A000364 -id:A000364 - cp's OEIS frontend

A242194 Least prime divisor of E_{2n} which does not divide any E_{2k} with k < n, or 1 if such a primitive prime divisor of E_{2*n} does not exist, where E_m denotes the m-th Euler number given by A122045.

1, 5, 61, 277, 19, 13, 47, 17, 79, 41737, 31, 2137, 67, 29, 15669721, 930157, 4153, 37, 23489580527043108252017828576198947741, 41, 137, 587, 285528427091, 5516994249383296071214195242422482492286460673697, 5639, 53, 2749, 5303, 1459879476771247347961031445001033, 6821509

Offset: 1

Zhi-Wei Sun, May 07 2014

Conjecture: a(n) is prime for any n > 1.
It is known that (-1)^n*E_{2*n} > 0 for all n = 0, 1, ....
See also A242193 for a similar conjecture involving Bernoulli numbers.

			a(4) = 277 since E_8 = 5*277 with 277 not dividing E_2*E_4*E_6, but 5 divides E_4 = 5.

Peter Luschny, Table of n, a(n) for n = 1..82, (a(1)..a(34) from Zhi-Wei Sun, a(35)..a(38) from Jean-François Alcover).
Z.-W. Sun, New observations on primitive roots modulo primes, arXiv preprint arXiv:1405.0290 [math.NT], 2014.

Cf. A000040, A000364, A122045, A242169, A242170, A242171, A242173, A242193, A242195.

e[n_]:=Abs[EulerE[2n]]
f[n_]:=FactorInteger[e[n]]
p[n_]:=p[n]=Table[Part[Part[f[n],k],1],{k,1,Length[f[n]]}]
Do[If[e[n]<2,Goto[cc]];Do[Do[If[Mod[e[i],Part[p[n],k]]==0,Goto[aa]],{i,1,n-1}];Print[n," ",Part[p[n],k]];Goto[bb];Label[aa];Continue,{k,1,Length[p[n]]}];Label[cc];Print[n," ",1];Label[bb];Continue,{n,1,30}]
(* Second program: *)
LPDtransform[n_, fun_] := Module[{}, d[p_, m_] := d[p, m] = AllTrue[ Range[m-1], ! Divisible[fun[#], p]&]; f[m_] := f[m] = FactorInteger[ fun[m]][[All, 1]]; SelectFirst[f[n], d[#, n]&] /. Missing[_] -> 1];
a[n_] := a[n] = LPDtransform[n, Function[k, Abs[EulerE[2k]]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 38}]  (* Jean-François Alcover, Jul 28 2019, non-optimized adaptation of Peter Luschny's Sage code *)

# uses[LPDtransform from A242193]
A242194list = lambda sup: [LPDtransform(n, lambda k: euler_number(2*k)) for n in (1..sup)]
print(A242194list(16)) # Peter Luschny, Jul 26 2019

A302584 a(n) = n! * [x^n] exp(n*x)/cos(x).

Original entry on oeis.org

1, 1, 5, 36, 357, 4500, 68857, 1239504, 25661545, 600655824, 15684383021, 452001644864, 14249852124365, 487836995500608, 18022519535240417, 714658089577017600, 30275849571771536977, 1364687729891761740032, 65213822241378992547925, 3293203845745202062590976

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

N. J. A. Sloane, Transforms

Cf. A000364, A003701, A062024, A115415, A115416, A302583, A302585, A302586, A302587.

Table[n! SeriesCoefficient[Exp[n x]/Cos[x], {x, 0, n}], {n, 0, 19}]
Table[(2 I)^n EulerE[n, (1 - I n)/2], {n, 0, 19}]

a(n) ~ n^n / cos(1). - Vaclav Kotesovec, Jun 08 2019

A302585 a(n) = n! * [x^n] exp(n*x)/cosh(x).

Original entry on oeis.org

1, 1, 3, 18, 165, 2000, 29855, 527632, 10762857, 248811264, 6428081979, 183537694208, 5739195739277, 195059957567488, 7159662639822615, 282252719348582400, 11894243092571825745, 533554809104057434112, 25384473065818477067123, 1276688324194885747474432

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

N. J. A. Sloane, Transforms

Cf. A000364, A062024, A119880, A119881, A155585, A302583, A302584, A302586, A302587.

