0,6

Or, triangle of coefficients (with exponents in increasing order) in polynomials Q_n(u) defined by d^n sec x / dx^n = Q_n(tan x)*sec x.

Interpolates between factorials and Euler (or secant) numbers. Related to Springer numbers.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998, p. 287.

Haigh, Gordon; A "natural" approach to Pick's theorem. Math. Gaz. 64 (1980), no. 429, 173-180.

Michael E. Hoffman, Derivative polynomials for tangent and secant, Amer. Math. Monthly, 102 (1995), 23-30.

Knuth, D. E. and Buckholtz, Thomas J., Computation of tangent, Euler and Bernoulli numbers. Math. Comp. 21 1967 663-688.

S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see pp. 445 and 469.

M.-P. Grosset and A. P. Veselov, Bernoulli numbers and solitons

Michael E. Hoffman, DERIVATIVE POLYNOMIALS, EULER POLYNOMIALS, AND ASSOCIATED INTEGER SEQUENCES

T(n, n) = n!; T(n, 0) = 0 if n = 2m + 1; T(n, 0) = A000364(m) if n = 2m.

Sum_{k>=0} T(m, k)*T(n, k) = T(m+n, 0).

Sum_{k>=0} T(n, k) = A001586(n): Springer numbers.

G.f.: Sum_{n >= 0} Q_n(u)*t^n/n! = 1/(cos t - u sin t).

The polynomials Q_0(u) through Q_6(u) (with exponents in descreasing order) are:

1

u

2*u^2+1

6*u^3+5*u

24*u^4+28*u^2+5

120*u^5+180*u^3+61*u

720*u^6+1320*u^4+662*u^2+61

Triangle begins:

1

0 1

1 0 2

0 5 0 6

5 0 28 0 24

0 61 0 180 0 120

61 0 662 0 1320 0 720

0 1385 0 7266 0 10920 0 5040

1385 0 24568 0 83664 0 100800 0 40320

0 50521 0 408360 0 1023120 0 1028160 0 362880

50521 0 1326122 0 6749040 0 13335840 0 11491200 0 3628800

0 2702765 0 30974526 0 113760240 0 185280480 0 139708800 0 39916800

2702765 0 98329108 0 692699304 0 1979524800 0 2739623040 0 1836172800 0 479001600

See A008294 for another version of this triangle.

Setting u=0,1,2,3,4 gives A000364, A001586, A156129, A156131, A156132.

Setting u=sqrt(2) gives A156134 and A156138; u=sqrt(3) gives A002437 and A002439.

Sequence in context: A054013 A048050 A078153 this_sequence A115333 A105523 A126120

Adjacent sequences: A104032 A104033 A104034 this_sequence A104036 A104037 A104038

nonn,easy,tabl,nice,new

Philippe DELEHAM ( kolotoko(AT)wanadoo.fr), Apr 06 2005

Entry revised by njas, Nov 06 2009