A323833 -id:A323833 - cp's OEIS frontend

A342195 a(n) = Sum_{k=0..floor(n/2)} A323833(n,k) if A323833 is read as a triangle.

0, 1, 1, -5, -8, 61, 130, -1385, -3680, 50521, 160816, -2702765, -10026368, 199360981, 844583440, -19391512145, -92369507840, 2404879675441, 12722897618176, -370371188237525, -2154662195222528, 69348874393137901, 440001333689382400, -15514534163557086905, -106615331831035289600, 4087072509293123892361

Offset: 0

Petros Hadjicostas, Mar 04 2021

sign

Because A323833(n,n/2) = 0 for n even (if A323833 is read as a triangle), we also have a(n) = Sum_{k=0..ceiling((n-1)/2)} A323833(n,k).

			a(3) = -2 - 3 = -5.
a(4) = -5 - 3 = -8.
a(5) = 16 + 21 + 24 = 61.
a(6) = 61 + 45 + 24 = 130.
a(7) = -272 - 333 - 378 - 402 = -1385.

Cf. A000111, A028296, A163747, A323833.

{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)^(n+floor(n/2))*b(n))
a(n) = sum(k=0, floor(n/2), sum(i=0, n-k, binomial(n-k,i)*(-1)^(k+i)*c(k+i)))

a(n) = Sum_{k=0..floor(n/2)} A323833(n-k,k) if A323833 is read as a square array (by upwards antidiagonals).
a(2*n+1) = -A028296(n+1).
a(n) = Sum_{k=0..floor(n/2)} Sum_{i=0..n-k} binomial(n-k,i) * (-1)^(k+i) * A163747(k+i).

A163747 Expansion of e.g.f. 2exp(x)(1-exp(x))/(1+exp(2*x)).

Original entry on oeis.org

0, -1, -1, 2, 5, -16, -61, 272, 1385, -7936, -50521, 353792, 2702765, -22368256, -199360981, 1903757312, 19391512145, -209865342976, -2404879675441, 29088885112832, 370371188237525, -4951498053124096, -69348874393137901

Offset: 0

Roger L. Bagula, Aug 03 2009

sign

The real part of the exponential expansion of 2*((1+i)/(1+i*exp(z))-1) = (-1-i)*z + (-1/2+i/2)*z^2 + (1/3+i/3)*z^3 + (5/24-5i/24)*z^4 + (-2/15-2i/15)*z^5 + ...  where i is the imaginary unit.
From Paul Curtz, Mar 12 2013: (Start)
a(n) is an autosequence of the first kind; a(n) and successive differences are:
0,   -1,   -1,      2,      5,    -16,    -61;
-1,   0,    3,      3,    -21,    -45,    333;
1,    3,    0,    -24,    -24,    378,    780;
2,   -3,  -24,      0,    402,    402, -11214;
-5, -21,   24,    402,      0, -11616, -11616;
-16, 45,  378,   -402, -11616,      0, 514608;
61, 333, -780, -11214,  11616, 514608,      0;
The main diagonal is A000004. The inverse binomial transform is the signed sequence.
The first two upper diagonals are A002832 (median Euler numbers) signed.
Sum of the antidiagonals: 0, -2, 0, 10, 0, ... = 2*A122045(n+1) (End)

Robert Israel, Table of n, a(n) for n = 0..485
Toufik Mansour, Howard Skogman, and Rebecca Smith, Passing through a stack k times with reversals, arXiv:1808.04199 [math.CO], 2018.
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 (negative of the zeroth column of matrix a_{n,k} on p. 18).

Variant: A163982.
Cf. A000004, A000111, A002832, A122045.
Minus the zeroth column of A323833.