Table[n! SeriesCoefficient[Exp[n x]/Cosh[x], {x, 0, n}], {n, 0, 19}]
Table[2^n EulerE[n, (n + 1)/2], {n, 0, 19}]

a(n) ~ n^n / cosh(1). - Vaclav Kotesovec, Jun 08 2019

A318146 Coefficients of the Omega polynomials of order 2, triangle T(n,k) read by rows with 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, -2, 3, 0, 16, -30, 15, 0, -272, 588, -420, 105, 0, 7936, -18960, 16380, -6300, 945, 0, -353792, 911328, -893640, 429660, -103950, 10395, 0, 22368256, -61152000, 65825760, -36636600, 11351340, -1891890, 135135

Offset: 0

Peter Luschny, Aug 22 2018

The name 'Omega polynomial' is not a standard name. The Omega numbers are the coefficients of the Omega polynomials, the associated Omega numbers are the weights of P(m, k) in the recurrence formula given below.

The signed Euler secant numbers appear as values at x=-1 and the signed Euler tangent numbers as the coefficients of x.

			Row n in the triangle below is the coefficient list of OmegaPolynomial(2, n). For other cases than m = 2 see the cross-references.
[0] [1]
[1] [0,        1]
[2] [0,       -2,         3]
[3] [0,       16,       -30,       15]
[4] [0,     -272,       588,     -420,       105]
[5] [0,     7936,    -18960,    16380,     -6300,      945]
[6] [0,  -353792,    911328,  -893640,    429660,  -103950,    10395]
[7] [0, 22368256, -61152000, 65825760, -36636600, 11351340, -1891890, 135135]

Peter Luschny, Table of the first 63 rows.

Variant is A088874 (unsigned).
T(n,1) = A000182(n), T(n,n) = A001147(n).
All row sums are 1, alternating row sums are A028296 (A000364).
A023531 (m=1), this seq (m=2), A318147 (m=3), A318148 (m=4).
Associated Omega numbers: A318254 (m=2), A318255 (m=3).
Coefficients of x for Omega polynomials of all orders are in A318253.

OmegaPolynomial := proc(m, n) local Omega;
Omega := m -> hypergeom([], [seq(i/m, i=1..m-1)], (z/m)^m):
series(Omega(m)^x, z, m*(n+1)):
sort(expand((m*n)!*coeff(%, z, n*m)), [x], ascending) end:
CL := p -> PolynomialTools:-CoefficientList(p, x):
FL := p -> ListTools:-Flatten(p):
FL([seq(CL(OmegaPolynomial(2, n)), n=0..8)]);
# Alternative:
ser := series(sech(z)^(-x), z, 24): row := n -> n!*coeff(ser, z, n):
seq(seq(coeff(row(2*n), x, k), k=0..n), n=0..6); # Peter Luschny, Jul 01 2019

OmegaPolynomial[m_,n_] :=  Module [{ },
S = Series[MittagLefflerE[m,z]^x, {z,0,10}];
Expand[(m n)! Coefficient[S,z,n]] ]
Table[CoefficientList[OmegaPolynomial[2,n],x], {n,0,7}] // Flatten
(* Second program: *)
T[n_, k_] := (2n)! SeriesCoefficient[Sech[z]^-x, {z, 0, 2n}, {x, 0, k}];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 23 2019, after Peter Luschny *)

def OmegaPolynomial(m, n):
    R = ZZ[x]; z = var('z')
    f = [i/m for i in (1..m-1)]
    h = lambda z: hypergeometric([], f, (z/m)^m)
    return R(factorial(m*n)*taylor(h(z)^x, z, 0, m*n + 1).coefficient(z, m*n))
[list(OmegaPolynomial(2, n)) for n in (0..6)]
# Recursion over the polynomials, returns a list of the first len polynomials:
def OmegaPolynomials(m, len, coeffs=true):
    R = ZZ[x]; B = [0]*len; L = [R(1)]*len
    for k in (1..len-1):
        s = x*sum(binomial(m*k-1, m*(k-j))*B[j]*L[k-j] for j in (1..k-1))
        B[k] = c = 1 - s.subs(x=1)
        L[k] = R(expand(s + c*x))
    return [list(l) for l in L] if coeffs else L
print(OmegaPolynomials(2, 6))

