A323834 -id:A323834 - cp's OEIS frontend

A000657 Median Euler numbers (the middle numbers of Arnold's shuttle triangle).

1, 1, 4, 46, 1024, 36976, 1965664, 144361456, 13997185024, 1731678144256, 266182076161024, 49763143319190016, 11118629668610842624, 2925890822304510631936, 895658946905031792553984, 315558279782214450517374976, 126780706777739389745128013824

Offset: 0

N. J. A. Sloane, Don Knuth

Also central terms of the triangle in A008280. - Reinhard Zumkeller, Nov 01 2013

Conjecture: taking the sequence modulo an integer k gives an eventually purely periodic sequence with period dividing phi(k). For example, the sequence taken modulo 9 begins [1, 1, 4, 1, 7, 4, 1, 7, 4, 1, 7, ...] with an apparent period [4, 1, 7] of length 3 = phi(9)/2 beginning at a(2). - Peter Bala, May 08 2023

V. I. Arnold, Springer numbers and Morsification spaces. J. Algebraic Geom. 1 (1992), no. 2, 197-214.
L. Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
V. I. Arnold, The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups, Uspekhi Mat. nauk., 47 (#1, 1992), 3-45 = Russian Math. Surveys, Vol. 47 (1992), 1-51.
Ange Bigeni and Evgeny Feigin, Symmetric Dellac configurations, arXiv:1808.04275 [math.CO], 2018.
D. Dumont, Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers, Adv. Appl. Math., 16 (1995), 275-296.
A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26.

Cf. A084938, A002832. For a signed version see A099023.
Related polynomials in A098277.
A diagonal of A323834.
Cf. A005799.

a000657 n = a008280 (2 * n) n  -- Reinhard Zumkeller, Nov 01 2013

Digits := 40: rr := array(1..40,1..40): rr[1,1] := 1: for i from 1 to 39 do rr[i+1,1] := subs(x=0,diff(1+tan(x),x$i)): od: for i from 2 to 40 do for j from 2 to i do rr[i,j] := rr[i,j-1]-(-1)^i*rr[i-1,j-1]: od: od: [seq(rr[2*i-1,i],i=1..20)];
# Alternatively after Alois P. Heinz in A000111:
b := proc(u, o) option remember;
`if`(u + o = 0, 1, add(b(o - 1 + j, u - j), j = 1..u)) end:
a := n -> b(n, n): seq(a(n), n = 0..15); # Peter Luschny, Oct 27 2017

max = 20; rr[1, 1] = 1; For[i = 1, i <= 2*max - 1, i++, rr[i + 1, 1] = D[1 + Tan[x], {x, i}] /. x -> 0]; For[i = 2, i <= 2*max, i++, For[j = 2, j <= i, j++, rr[i, j] = rr[i, j - 1] - (-1)^i*rr[i - 1, j - 1]]]; Table[rr[2*i - 1, i], {i, 1, max}] (* Jean-François Alcover, Jul 10 2012, after Maple *)
T[n_,0] := KroneckerDelta[n,0]; T[n_,k_] := T[n,k]=T[n,k-1]+T[n-1,n-k]; Table[T[2n,n], {n,0,16}] (* Oliver Seipel, Nov 24 2024, after Peter Luschny *)

a(n):=(-1)^(n)*sum(binomial(n,k)*euler(n+k),k,0,n); /* Vladimir Kruchinin, Apr 06 2015 */

# Algorithm of L. Seidel (1877)
def A000657_list(n) :
    R = []; A = {-1:0, 0:1}
    k = 0; e = 1
    for i in (0..n) :
        Am = 0; A[k + e] = 0; e = -e
        for j in (0..i) :
            Am += A[k]; A[k] = Am; k += e
        if e < 0 :
            R.append(A[0])
    return R
A000657_list(30)  # Peter Luschny, Apr 02 2012

