){i,i} = 2, i>0, A(n){i,j} = 1, otherwise (starts with (0,0) position).
The triangle can be generated by the matrix multiplication A007318 * A114219s, where A114219s = 0; 0,1; 0,1,1; 0,-1,2,1; 0,1,-2,3,1; 0,-1,2,-3,4,1; ... = A097807 * A128229 is a signed variant of A114219. - Gary W. Adamson, Feb 20 2007

Examples

			From _Roger L. Bagula_, Oct 21 2008: (Start)
The triangle begins:
  1;
  1,  1;
  1,  3,   1;
  1,  5,   5,   1;
  1,  7,  12,   7,   1;
  1,  9,  22,  22,   9,   1;
  1, 11,  35,  50,  35,  11,   1;
  1, 13,  51,  95,  95,  51,  13,   1;
  1, 15,  70, 161, 210, 161,  70,  15,   1;
  1, 17,  92, 252, 406, 406, 252,  92,  17,  1;
  1, 19, 117, 372, 714, 882, 714, 372, 117, 19, 1; ... (End)

Links

Crossrefs

Row sums are A045623.
Columns include: A000326, A002412, A002418, A005408.
Cf. A000045, A208354.

Programs

  • Magma
    A103450:= func< n,k | k eq 0 select 1 else Binomial(n, k)*(k*(n-k) + n)/n >;
[A103450(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 17 2021
  • Mathematica
    (* First program *)
p[x_, n_]:= p[x, n]= If[n==0, 1, (-1+x)^(n-2)*(1 -(n+1)*x +x^2)];
T[n_, k_]:= T[n,k]= (-1)^(n+k)*SeriesCoefficient[p[x, n], {x, 0, k}];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* Roger L. Bagula and Gary W. Adamson, Oct 21 2008 *)(* corrected by G. C. Greubel, Jun 17 2021 *)
(* Second program *)
T[n_, k_]:= If[k==0, 1, Binomial[n, k]*(n*(k+1) -k^2)/n];
Table[T[n, k], {n,0,16}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 17 2021 *)
  • Sage
    def A103450(n, k): return 1 if (k==0) else binomial(n, k)*(k*(n-k) + n)/n
flatten([[A103450(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 17 2021

Formula

T(n, k) = binomial(n-1, k-1)*(k*(n-k) + n)/k with T(n, 0) = 1.
T(n, k) = T(n-1, k-1) + T(n-1, k) + binomial(n-2, k-1) with T(n, 0) = 1.
Column k is generated by (1+k*x)*x^k/(1-x)^(k+1).
Rows are coefficients of the polynomials P(0, x) = 1, P(n, x) = (1+x)^(n-2)*(1 +(n+1)*x + x^2) for n>0.
T(n,k) = Sum_{j=0..n} binomial(k, k-j)*binomial(n-k, j)*(j+1). - Paul Barry, Oct 28 2006
A signed version arises from the coefficients of the polynomials defined by: p(x, 0) = 1, p(x, 1) = (-1 +x), p(x, 2) = (1 -3*x +x^2), p(x,n) = (-1 +x)^(n-2)*(1 - (n + 1)*x + x^2); T(n, k) = (-1)^(n+k)*coefficient of x^k of ( p(x,n) ). - Roger L. Bagula and Gary W. Adamson, Oct 21 2008
T(2*n+1, n) = A141222(n). - Emanuele Munarini, Jun 01 2012 [corrected by Werner Schulte, Nov 27 2021]
G.f.: is 1 / ( (1-q*x/(1-x)) * (1-x/(1-q*x)) ). - Joerg Arndt, Aug 27 2013
Sum_{k=0..floor(n/2)} T(n-k, k) = (1/5)*((-n+5)*Fibonacci(n+1) + (3*n- 2)*Fibonacci(n)) = A208354(n). - G. C. Greubel, Jun 17 2021
T(2*n, n) = A000984(n) * (n + 2) / 2 for n >= 0. - Werner Schulte, Nov 27 2021
Showing 1-3 of 3 results.

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