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.

A211232 Irregular triangle read by rows: T(n,k) is the k-th generalized Eulerian number of order n and degree 2, for n >= 1 (the rows start at k=1).

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 7, 0, -7, -1, 1, 12, -12, -56, -12, 12, 1, 1, 21, -67, -284, 0, 284, 67, -21, -1, 1, 38, -273, -1170, 753, 3408, 753, -1170, -273, 38, 1, 1, 71, -982, -4241, 8562, 29055, 0, -29055, -8562, 4241, 982, -71, -1, 1, 136, -3314, -13888, 66335, 199616, -106113, -464880, -106113, 199616, 66335, -13888, -3314, 136, 1
Offset: 1

Views

Author

N. J. A. Sloane, Apr 05 2012

Keywords

Examples

			Triangle begins
  1,  2;
  1,  4,    1;
  1,  7,    0,    -7,  -1;
  1, 12,  -12,   -56, -12,   12,   1;
  1, 21,  -67,  -284,   0,  284,  67,   -21,   -1;
  1, 38, -273, -1170, 753, 3408, 753, -1170, -273, 38, 1;
  ...

Links

Crossrefs

Row sums of even rows are A047681; row sums of odd rows are zero for n > 1.
Cf. A008292, A211233, A211234, A211235.

Programs

  • PARI
    T(n,r=2)={my(R=vector(n)); R[1]=[1..r]; for(n=2, n, my(u=R[n-1]); R[n]=vector(r*n-1, k, sum(j=0, r, (k - j*n)*if(k>j && k-j<=#u, u[k-j], 0)))); R}
{ my(A=T(7)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, May 18 2020

Formula

From Andrew Howroyd, May 18 2020: (Start)
T(n,k) = k*T(n-1,k) - (n-k)*T(n-1,k-1) - (2*n-k)*T(n-1,k-2) for n > 1.
A047681(n) = Sum_{k>=1} T(2*n, k).
(End)

Extensions

Terms a(38) and beyond from Andrew Howroyd, May 18 2020

A211234 Triangle read by rows: T(n,k) is the k-th generalized Eulerian number of order n and degree 4, n >= 1.

Original entry on oeis.org

1, 2, 3, 4, 1, 4, 10, 20, 10, 4, 1, 1, 7, 27, 77, 57, 0, -57, -77, -27, -7, -1, 1, 12, 69, 272, 221, -272, -1084, -1688, -1084, -272, 221, 272, 69, 12, 1, 1, 21, 176, 936, 625, -3288, -11868, -21023, -16223, 0, 16223, 21023, 11868, 3288, -625, -936, -176, -21, -1
Offset: 1

Views

Author

N. J. A. Sloane, Apr 05 2012

Keywords

Examples

			Triangle begins:
  1, 2,  3,  4;
  1, 4, 10, 20, 10, 4,   1;
  1, 7, 27, 77, 57, 0, -57, -77, -27, -7, -1;
  ...

Links

Crossrefs

Row sums of even rows are A047683; row sums of odd rows are zero for n > 1.
Cf. A008292, A211232, A211233, A211235.

Programs

  • PARI
    T(n,r=4)={my(R=vector(n)); R[1]=[1..r]; for(n=2, n, my(u=R[n-1]); R[n]=vector(r*n-1, k, sum(j=0, r, (k - j*n)*if(k>j && k-j<=#u, u[k-j], 0)))); R}
{ my(A=T(5)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, May 18 2020

Formula

A047683(n) = Sum_{k>=1} T(2*n, k). - Andrew Howroyd, May 18 2020

Extensions

More terms from Franck Maminirina Ramaharo, Nov 30 2018
a(20) corrected by Andrew Howroyd, May 18 2020

A047682 a(n) = 4^(2*n)*(4^(2*n)-1)*Bernoulli(2*n)/(2*n).

Original entry on oeis.org

20, -544, 66560, -17895424, 8329625600, -5937093935104, 6004799480791040, -8176700159200067584, 14421891569272362106880, -31983597922505761818148864, 87107695717210805652024197120, -285816431841945942589104606674944, 1112035502745012017553352712530165760
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 283.

Links

Crossrefs

Column 3 of A241066.
Row sums of even-indexed rows of A211233.

Programs

  • Mathematica
    f[n_]:=Module[{c=2n,x},x=4^c;(x(x-1)BernoulliB[c])/c]; Array[f,20] (* Harvey P. Dale, Feb 19 2013 *)
  • PARI
    a(n) = 4^(2*n)*(4^(2*n)-1)*bernfrac(2*n)/(2*n) \\ Andrew Howroyd, May 18 2020

Extensions

Offset corrected and terms a(13) and beyond from Andrew Howroyd, May 18 2020
Showing 1-3 of 3 results.

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

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