Row sums of triangle, read by rows, [0, 1, 4, 9, 16, 25, 36, 49, ...] DELTA [1, 2, 6, 5, 11, 8, 16, 11, 21, 14, ...] where DELTA is Deléham's operator defined in A084938.
G.f.: Sum_{n>=0} a(n)*x^n = 1/(1-1*1x/(1-1*3x/(1-2*5x/(1-2*7x/(1-3*9x/...))))). - Ralf Stephan, Sep 09 2004
G.f.: 1/G(0) where G(k) = 1 - x*(8*k^2+4*k+1) - x^2*(k+1)^2*(4*k+1)*(4*k+3)/G(k+1); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 05 2013
G.f.: G(0)/(1-x), where G(k) = 1 - x^2*(k+1)^2*(4*k+1)*(4*k+3)/( x^2*(k+1)^2*(4*k+1)*(4*k+3) - (1 - x*(8*k^2+4*k+1))*(1 - x*(8*k^2+20*k+13))/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Feb 01 2014
a(n) = (-1)^(n)*Sum_{k=0..n} C(n,k)*Euler(n+k). - Vladimir Kruchinin, Apr 06 2015
a(n) ~ 2^(4*n+5/2) * n^(2*n+1/2) / (exp(2*n) * Pi^(2*n+1/2)). - Vaclav Kotesovec, Apr 06 2015
Conjectural e.g.f. as a continued fraction: 1/(1 - (1 - exp(-2*t))/(2 - (1 - exp(-4*t))/(1 - (1 - exp(-6*t))/(2 - (1 - exp(-8*t))/(1 - ... )))) = 1 + t + 4*t^2/2! + 46*t^3/3! + .... Cf. A005799. - Peter Bala, Dec 26 2019

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 12 2001

Corrected by Sean A. Irvine, Dec 22 2010

A323833 A Seidel matrix A(n,k) read by antidiagonals upwards.

Original entry on oeis.org

0, 1, 1, 1, 0, -1, -2, -3, -3, -2, -5, -3, 0, 3, 5, 16, 21, 24, 24, 21, 16, 61, 45, 24, 0, -24, -45, -61, -272, -333, -378, -402, -402, -378, -333, -272, -1385, -1113, -780, -402, 0, 402, 780, 1113, 1385, 7936, 9321, 10434, 11214, 11616, 11616, 11214, 10434, 9321, 7936

Offset: 0

N. J. A. Sloane, Feb 03 2019

The first row is a signed version of the Euler numbers A000111.

Other rows are defined by A(n+1,k) = A(n,k) + A(n,k+1).

			Triangular array T(n,k) = A(n-k,k) (n >= 0, k = 0..n), read from the antidiagonals upwards of square array A:
     0;
     1,    1;
     1,    0,   -1;
    -2,   -3,   -3,   -2;
    -5,   -3,    0,    3,    5;
    16,   21,   24,   24,   21,   16;
    61,   45,   24,    0,  -24,  -45,  -61;
  -272, -333, -378, -402, -402, -378, -333, -272;
  ...
From _Petros Hadjicostas_, Mar 04 2021: (Start)
Square array A(n,k) (n, k >= 0) begins:
   0,  1,   -1,   -2,     5,    16,     -61,    -272,     1385, ...
   1,  0,   -3,    3,    21,   -45,    -333,    1113,     9321, ...
   1, -3,    0,   24,   -24,  -378,     780,   10434,   -33264, ...
  -2, -3,   24,    0,  -402,   402,   11214,  -22830,  -480162, ...
  -5, 21,   24, -402,     0, 11616,  -11616, -502992,  1017600, ...
  16, 45, -378, -402, 11616,     0, -514608,  514608, 31880016, ...
  ... (End)

Alois P. Heinz, Antidiagonals n = 0..140, flattened
A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites classiques de nombres, Adv. Appl. Math. 17 (1996), 1-26. See Section 6 (matrix a_{n,k} on p. 18).

Cf. A000111, A002832 (next-to-main diagonal), A163747, A323834.