OmegaPolynomial(m, n) = (m*n)! [z^n] E(m, z)^x where E(m, z) is the Mittag-Leffler function.
OmegaPolynomial(m, n) = (m*n)!*[z^(n*m)] H(m, z)^x where H(m, z) = hypergeom([], [seq(i/m, i=1..m-1)], (z/m)^m).
The Omega polynomials can be computed by the recurrence P(m, 0) = 1 and for n >= 1 P(m, n) = x * Sum_{k=0..n-1} binomial(m*n-1, m*k)*T(m, n-k)*P(m, k) where T(m, n) are the generalized tangent numbers A318253. A separate computation of the T(m, n) can be avoided, see the Sage implementation below for the details.
T(n, k) = [x^k] (2*n)! [z^(2*n)] sech(z)^(-x). - Peter Luschny, Jul 01 2019

A002438 Multiples of Euler numbers.

Original entry on oeis.org

1, 5, 205, 22265, 4544185, 1491632525, 718181418565, 476768795646785, 417370516232719345, 465849831125196593045, 645702241048404020542525, 1088120580608731523115639305, 2190881346273790815462670984105

Offset: 1

N. J. A. Sloane

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 75.
Glaisher, J. W. L.; Messenger of Math., 28 (1898), 36-79, see esp. p. 51.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..100
P. Bala, A triangle for calculating A002438
P. Bala, Some S-fractions related to the expansions of sin(ax)/cos(bx) and cos(ax)/cos(bx)
Matthieu Josuat-Vergès and Jang Soo Kim, Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity, arXiv:1101.5608 [math.CO] (2011)

Cf. A000364, A086646, A255884.

a[n_] := (1+9^(n-1))*Abs[EulerE[2*(n-1)]]/2; Table[ a[n], {n, 1, 13}](* Jean-François Alcover, Feb 10 2012 *)

A002438(n)=A000364(n-1)*(9^(n-1)+1)\2 \\ - M. F. Hasler, Jul 21 2013

a(n) = A000364(n-1) * (9^(n-1) + 1)/2.
a(n+1) = Sum_{k = 0..n} A086646(n, k)*(-4)^k*9^(n-k). - Philippe Deléham, Aug 26 2005
From Peter Bala, Mar 13 2015: (Start)
a(n+1) = (-1)^n*6^(2*n)*E(2*n,1/6).
Assuming an offset of 0, the e.g.f. is cos(2*x)/cos(3*x) = 1 + 5*x + 205*x^2/2! + 22265*x^3/3! + 4544185*x^4/4! + ....
O.g.f. as a continued fraction: x/(1 - (3^2 - 2^2)*x/(1 - 6^2*x/(1 - (9^2 - 2^2)*x/(1 - 12^2*x/(1 - ... ))))) = x + 5*x^2 + 205*x^3 + 22265*x^4 + 4544185*x^5 + .... See Josuat-Vergès and Kim, p. 23. Cf. A086646.
The expansion of exp( Sum_{n >= 1} a(n+1)*x^n/n ) = exp( 5*x + 205*x^2/2 + 22265*x^3/3 + 4544185 *x^4/4 + ... ) appears to have integer coefficients. See A255884.
(End)
From Peter Bala, Nov 10 2015: (Start)
O.g.f. A(x) = 1/(1 + x - 6*x/(1 - 30*x/(1 + x - 84*x/(1 - 132*x/(1 + x - ... - 6*n*(6*n - 5)*x/(1 - 6*n*(6*n - 1)*x/(1 + x - ))))))).
A(x) = 1/(1 + 25*x - 30*x/(1 - 6*x/(1 + 25*x - 132*x/(1 - 84*x/(1 + 25*x - ... - 6*n*(6*n - 1)*x/(1 - 6*n*(6*n - 5)*x/(1 + 25*x - ))))))). (End)

More terms from Herman P. Robinson

More terms from Jon E. Schoenfield, May 09 2010

A012816 E.g.f. arctan(sec(x)*sinh(x)) (odd powers only).

