A060082 -id:A060082 - cp's OEIS frontend

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

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

A076552 a(n) = (-1)^(n+1)/3/(2n+1) * Sum_{k=0..n} 16^kB(2k)C(2n+1,2k) where B(k) denotes the k-th Bernoulli number.

Original entry on oeis.org

1, 1, 21, 461, 16841, 900921, 66453661, 6463837381, 801626558481, 123457062745841, 23116291464379301, 5171511387852362301, 1362357503097707964121, 417419880467876621822761, 147181297749674368184560941, 59173130526513096478888263221

Offset: 1

Benoit Cloitre, Oct 19 2002

Terms are of form 10k+1.

Eric Weisstein's World of Mathematics, Euler Polynomial.

Cf. A000364, A060082.

max = 28; CoefficientList[Series[1/3-2*Sin[x]/(3*Tan[2*x]), {x, 0, max}], x^2] * Range[0, max, 2]! // Rest (* Jean-François Alcover, Apr 08 2015, after Vladimir Kruchinin *)

a(n)=(-1)^(n+1)/3/(2*n+1)*sum(k=0,n,16^k*bernfrac(2*k)*binomial(2*n+1,2*k))

From Peter Bala, Jul 26 2013: (Start)
It appears that a(n) = 1/3*(A000364(n) - 2*(-1)^n). See A060082.
Conjectural e.g.f. with offset 0 (checked up to a(14)): 1/3*(2 - cos(x)^2 + 2*cos(x)^4)/cos(x)^3 = 1 + x^2/2! + 21*x^4/4! + 461*x^6/6! + .... (End)
G.f.: 1/(Q(0)*3*x) + 2/(3*x^2*(1+x)) - 2/(3*x^2) + 1/(3*x), where Q(k) = 1 - x*(k+1)^2/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Sep 19 2013
a(n) = (2n)! * [x^(2n)] 1/3-2*sin(x)/(3*tan(2*x)). - Vladimir Kruchinin, Apr 08 2015
Conjecture: a(n) = -1/3*(-4)^n*E(2*n,-1/2), where E(n,x) is the n-th Euler polynomial. - Peter Bala, Sep 25 2016

A004173 Triangle of coefficients of Euler polynomials E_2n(x) (exponents in decreasing order).

Original entry on oeis.org

1, 1, -1, 0, 1, -2, 0, 1, 0, 1, -3, 0, 5, 0, -3, 0, 1, -4, 0, 14, 0, -28, 0, 17, 0, 1, -5, 0, 30, 0, -126, 0, 255, 0, -155, 0, 1, -6, 0, 55, 0, -396, 0, 1683, 0, -3410, 0, 2073, 0, 1, -7, 0, 91, 0, -1001, 0, 7293, 0, -31031, 0, 62881, 0, -38227, 0, 1, -8, 0, 140, 0

Offset: 0

N. J. A. Sloane, corrected Jan 09 2007

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.

T. D. Noe, Rows n=0..50 of triangle, flattened
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].
H. Pan and Z. W. Sun, New identities involving Bernoulli and Euler polynomials, arXiv:math/0407363 [math.NT], 2004.
Eric Weisstein's World of Mathematics, Euler Polynomial.

Cf. A060082

Flatten[Table[Reverse @ CoefficientList[EulerE[2n, x], x] , {n, 0, 8}]] (* Jean-François Alcover, Jul 21 2011 *)

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

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A076552 a(n) = (-1)^(n+1)/3/(2n+1) * Sum_{k=0..n} 16^kB(2k)C(2n+1,2k) where B(k) denotes the k-th Bernoulli number.

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A004173 Triangle of coefficients of Euler polynomials E_2n(x) (exponents in decreasing order).

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cp's OEIS Frontend

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

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Author

Keywords

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Mathematica

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Formula

A076552 a(n) = (-1)^(n+1)/3/(2n+1) * Sum_{k=0..n} 16^k*B(2k)*C(2n+1,2k) where B(k) denotes the k-th Bernoulli number.

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PARI

Formula

A004173 Triangle of coefficients of Euler polynomials E_2n(x) (exponents in decreasing order).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

A076552 a(n) = (-1)^(n+1)/3/(2n+1) * Sum_{k=0..n} 16^kB(2k)C(2n+1,2k) where B(k) denotes the k-th Bernoulli number.