A323833 := proc(n,k)
    option remember;
    local i ;
    if k =0 then
        -A163747(n) ;
    elif n =0 then
        (-1)^k*A163747(k) ;
    elif k =n then
        0 ;
    else
        add(binomial(n,i)*procname(0,k+i), i=0..n) ;
    end if;
end proc:
seq(seq(A323833(d-k,k),k=0..d),d=0..12) ; # R. J. Mathar, Jun 11 2025

{b(n) = local(v=[1], t); if( n<0, 0, for(k=2, n+2, t=0; v = vector(k, i, if( i>1, t+= v[k+1-i]))); v[2])}; \\ Michael Somos's PARI program for A000111.
c(n) = if(n==0, 0, (-1)^floor(n/2)*b(n))
A(n, k) = sum(i=0, n, binomial(n, i)*c(k+i)) \\ Petros Hadjicostas, Mar 04 2021

From Petros Hadjicostas, Mar 04 2021: (Start)
Formulas about the square array A(n,k) (n,k > 0):
A(n,0) = -A163747(n) = (-1)^(n+1)*A(0,n) = if(n==0, 0, (-1)^floor(n/2)*A000111(n)).
A(n,n) = 0 and A(n,k) + (-1)^(n+k)*A(k,n) = 0.
A(n, k) = Sum_{i=0..n} binomial(n, i)*A(0,k+i).
Joint e.g.f.: Sum_{n,k >= 0} A(n,k)*(x^n/n!)*(y^k/k!) = 2*exp(-y)*(1 - exp(-x - y)) / (1 + exp(-2*(x + y))) = 2*exp(x)*(exp(x+y) - 1) / (exp(2*(x+y)) + 1).
Formulas about the triangular array T(n,k) = A(n-k,k) (0 <= k <= n):
T(n+1,k+1) = T(n+1,k) - T(n,k).
T(n,k) = -(-1)^n*T(n,n-k).
T(n,k) = Sum_{i=0..n-k} binomial(n-k,i)*T(k+i,k+i) for k=0..n with initial condition T(n,n) = (-1)^n*A163747(n). (End)

More terms from Alois P. Heinz, Feb 09 2019

A342161 Expansion of the exponential generating function (tanh(x) - sech(x) + 1) * exp(x).

Original entry on oeis.org

0, 1, 3, 4, -3, -14, 63, 274, -1383, -7934, 50523, 353794, -2702763, -22368254, 199360983, 1903757314, -19391512143, -209865342974, 2404879675443, 29088885112834, -370371188237523, -4951498053124094, 69348874393137903, 1015423886506852354, -15514534163557086903

Offset: 0

Petros Hadjicostas, Mar 03 2021

sign

A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites classiques de nombres, Adv. Appl. Math. 17 (1996), 1-26. See Section 6 (zeroth column of matrix b_{n,k} on p. 19).

Cf. A000111, A323834.

series((2*exp(x)-2)/(exp(-2*x)+1),x,30):seq(n!*coeff(%,x,n),n=0..24); # Peter Luschny, Mar 05 2021

my(x='x+O('x^30)); concat(0, Vec(serlaplace((-1/cosh(x) + tanh(x) + 1)*exp(x)))) \\ Michel Marcus, Mar 05 2021

def A323834List(prec):
    R. = PowerSeriesRing(QQ, default_prec=prec)
    f = (2*exp(2*x)*(exp(x) - 1))/(exp(2*x) + 1)
    return f.egf_to_ogf().list()
print(A323834List(25)) # Peter Luschny, Mar 05 2021

a(n) = A323834(n, 0).
a(n) = n! [x^n] (tanh(x) - sech(x) + 1) * exp(x).
a(n) = Sum_{i=1..n} binomial(n,i) * (-1)^floor((i-1)/2) * A000111(i).

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A000657 Median Euler numbers (the middle numbers of Arnold's shuttle triangle).

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A323833 A Seidel matrix A(n,k) read by antidiagonals upwards.

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Maple

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A342161 Expansion of the exponential generating function (tanh(x) - sech(x) + 1) * exp(x).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

SageMath

Formula