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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A214445 a(n) = Euler(2*n)*binomial(4*n,2*n).

Original entry on oeis.org

1, -6, 350, -56364, 17824950, -9334057876, 7308698191340, -7997684730384600, 11655857682806336550, -21824608434847162167300, 51054382673481634917970500, -145916894745749901373155951720, 500306549034304293474784779805500, -2026855002861172152744641068895033544
Offset: 0

Views

Author

Peter Luschny, Jul 18 2012

Keywords

Comments

Central column of the nonzero coefficients of the Swiss-Knife polynomials, a(n) = A153641(4*n, 2*n).

Crossrefs

Cf. A086646, A153641.

Programs

  • Sage
    def A214445(n) : return binomial(4*n,2*n)*euler_number(2*n)

Formula

a(n) = (-1)^n * A086646(2n,n). - Alois P. Heinz, Apr 27 2023

A117414 An Euler triangle.

Original entry on oeis.org

1, 0, 1, 0, 4, 1, 0, 48, 12, 1, 0, 1088, 272, 24, 1, 0, 39680, 9920, 880, 40, 1, 0, 2122752, 530688, 47104, 2160, 60, 1, 0, 156577792, 39144448, 3474688, 159488, 4480, 84, 1, 0, 15230058496, 3807514624, 337979392, 15514880, 436352, 8288, 112, 1
Offset: 0

Views

Author

Paul Barry, Mar 13 2006

Keywords

Comments

Conjecture: row sums are the Euler numbers A000364. Second column is A024255. Third column is A117415.
Here, w = w_1,w_2,...,w_(2n) is an alternating permutation if w_1 < w_2 > w_3 < ... > w_(2n-1) < w_2n. An alternating permutation is cyclically alternating if w_1 < w_(2n). Define the cyclically alternating decomposition of w in the following manner: From the set {w_2,w_4,w_6,...,w_(2n)} find the largest i such that w_(2i) > w_1. Then w_1,w_2,...,w_(2i) is the first component in the cyclically alternating decomposition of w. Repeat the process with the set {w_(2i+1),w_(2i+2),...,w_(2n)} to find the successive components. Conjecture: T(n,k) is the number of alternating permutations of [2n] with exactly k cyclically alternating components. - Geoffrey Critzer, Apr 26 2023

Examples

			Triangle begins:
  1;
  0,       1;
  0,       4,      1;
  0,      48,     12,     1;
  0,    1088,    272,    24,    1;
  0,   39680,   9920,   880,   40,  1;
  0, 2122752, 530688, 47104, 2160, 60, 1;
  ...

Links

Crossrefs

Cf. A000364, A024255, A117415, A086646, A000680.

Programs

  • Mathematica
    nn = 6; B[n_] := (2 n)!/2^n; e[z_] := Sum[z^n/B[n], {n, 0, nn}];
Map[Select[#, # > 0 &] &,Table[B[n], {n, 0, nn}] CoefficientList[
Series[e[(u - 1) z] 1/e[-z], {z, 0, nn}], {z, u}]] // Grid (* Geoffrey Critzer, Apr 26 2023 *)

Formula

From Geoffrey Critzer, Apr 26 2023: (Start)
Sum_{n>=0} Sum_{k=0..n} T(n,k)*u^k*z^n/A000680(n) = E((u-1)*z)/E(-z) Where E(z) = Sum_{n>=0} z^n/A000680(n).
Sum_{k=0..n} T(n,k)*k = A086646(n,1). (End)
Previous Showing 11-12 of 12 results.

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

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