Original entry on oeis.org

1, 2, -20, -488, 22160, 1616672, -172976960, -25518205568, 4964227109120, 1231298393825792, -379260096755225600, -142026494757146421248, 63547531933929827962880, 33481297996129270926221312, -20517021964757071715832381440, -14468510293983989090015078678528

Offset: 0

Patrick Demichel (patrick.demichel(AT)hp.com)

sign

The unsigned sequence {|a(n)|}n>=1 = [2,20,488,22160,...] enumerates binary increasing trees on 2*n vertices with a perfect matching (Kuba and Wagner).

			arctan(sec(x)*sinh(x)) = x+2/3!*x^3-20/5!*x^5-488/7!*x^7+22160/9!*x^9...

Alois P. Heinz, Table of n, a(n) for n = 0..228
M. Kuba and S. Wagner, Perfect matchings and k-indecomposabilty of increasing trees, Seminaire Lotharingien de Combinatoire 57 (2007), Article B57a.

Bisection (odd part) of A009342.

a:= n-> (2*n+1)! *
    coeff(series(arctan(sec(x)*sinh(x)), x, 2*(n+1)), x, 2*n+1):
seq(a(n), n=0..20);

With[{nn=40},Take[CoefficientList[Series[ArcTan[Sec[x]Sinh[x]],{x,0,nn}],x] Range[0,nn-1]!,{2,-1,2}]] (* Harvey P. Dale, Jul 25 2024 *)

1/cosh(x*sqrt(2)) = 1 - 2x^2/2! + 20*x^4/4! - 488*x^6/6! +-...
a(n) = (-1)^[n/2]*2^n*A000364(n). - Philippe Deléham, Jun 16 2007
G.f. (for the unsigned sequence): 1/G(0) where G(k) =  1 - 2*x*(k+1)^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 12 2013
G.f. (for the unsigned sequence): Q(0), where Q(k) = 1 - 2*x*(k+1)^2/(2*x*(k+1)^2 - 1/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 10 2013
E.g.f.(for the unsigned sequence, odd powers only): 1 + T(0)*x^2 /(1-x^2), where T(k) = 1 - 2*x^2*(2*k+1)*(2*k+2)/( 2*x^2*(2*k+1)*(2*k+2) + ((2*k+1)*(2*k+2)-2*x^2)*((2*k+3)*(2*k+4)-2*x^2)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 14 2013

A060074 Triangle A060058 by diagonals.

Original entry on oeis.org

1, 1, 1, 5, 5, 1, 61, 61, 14, 1, 1385, 1385, 331, 30, 1, 50521, 50521, 12284, 1211, 55, 1, 2702765, 2702765, 663061, 68060, 3486, 91, 1, 199360981, 199360981, 49164554, 5162421, 281210, 8526, 140, 1

Offset: 0

Wolfdieter Lang, Mar 16 2001

Row sums give A060059. Columns give A000364 (Euler numbers), A000364, A060075-78 for m=0,..,5.

Triangle can be used to express the Euler numbers E(n)=A000364(n), n >= 2, in terms of the numbers A060080 (scaled sums of squares), according to E(n+2)= sum(a(n,m)*A060080(m+2),m=0..n).

			{1}; {1,1}; {5,5,1}; {61,61,14,1}; ...

a(n, m)= a(n-1, m-1)+(m+1)^2*a(n, m+1), a(n, -1) := 0, a(0, 0)=1, a(n, m)=0 if n

a(n, m)=A060058(n, n-m).

A060081 Exponential Riordan array (sech(x), tanh(x)).

