0,8

Coefficients of polynomials H(n,x) related to Euler polynomials through H(n,x(x-1)) = E(2n,x).

J. M. Hammersley, An undergraduate exercise in manipulation, The Mathematical Scientist, 14 (1989) 1-23.

A. F. Horadam, Generation of Genocchi polynomials of first order by recurrence relation, Fib. Quart. 2 (1992), 239-243.

D. Dumont and J. Zeng, Polynomes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.

Ira M. Gessel and X. G. Viennot, Determinants, paths, and plane partitions, 1989, p. 27, eqn 12.1

E.g.f.: Sum((n, k=0..inf) T(n, k) t^k x^(2n)/(2n)! = cosh(sqrt(1+4t) x/2)/ cosh(x/2).

T(k, n) = Sum[i=0..n-k, A028296(i)/4^(n-k)*C(2n, 2i)*C(n-l, n-k-l)], or 0 if n<k.

Polynomial recurrences: x^n = Sum[0<=2i<=n, C(n, 2i)*H(n-i, x)]; (1/4+x)^n = Sum[m=0..n, C(2n, 2m)*(1/4)^(n-m)*H(m, x)].

Dumont/Zeng give a continued fraction and other formulae.

Triangle T(n, k) read by rows; given by [0, 1, 2, 4, 6, 9, 12, 16, ...] DELTA A000035, where DELTA is Deleham's operator defined in A084938.

Sum_{k, 0<=k<=n}(-4)^(n-k)*T(n,k)=A000364(n)(Euler numbers) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 25 2006

{1}, {0,1}, {0,-1,1}, {0,3,-3,1}, {0,-17,17,-6,1}, ...

(PARI) { S2(n, k) = (1/k!)*sum(i=0, k, (-1)^(k-i)*binomial(k, i)*i^n) }{ Eu(n) = sum(m=0, n, (-1)^m*m!*S2(n+1, m+1)*(-1)^floor(m/4)*2^-floor(m/2)*((m+1)%4!=0)) } T(n, k)=if(n<k, 0, sum(l=0, n-k, Eu(2*l)/2^(2*(n-k))*binomial(2*n, 2*l)*binomial(n-l, n-k-l))) (from R. Stephan)

Sum_{k>=0} (-1)^(n+k)*2^(n-k)*T(n, k) = A005647(n). Sum_{k>=0} (-1)^(n+k)*2^(2n-k)*T(n, k) = A000795(n). Sum_{k>=0} (-1)^(n+k)*T(n, k) = A006846(n), where A006846 = Hammersley's polynomial p_n(1). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 26 2004.

Column sequences (without leading zeros) give, for k=1..10: A065547 (twice), A095652-9.

Cf. A000795, A005647, A000035.

See A085707 for unsigned and transposed version.

See A098435 for negative values of n, k.

Sequence in context: A092747 A122850 A132062 this_sequence A065551 A059441 A059790

Adjacent sequences: A065544 A065545 A065546 this_sequence A065548 A065549 A065550

sign,tabl

Wouter Meeussen (wouter.meeussen(AT)pandora.be), Dec 02 2001

Edited by Ralf Stephan (ralf(AT)ark.in-berlin.de), Sep 08 2004