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 A104033 #28 Jan 22 2020 05:16:17
%S A104033 1,-3,1,25,-10,1,-427,175,-21,1,12465,-5124,630,-36,1,-555731,228525,
%T A104033 -28182,1650,-55,1,35135945,-14449006,1782495,-104676,3575,-78,1,
%U A104033 -2990414715,1229758075,-151714563,8912475,-305305,6825,-105,1,329655706465,-135565467080,16724709820,-982532408
%N A104033 Triangle, read by rows, equal to the matrix inverse of triangle A103327, where A103327(n,k) = binomial(2*n+1,2*k+1).
%C A104033 Column 0 equals signed A009843 (expansion of x/cosh(x)). Row sums form signed A000182 (expansion of tanh(x)).
%C A104033 The matrix logarithm is L(n,k) = -(-1)^(n-k)*A000182(n-k)*A103327(n,k), where A000182 = tangent numbers.
%C A104033 Let E(y) = cosh(sqrt(y)) = 1 + 3*y/3! + 5*y^2/5! + 7*y^3/7! + .... so that 1/E(y) = 1 - 3*y/3! + 25*y^2/5! - 427*y^3/7! + .... Then this triangle is the generalized Riordan array (1/E(y), y) with respect to the sequence (2*n+1)! as defined in Wang and Wang. - _Peter Bala_, Aug 06 2013
%H A104033 W. Wang and T. Wang, <a href="https://doi.org/10.1016/j.disc.2007.12.037">Generalized Riordan array</a>, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.
%F A104033 Column k: Sum_{j=0..n} C(2*n+1, 2*j+1) * T(j, k) = 0 (n>k), or 1 (n=k).
%F A104033 Row n: Sum_{j=0..n} T(n, j) * C(2*j+1, 2*k+1) = 0 (k<n), or 1 (k=n).
%F A104033 Sum_{k=0..n} T(n, k) * 4^k = 1 for n >= 0.
%F A104033 T(n, k) = (-1)^(n-k)*A000364(n-k)*A103327(n, k), where A000364 = Euler numbers.
%F A104033 Sum_{k=0..n} (-1)^(n-k)*T(n, k) = A002084(n). - _Philippe Deléham_, Aug 27 2005
%F A104033 From _Peter Bala_, Aug 06 2013: (Start)
%F A104033 Generating function: 1/sqrt(x)*sinh(sqrt(x)*t)/cosh(t) = t + (-3 + x)*t^3/3! + (25 - 10*x + x^2)*t^5/5! + ....
%F A104033 Recurrence equation for the row polynomials: R(n,x) = x^n - Sum_{k = 0..n-1} binomial(2*n+1,2*k+1)*R(k,x) with initial value R(0,x) = 1.
%F A104033 It appears that for arbitrary nonzero complex x we have
%F A104033 lim_{n -> inf} R(n,x^2)/R(n,0) = (1/(Pi/2*x))*sin(Pi/2*x).
%F A104033 A stronger result than pointwise convergence may hold: the convergence may be uniform on compact subsets of the complex plane. This would explain the observation that the real zeros of the polynomials R(n,x) seem to converge to the even squares 4, 16, 36, ... as n increases. Some numerical examples are given below. Cf. A055133, A086646 and A103364.
%F A104033 If p = 2*n + 1 is a prime then all the entries in row n are divisible by p, apart from T(n,n) = 1. Thus the row sum is congruent to 1 modulo p.
%F A104033 Row sums R(n,1) = (-1)^n*A000182(n+1).
%F A104033 R(n,4) = 1; R(n,16) = (1/2)*( 3^(2*n+1) - 1 ) = A096053(n);
%F A104033 R(n,36) = (1/3)*( 5^(2*n+1) - 3^(2*n+1) + 1 );
%F A104033 R(n,64) = (1/4)*( 7^(2*n+1) - 5^(2*n+1) + 3^(2*n+1) - 1 ). (End)
%e A104033 Rows begin:
%e A104033 1;
%e A104033 -3, 1;
%e A104033 25, -10, 1;
%e A104033 -427, 175, -21, 1;
%e A104033 12465, -5124, 630, -36, 1;
%e A104033 -555731 ,228525, -28182, 1650, -55, 1;
%e A104033 35135945, -14449006, 1782495, -104676, 3575, -78, 1;
%e A104033 -2990414715, 1229758075, -151714563, 8912475, -305305, 6825, -105, 1;
%e A104033 329655706465, -135565467080, 16724709820, -982532408, 33669350, -754936, 11900, -136, 1; ...
%e A104033 From _Peter Bala_, Aug 06 2013: (Start)
%e A104033 The real zeros of the row polynomials R(n,x) seem to converge to the even squares as n increases.
%e A104033 Polynomial | Real zeros to 6 decimal places
%e A104033 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
%e A104033 R(5,x) | 3.999986
%e A104033 R(10,x) | 4.000000, 15.999978
%e A104033 R(15,x) | 4.000000, 16.000000, 35.999992, 64.414273, 76.998346
%e A104033 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
%e A104033 (End)
%o A104033 (PARI) {T(n,k) = if(n<k || k<0,0, ((matrix(n+1,n+1,m,j, if(m>=j, binomial(2*m-1,2*j-1))))^-1)[n+1,k+1])}
%o A104033 for(n=0,10,for(k=0,n, print1(T(n,k),", "));print(""))
%o A104033 (PARI) {T(n,k) = binomial(2*n+1,2*k+1) * polcoeff(1/cosh(x+x*O(x^(2*n))), 2*n-2*k) * (2*n-2*k)!}
%o A104033 for(n=0,10,for(k=0,n, print1(T(n,k),", "));print(""))
%Y A104033 Cf. A000364, A103327, A009843, A000182 (unsigned row sums), A055133, A086645, A086646, A096053, A103364.
%K A104033 sign,tabl
%O A104033 0,2
%A A104033 _Paul D. Hanna_, Feb 28 2005
%E A104033 Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jun 08 2007