Original entry on oeis.org

1, 0, 1, -1, 0, 1, 0, -5, 0, 1, 5, 0, -14, 0, 1, 0, 61, 0, -30, 0, 1, -61, 0, 331, 0, -55, 0, 1, 0, -1385, 0, 1211, 0, -91, 0, 1, 1385, 0, -12284, 0, 3486, 0, -140, 0, 1, 0, 50521, 0, -68060, 0, 8526, 0, -204, 0, 1, -50521, 0, 663061

Offset: 0

Wolfdieter Lang, Mar 29 2001

Previous name was: "Triangle of coefficients (lower triangular matrix) of certain (binomial) convolution polynomials related to 1/cosh(x) and tanh(x). Use trigonometric functions for the unsigned version".
Row sums give A009265(n) (signed); A009244(n) (unsigned). Column sequences without interspersed zeros and unsigned: A000364 (Euler), A000364, A060075-8 for m=0,...,5.
a(n,m) = ((-1)^((n-m)/2))*ay(m+1,(n-m)/2) if n-m is even, else 0; where the rectangular array ay(n,m) is defined in A060058 Formula.
The row polynomials p(n,x) appear in a problem of thermo field dynamics (Bogoliubov transformation for the harmonic Bose oscillator). See the link to a .ps.gz file where they are called R_{n}(x).
The inverse of this Sheffer matrix with elements a(n,m) is the Sheffer matrix A060524. This Sheffer triangle appears in the Moyal star product of the harmonic Bose oscillator: x^{*n} = Sum_{m=0..n} a(n,m) x^m with x = 2 (bar a) a/hbar. See the Th. Spernat link, pp. 28, 29, where the unsigned version is used for y=-ix. - Wolfdieter Lang, Jul 22 2005
In the umbral calculus (see Roman reference under A048854) the p(n,x) are called Sheffer for (g(t)=1/cosh(arctanh(t)) = 1/sqrt(1-t^2), f(t)=arctanh(t)).
p(n,x) := Sum_{m=0..n} a(n,m)*x^m, n >= 0, are monic polynomials satisfying p(n,x+y) = Sum_{k=0..n} binomial(n,k)*p(k,x)*q(n-k,y) (binomial, also called exponential, convolution polynomials) with the row polynomials of the associated triangle q(n,x) := Sum_{m=0..n} A111593(n,m)*x^m. E.g.f. for p(n,x) is exp(x*tanh(z))*cosh(z)(signed). [Corrected by Wolfdieter Lang, Sep 12 2005]
Exponential Riordan array [sech(x), tanh(x)]. Unsigned triangle is [sec(x), tan(x)]. - Paul Barry, Jan 10 2011

			p(3,x) = -5*x + x^3.
Exponential convolution together with A111593 for row polynomials q(n,x), case n=2: -1+(x+y)^2 = p(2,x+y) = 1*p(0,x)*q(2,y) + 2*p(1,x)*q(1,y) + 1*p(2,x)*q(0,y) = 1*1*y^2 + 2*x*y + 1*(-1+x^2)*1.
Triangle begins:
  1,
  0, 1,
  -1, 0, 1,
  0, -5, 0, 1,
  5, 0, -14, 0, 1,
  0, 61, 0, -30, 0, 1,
  -61, 0, 331, 0, -55, 0, 1,
  0, -1385, 0, 1211, 0, -91, 0, 1,
  1385, 0, -12284, 0, 3486, 0, -140, 0, 1,
  0, 50521, 0, -68060, 0, 8526, 0, -204, 0, 1,
  -50521, 0, 663061, 0, -281210, 0, 18522, 0, -285, 0, 1,
  ...
As a right-aligned triangle:
                                                       1;
                                                    0, 1;
                                                -1, 0, 1;
                                           0,   -5, 0, 1;
                                        5, 0,  -14, 0, 1;
                                 0,    61, 0,  -30, 0, 1;
                            -61, 0,   331, 0,  -55, 0, 1;
                     0,   -1385, 0,  1211, 0,  -91, 0, 1;
               1385, 0,  -12284, 0,  3486, 0, -140, 0, 1;
          0,  50521, 0,  -68060, 0,  8526, 0, -204, 0, 1;
  -50521, 0, 663061, 0, -281210, 0, 18522, 0, -285, 0, 1;
  ...
Production matrix begins
   0,   1;
  -1,   0,   1;
   0,  -4,   0,   1;
   0,   0,  -9,   0,   1;
   0,   0,   0, -16,   0,   1;
   0,   0,   0,   0, -25,   0,   1;
   0,   0,   0,   0,   0, -36,   0,   1;
   0,   0,   0,   0,   0,   0, -49,   0,   1;
   0,   0,   0,   0,   0,   0,   0, -64,   0,   1;
- _Paul Barry_, Jan 10 2011