A163747 := proc(n) exp(t)*(1-exp(t))/(1+exp(2*t)) ; coeftayl(%,t=0,n) ; 2*%*n! ; end proc: # R. J. Mathar, Sep 11 2011
seq((euler(n) - 2^n*(2*euler(n,1)+euler(n,3/2)))/2 + 1, n=0..30); # Robert Israel, May 24 2016
egf := (2 - 2*I)/(exp(-x) + I); ser := series(egf, x, 24):
seq(n!*Re(coeff(ser, x, n)), n = 0..22); # Peter Luschny, Aug 09 2021

f[t_] = (1 + I)/(1 + I*Exp[t]) - 1;
Table[Re[2*n!*SeriesCoefficient[Series[f[t], {t, 0, 30}], n]], {n, 0, 30}]
max = 20; Clear[g]; g[max + 2] = 1; g[k_] := g[k] = 1 - x + (4*k+3)*(k+1)*x^2 /( 1 + (4*k+5)*(k+1)*x^2 / g[k+1]); gf = -x/g[0]; CoefficientList[Series[gf, {x, 0, max}], x] (* Vaclav Kotesovec, Jan 22 2015, after Sergei N. Gladkovskii *)
Table[(EulerE[n] - 2^n (2 EulerE[n, 1] + EulerE[n, 3/2]))/2 + 1, {n, 0, 20}] (* Benedict W. J. Irwin, May 24 2016 *)

G.f.: -x/W(0), where W(k) = 1 - x + (4*k+3)*(k+1)*x^2 / (1 + (4*k+5)*(k+1)*x^2 / W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 22 2015
a(n) ~ n! * (cos(Pi*n/2) - sin(Pi*n/2)) * 2^(n+2) / Pi^(n+1). - Vaclav Kotesovec, Apr 23 2015
a(n) = (A122045(n) - 2^n(2*Euler(n,1) + Euler(n,3/2)))/2 + 1, where Euler(n,x) is the n-th Euler polynomial. - Benedict W. J. Irwin, May 24 2016
a(n) = 2*4^n*(HurwitzZeta(-n, 1/4) - HurwitzZeta(-n, 3/4)) + HurwitzZeta(-n, 1)*(4^(n+1) - 2^(n+1)). - Peter Luschny, Jul 21 2020
a(n) = 2^n*(Euler(n, 1/2) - Euler(n, 1)). - Peter Luschny, Mar 19 2021
a(n) = ((-2)^(n + 1)*(1 - 2^(n + 1))*Bernoulli(n + 1))/(n + 1) + Euler(n). - Peter Luschny, May 06 2021
a(n) = n!*Re([x^n]((2 - 2*i)/(i + exp(-x)))). - Peter Luschny, Aug 09 2021

A002832 Median Euler numbers.

Original entry on oeis.org

1, 3, 24, 402, 11616, 514608, 32394624, 2748340752, 302234850816, 41811782731008, 7106160248346624, 1455425220196234752, 353536812021243273216, 100492698847094242603008, 33045185784774350171111424

Offset: 1

N. J. A. Sloane, Dec 11 1996

nonn

There are two kinds of Euler median numbers, the 'right' median numbers (this sequence), and the 'left' median numbers (A000657).

Apparently all terms (except the initial 1) have 3-valuation 1. - F. Chapoton, Aug 02 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Ange Bigeni and Evgeny Feigin, Symmetric Dellac configurations, arXiv:1808.04275 [math.CO], 2018.
Kwang-Wu Chen, An Interesting Lemma for Regular C-fractions, J. Integer Seqs., Vol. 6, 2003.
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 polynômes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26. (In French, with a summary in English on p. 1).
R. C. Read, Letter to N. J. A. Sloane, 1992

Cf. A000657.
See related polynomials in A098277.
A diagonal of A323833.

rr := array(1..40,1..40):rr[1,1] := 0:for i from 1 to 39 do rr[i+1,1] := (subs(x=0,diff((exp(x)-1)/cosh(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]-rr[i-1,j-1]:od:od: seq(rr[2*i-1,i-1],i=2..20); # Barbara Haas Margolius (margolius(AT)math.csuohio.edu) Feb 16 2001, corrected by R. J. Mathar, Dec 22 2010
# alternative
A002832 := proc(n)
    abs(A323833(n-1,n)) ;
end proc:
seq(A002832(n),n=1..40) ; # R. J. Mathar, Jun 11 2025

max = 20; rr[1, 1] = 0; For[i = 1, i <= 2*max - 1, i++, rr[i + 1, 1] = D[(Exp[x] - 1)/Cosh[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] - rr[i - 1, j - 1]]]; Table[(-1)^i*rr[2*i - 1, i - 1], {i, 2, max}] (* Jean-François Alcover, Jul 10 2012, after Maple *)

