A275801 -id:A275801 - cp's OEIS frontend

A005799 Generalized Euler numbers of type 2^n.

1, 1, 2, 10, 104, 1816, 47312, 1714000, 82285184, 5052370816, 386051862272, 35917232669440, 3996998043812864, 524203898507631616, 80011968856686405632, 14061403972845412526080, 2818858067801804443910144

Offset: 0

N. J. A. Sloane, Simon Plouffe

Also, a(n) equals the number of alternating permutations (p(1),...,p(2n)) of the multiset {1,1,2,2,...,n,n} satisfying p(1) <= p(2) > p(3) <= p(4) > p(5) <= ... <= p(2n). Hence, A275801(n) <= a(n) <= A275829(n). - Max Alekseyev, Aug 10 2016
This is the BinomialMean transform of A000364 (see A075271 for definition of transform). - John W. Layman, Dec 04 2002
This sequence appears to be middle column in Poupard's triangle A008301.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Peter Bala, Some S-fractions related to the expansions of sin(ax)/cos(bx) and cos(ax)/cos(bx)
Shishuo Fu, Zhicong Lin, and Zhi-Wei Sun, Proof of several conjectures relating permanents to Combinatorial sequences, arXiv:2109.11506v3 [math.CO], 2021-2023.
Shishuo Fu, Zhicong Lin, and Zhi-Wei Sun, Permanent identities, combinatorial sequences, and permutation statistics, Advances in Applied Mathematics, Volume 163, Part A, 102789 (2025).
Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), no. 2, 257-285.
H. Prodinger, On Touchard's continued fraction and extensions: combinatorics-free, self-contained proofs , arXiv:1102.5186 [math.CO], 2011.
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Right edge of triangle A210108.

Cf. A000364, A008301, A275801, A275829, A154283, A000657.

T := proc(n, k) option remember;
if n < 0 or k < 0 then 0
elif n = 0 then euler(k, 1)
else T(n-1, k+1) - T(n-1, k) fi end:
a := n -> (-2)^n*T(n, n); seq(a(n), n=0..16); # Peter Luschny, Aug 23 2017

a[n_] := Sum[Binomial[n, i]Abs[EulerE[2i]], {i, 0, n}]/2^n

a(n) = (1/2^n) * Sum_{i=0..n} binomial(n, i) * A000364(i).
From Sergei N. Gladkovskii, Dec 27 2012, Oct 11 2013, Oct 27 2013, Jan 08 2014: (Start) Continued fractions:
G.f.: A(x) = 1/G(0) where G(k) = 1 - x*(k+1)*(2*k+1)/(1 - x*(k+1)*(2*k+1)/G(k+1)).
G.f.: Q(0)/(1-x), where Q(k) = 1 - x^2*(k+1)^2*(2*k+1)^2/(x^2*(k+1)^2*(2*k+1)^2 - (4*x*k^2 + 2*x*k + x - 1)*( 4*x*k^2 + 10*x*k + 7*x - 1)/Q(k+1)).
G.f.: R(0), where R(k) = 1 - x*(2*k+1)*(k+1)/(x*(2*k+1)*(k+1) - 1/(1 - x*(2*k+1)*(k+1)/(x*(2*k+1)*(k+1) - 1/R(k+1)))).
G.f.: 2/(x*Q(0)), where Q(k) = 2/x - 1 - (2*k+1)^2/(1 - (2*k+2)^2/Q(k+1)). (End)
a(n) ~ 2^(3*n+3) * n^(2*n+1/2) / (exp(2*n) * Pi^(2*n+1/2)). - Vaclav Kotesovec, May 30 2015
a(n) = 2^n * Sum_{k=0..n} (-1)^k*binomial(n, k)*euler(n+k, 1). - Peter Luschny, Aug 23 2017
From Peter Bala, Dec 21 2019: (Start)
O.g.f. as a continued fraction: 1/(1 - x/(1 - x/(1 - 6*x/(1 - 6*x/(1 - 15*x/(1 - 15*x/(1 - ... - n*(2*n-1)*x/(1 - n*(2*n-1)*x/(1 - ...))))))))) - apply Bala, Proposition 3, with a = 0, b = 1 and replace x with x/2.
Conjectures:
E.g.f. as a continued fraction: 2/(2 - (1-exp(-4*t))/(2 - (1-exp(-8*t))/(2 - (1-exp(-12*t))/(2 - ... )))) = 1 + t + 2*t^2/2! + 10*t^3/3! + 104*t^4/4! + ....
Cf. A000657. [added April 18 2024: for a proof of this conjecture see Fu et al., Section 4.3.]
a(n) = (-2)^(n+1)*Sum_{k = 0..floor((n-1)/2)} binomial(n,2*k+1)*(2^(2*n-2*k) - 1)*Bernoulli(2*n-2*k)/(2*n-2*k) for n >= 1. (End)