W. Lang, Two normal ordering problems and certain Sheffer polynomials, in Difference Equations, Special Functions and Orthogonal Polynomials, edts. S. Elaydi et al., World Scientific, 2007, pages 354-368. [From Wolfdieter Lang, Feb 06 2009]

Paul Barry, Exponential Riordan arrays and permutation enumeration,Journal of Integer Sequences, Vol. 13 (2010)
Wolfdieter Lang, Thermo field dynamics, exercise 29. WS 2008/2009 (in German)
Th. Spernat, Diplomarbeit 2004 (in German) (with permission)

riordan := (d,h,n,k) -> coeftayl(d*h^k,x=0,n)*n!/k!:
A060081 := (n,k) -> riordan(sech(x),tanh(x),n,k):
seq(print(seq(A060081(n,k),k=0..n)),n=0..5); # Peter Luschny, Apr 15 2015

max = 12; t = Transpose[ Table[ PadRight[ CoefficientList[ Series[ Tanh[x]^m/m!/Cosh[x], {x, 0, max}], x], max + 1, 0]*Table[k!, {k, 0, max}], {m, 0, max}]]; Flatten[ Table[t[[n, k]], {n, 1, max}, {k, 1, n}]] (* Jean-François Alcover, Sep 29 2011 *)

def A060081_triangle(dim): # computes unsigned T(n, k).
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+(k+1)^2*M[n-1,k+1]
    return M
A060081_triangle(9) # Peter Luschny, Sep 19 2012

E.g.f. for column m: (((tanh(x))^m)/m!)/cosh(x), m >= 0. Use trigonometric functions for unsigned case.
a(n, m) = a(n-1, m-1)-((m+1)^2)*a(n-1, m+1); a(0, 0)=1; a(n, -1) := 0, a(n, m)=0 if n < m. Use sum of the two recursion terms for unsigned case.
a(n, k) = (1/(k+1)!)*Sum_{q=0..n} C(n,q)*((-1)^(n-q)+1)*((-1)^(q-k)+1)*Sum_{j=0..q-k} C(j+k,k)*(j+k+1)!*2^(q-j-k-2)*(-1)^j*Stirling2(q+1,j+k+1). - Vladimir Kruchinin, Feb 12 2019

New name (using a comment from Paul Barry) from Peter Luschny, Apr 15 2015

A060083 Coefficients of even-indexed Euler polynomials (rising powers without zeros).

Original entry on oeis.org

1, -1, 1, 1, -2, 1, -3, 5, -3, 1, 17, -28, 14, -4, 1, -155, 255, -126, 30, -5, 1, 2073, -3410, 1683, -396, 55, -6, 1, -38227, 62881, -31031, 7293, -1001, 91, -7, 1, 929569, -1529080, 754572, -177320, 24310, -2184, 140, -8, 1, -28820619

Offset: 0

Wolfdieter Lang, Mar 29 2001

E(2*n,1/2)*(-4)^n = A000364(n) (signless Euler numbers without zeros).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 809.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
J. Cigler, q-Fibonacci polynomials and q-Genocchi numbers, arXiv:0908.1219
H.-C. Herbig, D. Herden, C. Seaton, On compositions with x^2/(1-x), arXiv preprint arXiv:1404.1022, 2014

A060082 (falling powers).
Matrix inverse is A102054. Column 0 is A001469 (Genocchi numbers).
Cf. A102054, A001469.

