A320842 Regular triangle whose rows are the coefficients of the Dominici expansion of f(t,x) = (1/2)*(1 - t^2)^(-x) with respect to t.

1, 7, 3, 127, 123, 30, 4369, 6822, 3579, 630, 243649, 532542, 439899, 162630, 22680, 20036983, 56717781, 64697499, 37155267, 10735470, 1247400, 2280356863, 7959325221, 11656842609, 9165745647, 4079027880, 973580580, 97297200, 343141433761, 1427877062076, 2563294235106, 2572662311496, 1558544277681, 569674791180, 116270210700, 10216206000

Offset: 1

Author

Matthew Miller, Dec 11 2018

Keywords

• nonn
• tabl

Comments

It appears that the first column (7, 127, 4369, ...) is from the sequence A002067.

It appears that the diagonal (3, 30, 630, ...) is from the sequence A007019.

It appears as though the unsigned row sum (10, 280, 15400, ...) is from the sequence A025035.

It appears as though the alternating sign row sum (sum(7, -3) = 4, sum(-127, 123, -30) = -34, ...) is from the sequence A002105.

This triangular array arises as the coefficients from terms in the inverse expansion of the function f(t,x) = (1/2)*(1 - t^2)^(-x) with respect to t evaluated at t = 0 for even values of the operation, using a method of Dominici's (nested derivatives, referenced below).

Without proof, appears to be related to computing the 'critical t-value' of Student's t-distribution. (conj.) Critical t-value t_(v, beta) is equal to: sqrt((v/(1-S^2)) - v) where S = (1/2)*Sum_{k>=1} (D^(2*k-2)[f](0)*(1/(2*k-1)!)*(B(1/2, v/2)*(1-2*beta))^(2*k-1)); where (1 - beta) is the confidence interval 'atta' (for a one-tailed distribution such that 'cumulative probability' = t_atta, where beta = 1-atta), x = 1 - (v/2), v: degrees of freedom, B(1/2, v/2) = gamma(1/2)*gamma(v/2)/gamma(1/2 + v/2), D^(2*k - 2)[f](0) is a polynomial function of 'x' whose coefficients are the terms of this sequence as computed using a method of Dominici's on f(t,x) with respect to t (referenced below).

Examples

			Given D^k[f]_(b) = (d/dt [f(t)*D^(k-1)[f](t)])_t = b where D^0[f](b) = 1, then for f(t,x) = (1/2)*(1 - t^2)^(-x) where f(0) = 1/2 one obtains: D^2[f]_(0) = -x/2, D^4[f]_(0) = (x/4)*(7*x - 3), D^6[f]_(0) = -(x/8)*(127*x^2 - 123*x + 30), etc., where b is an arbitrary constant.
Triangle begins:
           1;
           7,          3;
         127,        123,          30;
        4369,       6822,        3579,        630;
      243649,     532542,      439899,     162630,      22680;
    20036983,   56717781,    64697499,   37155267,   10735470,   1247400;
  2280356863, 7959325221, 11656842609, 9165745647, 4079027880, 973580580, 97297200;
         ...
		

Links

• Diego Dominici, Nested derivatives: a simple method for computing series expansions of inverse functions, International Journal of Mathematics and Mathematical Sciences, Volume 2003, Issue 58, Pages 3699-3715.
• Wikipedia, Student's t-distribution

Crossrefs

Cf. A002067, A007019, A025035, A002105.