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A060083
Coefficients of even-indexed Euler polynomials (rising powers without zeros).
7
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
0,5
COMMENTS
E(2*n,1/2)*(-4)^n = A000364(n) (signless Euler numbers without zeros).
REFERENCES
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.
LINKS
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.-C. Herbig, D. Herden, C. Seaton, On compositions with x^2/(1-x), arXiv preprint arXiv:1404.1022, 2014
FORMULA
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
MATHEMATICA
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 *)
PROG
(PARI) {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)
CROSSREFS
A060082 (falling powers).
Matrix inverse is A102054. Column 0 is A001469 (Genocchi numbers).
Sequence in context: A201377 A368070 A322942 * A069931 A373534 A209152
KEYWORD
sign,easy,tabl
AUTHOR
Wolfdieter Lang, Mar 29 2001
STATUS
approved