t[n_, k_] := Binomial[2*n, 2*k]*2*(n - k)*EulerE[2*(n - k) - 1, 0]/(2*k + 1); t[n_, n_] = 1; Table[t[n, k], {n, 0, 9}, {k, 0, n }] // Flatten (* Jean-François Alcover, Jul 03 2013 *)

{T(n,k)=local(X=x+x*O(x^(2*n)),Y=y+y*O(y^(2*k+1))); (2*n)!*polcoeff(polcoeff((cosh(X*Y)*(Y-1)+ exp(X*Y)/(exp(X)+1)+exp(-X*Y)/(exp(-X)+1))/Y,2*n,x),2*k,y)} (Hanna)

E(2*n, x)= sum(a(n, m)*x^(2*m+1), m=0..n-1) + x^(2*n), n >= 1; E(0, x)=1.
T(n, k) = A102054(n, k+1) - A102054(n+1, k+1), where A102054 is matrix inverse. E.g.f.: A(x^2, y^2) = [cosh(xy)*(y-1) + exp(xy)/(exp(x)+1) + exp(-xy)/(exp(-x)+1)]/y. - Paul D. Hanna, Dec 28 2004
T(n,k) = 1/(2*k+1)*binomial(2*n,2*k)*A001469(n-k) for 0 <= k <= n-1.
Let F(n,x) = Sum_{k=0..n-1} binomial(n-k-1,k)*x^k be a Fibonacci polynomial (see A011973 for coefficients). Then F(2*n,x) = -Sum_{k=0..n-1} T(n,k)*F(2*k+1,x). For example, F(8,x) = -17*F(1,x) + 28*F(3,x) - 14*F(5,x) + 4*F(7,x). See Cigler, Corollary 1.3. - Peter Bala, Mar 14 2012

A086872 Triangle T(n, k) read by rows; given by [1, 2, 3, 4, 5, 6, ..] DELTA [1, 4, 9, 16, 25, 36, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 3, 8, 5, 15, 75, 121, 61, 105, 840, 2478, 3128, 1385, 945, 11025, 51030, 115350, 124921, 50521, 10395, 166320, 1105335, 3859680, 7365633, 7158128, 2702765

Offset: 0

Philippe Deléham, Aug 20 2003, Aug 17 2007

			Triangle begins:
1;
1, 1;
3, 8, 5;
15, 75, 121, 61;
105, 840, 2478, 3128, 1385;
945, 11025, 51030, 115350, 124921, 50521;
10395, 166320, 1105335, 3859680, 7365633, 7158128, 2702765 ; ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000182 (row sums), A000364 (first diagonal), A001147 (first column), A084938, A261065 (2nd column).

Sum( k>=0, T(n, k)*(-1)^k ) = 0; if n>0.
Sum( k>=0, T(n, k)*(-1/2)^k ) = (1/2)^n.
Sum_{k, 0<=k<=n}T(n,k)*x^(n-k) = (-1)^n*A121822(n), (-1)^n*A092812(n), (-1)^n*A054879(n), A009117(n), A033999(n), A000007(n), A000364(n), A000182(n+1) for x = -6, -5, -4, -3, -2, -1, 0, 1 respectively .

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A242194 Least prime divisor of E_{2*n} which does not divide any E_{2*k} with k < n, or 1 if such a primitive prime divisor of E_{2*n} does not exist, where E_m denotes the m-th Euler number given by A122045.

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A302584 a(n) = n! * [x^n] exp(n*x)/cos(x).

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A302585 a(n) = n! * [x^n] exp(n*x)/cosh(x).

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A318146 Coefficients of the Omega polynomials of order 2, triangle T(n,k) read by rows with 0<=k<=n.

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A002438 Multiples of Euler numbers.

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A012816 E.g.f. arctan(sec(x)*sinh(x)) (odd powers only).

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A060074 Triangle A060058 by diagonals.

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A060081 Exponential Riordan array (sech(x), tanh(x)).

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A242194 Least prime divisor of E_{2n} which does not divide any E_{2k} with k < n, or 1 if such a primitive prime divisor of E_{2*n} does not exist, where E_m denotes the m-th Euler number given by A122045.