This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A320842 #87 Jan 23 2019 09:17:52
%S A320842 1,7,3,127,123,30,4369,6822,3579,630,243649,532542,439899,162630,
%T A320842 22680,20036983,56717781,64697499,37155267,10735470,1247400,
%U A320842 2280356863,7959325221,11656842609,9165745647,4079027880,973580580,97297200,343141433761,1427877062076,2563294235106,2572662311496,1558544277681,569674791180,116270210700,10216206000
%N 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.
%C A320842 It appears that the first column (7, 127, 4369, ...) is from the sequence A002067.
%C A320842 It appears that the diagonal (3, 30, 630, ...) is from the sequence A007019.
%C A320842 It appears as though the unsigned row sum (10, 280, 15400, ...) is from the sequence A025035.
%C A320842 It appears as though the alternating sign row sum (sum(7, -3) = 4, sum(-127, 123, -30) = -34, ...) is from the sequence A002105.
%C A320842 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).
%C A320842 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).
%H A320842 Diego Dominici, <a href="http://dx.doi.org/10.1155/S0161171203303291">Nested derivatives: a simple method for computing series expansions of inverse functions</a>, International Journal of Mathematics and Mathematical Sciences, Volume 2003, Issue 58, Pages 3699-3715.
%H A320842 Wikipedia, <a href="https://en.wikipedia.org/wiki/Student%27s_t-distribution">Student's t-distribution</a>
%e A320842 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.
%e A320842 Triangle begins:
%e A320842 1;
%e A320842 7, 3;
%e A320842 127, 123, 30;
%e A320842 4369, 6822, 3579, 630;
%e A320842 243649, 532542, 439899, 162630, 22680;
%e A320842 20036983, 56717781, 64697499, 37155267, 10735470, 1247400;
%e A320842 2280356863, 7959325221, 11656842609, 9165745647, 4079027880, 973580580, 97297200;
%e A320842 ...
%Y A320842 Cf. A002067, A007019, A025035, A002105.
%K A320842 nonn,tabl
%O A320842 1,2
%A A320842 _Matthew Miller_, Dec 11 2018