G.f.: Sum_{n>=0} a(n)*x^n = 1/(1-1*3x/(1-1*5x/(1-2*7x/(1-2*9x/(1-3*11x/...))))).
G.f.: -1/G(0) where G(k)= x*(8*k^2+8*k+3) - 1 - (4*k+5)*(4*k+3)*(k+1)^2*x^2/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Aug 08 2012
a(n) ~ 2^(4*n+3/2) * n^(2*n-1/2) / (exp(2*n) * Pi^(2*n-1/2)). - Vaclav Kotesovec, Apr 23 2015

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 16 2001

Terms corrected by R. J. Mathar, Dec 22 2010

A323834 A Seidel matrix A(n,k) read by antidiagonals downwards.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, -2, -1, 1, 4, -5, -7, -8, -7, -3, 16, 11, 4, -4, -11, -14, 61, 77, 88, 92, 88, 77, 63, -272, -211, -134, -46, 46, 134, 211, 274, -1385, -1657, -1868, -2002, -2048, -2002, -1868, -1657, -1383, 7936, 6551, 4894, 3026, 1024, -1024, -3026, -4894, -6551, -7934, 50521, 58457, 65008, 69902, 72928, 73952, 72928, 69902, 65008, 58457, 50523

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).

			Read as triangle T(n,k) = A(k, n-k) (n >= 0, k = 0..n), the first few antidiagonals of the square array A are:
     0,
     1,    1,
     1,    2,    3,
    -2,   -1,    1,   4,
    -5,   -7,   -8,  -7,  -3,
    16,   11,    4,  -4, -11, -14,
    61,   77,   88,  92,  88,  77,  63,
  -272, -211, -134, -46,  46, 134, 211, 274,
  ...
From _Petros Hadjicostas_, Mar 02 2021: (Start)
Square array A(n,k) (with rows n >= 0 and columns k >= 0) begins:
     0,   1,   1,    -2,    -5,    16,     61,     -272,    -1385, ...
     1,   2,  -1,    -7,    11,    77,   -211,    -1657,     6551, ...
     3,   1,  -8,     4,    88,  -134,  -1868,     4894,    65008, ...
     4,  -7,  -4,    92,   -46, -2002,   3026,    69902,  -179806, ...
    -3, -11,  88,    46, -2048,  1024,  72928,  -109904, -3784448, ...
   -14,  77, 134, -2002, -1024, 73952, -36976, -3894352,  5860016, ...
   ... (End)

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 b_{n,k} on p. 19).

Cf. A000111, A000657 (next-to-main diagonal), A323833.

{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-1)/2)*b(n))
A(n, k) = sum(i=0, n, binomial(n, i)*c(k+i)) \\ Petros Hadjicostas, Mar 02 2021

From Petros Hadjicostas, Mar 02 2021: (Start)
Formulas for the square array A(n,k) (n, k >= 0):
A(0,k) = (-1)^floor((k-1)/2)*A000111(k) for k > 0 with A(0,0) = 0.
A(n,k) = Sum_{i=0..n} binomial(n, i)*A(0,k+i) for n, k >= 0.
A(n,n)/2 = A(n+1,n) = +/- A000657(n) for n > 0.
Bivariate e.g.f.: Sum_{n,k >= 0} A(n,k)*(x^n/n!)*(y^k/k!) = (-sech(x + y) + tanh(x + y) + 1)*exp(x).
Formulas for the triangular array T(n,k) = A(k,n-k) (n >= 0, 0 <= k <= n):
T(n,k) = T(n-1,k-1) + T(n,k-1) for 1 <= k <= n with T(n,0) = (-1)^floor((n-1)/2) * A000111(n) for n > 0 and T(0,0) = 0.
T(n,k) = Sum_{i=0..k} binomial(k,i)*T(n-k+i,0) for 0 <= k <= n. (End)

Typo corrected by and more terms from Petros Hadjicostas, Mar 02 2021

cp's OEIS Frontend

A342195 a(n) = Sum_{k=0..floor(n/2)} A323833(n,k) if A323833 is read as a triangle.

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A163747 Expansion of e.g.f. 2*exp(x)*(1-exp(x))/(1+exp(2*x)).

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A002832 Median Euler numbers.

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A323834 A Seidel matrix A(n,k) read by antidiagonals downwards.

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A163747 Expansion of e.g.f. 2exp(x)(1-exp(x))/(1+exp(2*x)).