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 A111594 #37 Jun 30 2025 10:17:13
%S A111594 1,0,1,0,0,1,0,2,0,1,0,0,8,0,1,0,24,0,20,0,1,0,0,184,0,40,0,1,0,720,0,
%T A111594 784,0,70,0,1,0,0,8448,0,2464,0,112,0,1,0,40320,0,52352,0,6384,0,168,
%U A111594 0,1,0,0,648576,0,229760,0,14448,0,240,0,1
%N A111594 Triangle of arctanh numbers.
%C A111594 Sheffer triangle associated to Sheffer triangle A060524.
%C A111594 For Sheffer triangles (matrices) see the explanation and S. Roman reference given under A048854.
%C A111594 The inverse matrix of A with elements a(n,m), n,m>=0, is given in A111593.
%C A111594 In the umbral calculus notation (see the S. Roman reference) this triangle would be called associated to (1,tanh(y)).
%C A111594 The row polynomials p(n,x):=sum(a(n,m)*x^m,m=0..n), together with the row polynomials s(n,x) of A060524 satisfy the exponential (or binomial) convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), n>=0.
%C A111594 Without the n=0 row and m=0 column and signed, this will become the Jabotinsky triangle A049218 (arctan numbers). For Jabotinsky matrices see the Knuth reference under A039692.
%C A111594 The row polynomials p(n,x) (defined above) have e.g.f. exp(x*arctanh(y)).
%C A111594 Exponential Riordan array [1, arctanh(x)] = [1, log(sqrt((1+x)/(1-x)))]. - _Paul Barry_, Apr 17 2008
%C A111594 Also the Bell transform of A005359. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 27 2016
%H A111594 Wolfdieter Lang, <a href="/A111594/a111594.txt">First 10 rows.</a>
%F A111594 E.g.f. for column m>=0: ((arctanh(x))^m)/m!.
%F A111594 a(n, m) = coefficient of x^n of ((arctanh(x))^m)/m!, n>=m>=0, else 0.
%F A111594 a(n, m) = a(n-1, m-1) + (n-2)*(n-1)*a(n-2, m), a(n, -1):=0, a(0, 0)=1, a(n, m)=0 for n<m.
%e A111594 Binomial convolution of row polynomials:
%e A111594 p(3,x)= 2*x+x^3; p(2,x)=x^2, p(1,x)= x, p(0,x)= 1,
%e A111594 together with those from A060524:
%e A111594 s(3,x)= 5*x+x^3; s(2,x)= 1+x^2, s(1,x)= x, s(0,x)= 1; therefore:
%e A111594 5*(x+y)+(x+y)^3 = s(3,x+y) = 1*s(0,x)*p(3,y) + 3*s(1,x)*p(2,y) + 3*s(2,x)*p(1,y) +1*s(3,x)*p(0,y) = 2*y+y^3 + 3*x*y^2 + 3*(1+x^2)*y + (5*x+x^3).
%e A111594 Triangle begins:
%e A111594 1;
%e A111594 0, 1;
%e A111594 0, 0, 1;
%e A111594 0, 2, 0, 1;
%e A111594 0, 0, 8, 0, 1;
%e A111594 0, 24, 0, 20, 0, 1;
%e A111594 0, 0, 184, 0, 40, 0, 1;
%e A111594 0, 720, 0, 784, 0, 70, 0, 1;
%e A111594 0, 0, 8448, 0, 2464, 0, 112, 0, 1;
%e A111594 ...
%p A111594 # The function BellMatrix is defined in A264428.
%p A111594 BellMatrix(n -> `if`(n::even, n!, 0), 10); # _Peter Luschny_, Jan 27 2016
%t A111594 rows = 10;
%t A111594 t = Table[If[EvenQ[n], n!, 0], {n, 0, rows}];
%t A111594 T[n_, k_] := BellY[n, k, t];
%t A111594 Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018, after _Peter Luschny_ *)
%o A111594 (Sage) # uses[riordan_array from A256893]
%o A111594 riordan_array(1, atanh(x), 9, exp=true) # _Peter Luschny_, Apr 19 2015
%Y A111594 Row sums: A000246.
%Y A111594 Cf. A005359, A049218, A060524, A111593.
%K A111594 nonn,easy,tabl
%O A111594 0,8
%A A111594 _Wolfdieter Lang_, Aug 23 2005