A098277 -id:A098277 - cp's OEIS frontend

A098278 D(n,0)/2^n, where D(n,x) is triangle A098277.

1, 1, 3, 21, 267, 5349, 154923, 6120741, 316271787, 20701782309, 1673934058923, 163850823271461, 19093313058395307, 2611858473935397669, 414452507370456337323, 75508557963926980473381

Offset: 0

Ralf Stephan, Sep 07 2004

nonn

This is related to formula (1.7) in Lazar and Wachs reference.

Apparently all terms (except the initial 1s) have 3-valuation 1. - F. Chapoton, Jul 31 2021

			G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 267*x^4 + 5349*x^5 + ...
where A(x) = 1 + x/(1+x) + 2!^2*x^2/((1+x)*(1+3*x)) + 3!^2*x^3/((1+x)*(1+3*x)*(1+6*x)) + 4!^2*x^4/((1+x)*(1+3*x)*(1+6*x)*(1+10*x)) + ... - _Paul D. Hanna_, Sep 05 2012

Ange Bigeni and Evgeny Feigin, Symmetric Dellac configurations, arXiv:1808.04275 [math.CO], 2018.
Alexander Lazar and Michelle L. Wachs, On the homogenized Linial arrangement: intersection lattice and Genocchi numbers, Séminaire Lotharingien de Combinatoire, 82B.93 (FPSAC 2019).
A. Randrianarivony and J. Zeng, Une famille de polynômes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26.

Cf. A000366.

d[0, ] = 1; d[n, x_] := d[n, x] = (x+1)(x+2)d[n-1, x+2]-x(x+1)d[n-1, x];
a[n_] := d[n, 0]/2^n;
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Oct 26 2018 *)

{a(n)=polcoeff(sum(m=0, n, m!^2*x^m/prod(k=1, m, 1+k*(k+1)/2*x +x*O(x^n))), n)} \\ Paul D. Hanna, Sep 05 2012

G.f.: Sum_{n>=0} a(n)*x^n = 1/(1-1*1*x/(1-1*2*x/(1-2*3*x/(1-2*4*x/...)))).
G.f.: Sum_{n>=0} n!^2 * x^n / Product_{k=1..n} (1 + k*(k+1)/2*x). - Paul D. Hanna, Sep 05 2012
G.f.: 1/G(0) where G(k) = 1 - x*(k+1)*(2*k+1)/(1 - x*(k+1)*(2*k+2)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 14 2013.
a(n+1) = Sum_{k=0..n} A098277(n,k)*(1/2)^k. - Philippe Deléham, Feb 08 2013

A098279 a(n) = D(n,1)/2^n, where D(n,x) is triangle A098277.

Original entry on oeis.org

1, 2, 10, 98, 1594, 38834, 1323658, 60134210, 3511695322, 256306614866, 22861774551466, 2446866564603362, 309483997093321210, 45666236465616727538, 7774748058886412485834

Offset: 0

Ralf Stephan, Sep 07 2004

nonn

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 98*x^3 + 1594*x^4 + 38834*x^5 +...
where
A(x) = 1 + 2!*x/(1+x) + 2!*3!*x^2/((1+x)*(1+3*x)) + 3!*4!*x^3/((1+x)*(1+3*x)*(1+6*x)) + 4!*5!*x^4/((1+x)*(1+3*x)*(1+6*x)*(1+10*x)) + ...  - Paul D. Hanna, Sep 05 2012

Vaclav Kotesovec, Table of n, a(n) for n = 0..250
Ange Bigeni, Enumerating the symplectic Dellac configurations, arXiv:1705.03804 [math.CO], 2017.
Ange Bigeni, Evgeny Feigin, Poincaré polynomials of the degenerate flag varieties of type C, arXiv:1804.10804 [math.CO], 2018.
Ange Bigeni, Evgeny Feigin, Symmetric Dellac configurations, arXiv:1808.04275 [math.CO], 2018.
Xin Fang and Ghislain Fourier, Torus fixed points in Schubert varieties and Genocchi numbers, arXiv:1504.03980 [math.RT], 2015.
A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites classiques de nombres, Adv. Appl. Math. 17 (1996), 1-26.

d[0, _] = 1;
d[n_, x_] := d[n, x] = (x+1)(x+2) d[n-1, x+2] - x(x+1) d[n-1, x];
a[n_] := d[n, 1]/2^n;
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 27 2018 *)

{a(n)=polcoeff(sum(m=0, n, m!*(m+1)!*x^m/prod(k=1, m, 1+k*(k+1)/2*x +x*O(x^n))), n)} \\ Paul D. Hanna, Sep 05 2012

G.f.: Sum_{n>=0} a(n)*x^n = 1/(1-1*2*x/(1-1*3*x/(1-2*4*x/(1-2*5*x/...)))).
G.f.: Sum_{n>=0} n!*(n+1)! * x^n / Product_{k=1..n} (1 + k*(k+1)/2*x). - Paul D. Hanna, Sep 05 2012
G.f.: 2*G(0) - 1 where G(k) =  1 + x*(2*k+1)*(4*k+1)/( 1 + x + 6*x*k + 8*x*k^2 - 2*x*(k+1)*(4*k+3)*(1 + x + 6*x*k + 8*x*k^2)/(2*x*(k+1)*(4*k+3) + (1 + 6*x + 14*x*k + 8*x*k^2)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 11 2013
a(n) ~ 2^(3*n+11/2) * n^(2*n+2) / (exp(2*n) * Pi^(2*n+3/2)). - Vaclav Kotesovec, Apr 23 2015

A098431 Constant terms of polynomials in A098277.

Original entry on oeis.org

1, 2, 12, 168, 4272, 171168, 9915072, 783454848, 80965577472, 10599312542208, 1714108476337152, 335566486059952128, 78206210287187177472, 21396344618478777704448, 6790389880757556630700032

Offset: 0

Ralf Stephan, Sep 07 2004

nonn

G.f. 1/G(0) where G(k) = 1 - x*(2*k+1)*(2*k+2)/(1 - (2*k+2)^2*x/G(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Aug 11 2012

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

Original entry on oeis.org

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

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

cp's OEIS Frontend

A098278 D(n,0)/2^n, where D(n,x) is triangle A098277.

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A098279 a(n) = D(n,1)/2^n, where D(n,x) is triangle A098277.

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A098431 Constant terms of polynomials in A098277.

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

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A002832 Median Euler numbers.

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