Edited by Dean Hickerson, Dec 10 2002

A275784 Number A(n,k) of up-down sequences with k copies each of 1,2,...,n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 4, 5, 1, 1, 0, 1, 12, 53, 16, 1, 1, 0, 1, 36, 761, 936, 61, 1, 1, 0, 1, 120, 12661, 87336, 25325, 272, 1, 1, 0, 1, 400, 229705, 9929000, 18528505, 933980, 1385, 1, 1, 0, 1, 1400, 4410665, 1267945800, 17504311533, 6376563600, 45504649, 7936, 1

Offset: 0

Alois P. Heinz, Aug 12 2016

			A(4,1) = 5: 1324, 1423, 2314, 2413, 3412.
A(3,2) = 4: 121323, 132312, 231213, 231312.
A(3,3) = 12: 121313232, 121323132, 121323231, 131213232, 132312132, 132323121, 231213132, 231213231, 231312132, 231323121, 232312131, 232313121.
A(2,4) = 1: 12121212.
Square array A(n,k) begins:
  1,   1,      1,          1,              1,              1, ...
  1,   1,      0,          0,              0,              0, ...
  1,   1,      1,          1,              1,              1, ...
  1,   2,      4,         12,             36,            120, ...
  1,   5,     53,        761,          12661,         229705, ...
  1,  16,    936,      87336,        9929000,     1267945800, ...
  1,  61,  25325,   18528505,    17504311533, 19126165462061, ...
  1, 272, 933980, 6376563600, 59163289699260, ...

Alois P. Heinz, Antidiagonals n = 0..15, flattened

Columns k=0-3 give: A000012, A000111, A275801, A276636.
Rows n=2-5 give: A000012, A241530, A036916, A276637.
Cf. A269129, A331562.

b:= proc(n, l) option remember; `if`(l=[], 1, `if`(irem(add(i,
      i=l), 2)=0, add(b(i, subsop(i=`if`(l[i]=1, [][], l[i]-1),
      l)), i=n+1..nops(l)), add(b(i-`if`(l[i]=1, 1, 0), subsop(
      i=`if`(l[i]=1, [][], l[i]-1), l)), i=1..n-1)))
    end:
A:= (n, k)->`if`(k=0, 1, b(`if`(irem(k*n, 2)=0, 0, n+1), [k$n])):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[n_, l_List] := b[n, l] = If[l == {}, 1, If[EvenQ[Total[l]], Sum[b[i, ReplacePart[l, i -> If[l[[i]] == 1, Nothing, l[[i]]-1]]], {i, n+1, Length[l]}], Sum[b[i - If[l[[i]] == 1, 1, 0], ReplacePart[l, i -> If[l[[i]] == 1, Nothing, l[[i]]-1]]], {i, 1, n-1}]]]; A[n_, k_] := If[k == 0, 1, b[If[EvenQ[k*n], 0, n+1], Array[k&, n]]]; Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 23 2017, adapted from Maple *)

A275829 Number of weakly alternating permutations of the multiset {1,1,2,2,...,n,n}.

Original entry on oeis.org

1, 1, 2, 12, 140, 2564, 68728, 2539632, 123686800, 7677924688, 591741636128, 55438330474944, 6204888219697856, 817697605612952384, 125322509904814743424, 22102340129003429880576, 4444468680409243484516608, 1010802175212828388101900544, 258152577318424951261637001728

Offset: 0

Max Alekseyev, Aug 11 2016

nonn

Number of permutations (p(1),...,p(2n)) of {1,1,2,2,...,n,n} satisfying p(1) <= p(2) >= p(3) <= p(4) >= p(5) <= ... <= p(2n).

a(n) >= A005799(n) >= A275801(n).

Max Alekseyev, Table of n, a(n) for n = 0..70

Cf. A000111, A001250, A000459, A004075, A005799, A114938, A137729, A137730, A137737, A137749, A275801.

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A005799 Generalized Euler numbers of type 2^n.

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A275784 Number A(n,k) of up-down sequences with k copies each of 1,2,...,n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A275829 Number of weakly alternating permutations of the multiset {1,1,2,2,...,n,n}.

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