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 A111593 #74 Jan 20 2025 20:24:06
%S A111593 1,0,1,0,0,1,0,-2,0,1,0,0,-8,0,1,0,16,0,-20,0,1,0,0,136,0,-40,0,1,0,
%T A111593 -272,0,616,0,-70,0,1,0,0,-3968,0,2016,0,-112,0,1,0,7936,0,-28160,0,
%U A111593 5376,0,-168,0,1,0,0,176896,0,-135680,0,12432,0,-240,0,1,0,-353792,0,1805056,0,-508640,0,25872
%N A111593 Triangle of tanh numbers.
%C A111593 Sheffer triangle associated to Sheffer triangle A060081.
%C A111593 For Sheffer triangles (matrices) see the explanation and S. Roman reference given under A048854.
%C A111593 In the umbral calculus (see the S. Roman reference) this triangle would be called associated for (1,arctanh(y)).
%C A111593 Without the n=0 row and m=0 column and unsigned, this is the Jabotinsky triangle A059419.
%C A111593 The inverse matrix of A with elements a(n,m), n,m>=0, is A111594.
%C A111593 The row polynomials p(n,x):=sum(a(n,m)*x^m,m=0..n), together with the row polynomials s(n,x) of A060081, 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 A111593 The row polynomials p(n,x) (defined above) have e.g.f. exp(x*tanh(y)).
%C A111593 Exponential Riordan array [1, tanh(x)], inverse of [1, arctanh(x)] which is A111594. - _Paul Barry_, May 30 2010
%C A111593 Also the Bell transform of A155585(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 26 2016
%H A111593 W. Lang, <a href="/A111593/a111593.txt">First 10 rows.</a>
%F A111593 E.g.f. for column m>=0: ((tanh(x))^m)/m!.
%F A111593 a(n, m) = coefficient of x^n of ((tanh(x))^m)/m!, n>=m>=0, else 0.
%F A111593 a(n, m) = a(n-1, m-1) - (m+1)*m*a(n-1, m+1), a(n, -1):=0, a(0, 0)=1, a(n, m)=0 for n<m.
%F A111593 T(n,m) = (Sum_{k=0..n-m} binomial(k+m-1,m-1)*(k+m)!*(-1)^k*2^(n-k-m)*stirling2(n,k+m))/m!, T(0,0)=1. - _Vladimir Kruchinin_, Jun 09 2011
%F A111593 With e.g.f. exp(x*tanh(t)) = sum(n>= 0, P(n,x)*t^n/n!), the lowering operator is L = arctanh(d/dx) = d/dx + (1/3)(d/dx)^3 + (1/5)(d/dx)^5 + ..., and the raising operator is R = x [1 - (d/dx)^2], where L P(n,x) = n P(n-1,x) and R P(n,x) = P(n+1,x), since the sequence is a binomial Sheffer sequence. - _Tom Copeland_, Oct 01 2015
%F A111593 The raising operator R = x - x D^2 in matrix form acting on an o.g.f. (formal power series) is the transpose of the production matrix M below. The linear term x is the diagonal of ones after transposition. The other transposed diagonal (A002378) comes from -x D^2 x^n = -n * (n-1) x^(n-1). Then P(n,x) = (1,x,x^2,..) M^n (1,0,0,..)^T. - _Tom Copeland_, Aug 17 2016
%e A111593 Binomial convolution of row polynomials: p(3,x)= -2*x+x^3; p(2,x)=x^2, p(1,x)= x, p(0,x)= 1, together with those from A060081:
%e A111593 s(3,x)= -5*x+x^3; s(2,x)= -1+x^2, s(1,x)= x, s(0,x)= 1;
%e A111593 therefore -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 A111593 From _Paul Barry_, May 30 2010: (Start)
%e A111593 Triangle begins:
%e A111593 1;
%e A111593 0, 1;
%e A111593 0, 0, 1;
%e A111593 0, -2, 0, 1;
%e A111593 0, 0, -8, 0, 1;
%e A111593 0, 16, 0, -20, 0, 1;
%e A111593 0, 0, 136, 0, -40, 0, 1;
%e A111593 0, -272, 0, 616, 0, -70, 0, 1;
%e A111593 0, 0, -3968, 0, 2016, 0, -112, 0, 1;
%e A111593 Production matrix begins:
%e A111593 0, 1;
%e A111593 0, 0, 1;
%e A111593 0, -2, 0, 1;
%e A111593 0, 0, -6, 0, 1;
%e A111593 0, 0, 0, -12, 0, 1;
%e A111593 0, 0, 0, 0, -20, 0, 1;
%e A111593 0, 0, 0, 0, 0, -30, 0, 1;
%e A111593 0, 0, 0, 0, 0, 0, -42, 0, 1;
%e A111593 0, 0, 0, 0, 0, 0, 0, -56, 0, 1; (End)
%p A111593 # The function BellMatrix is defined in A264428.
%p A111593 BellMatrix(n -> 2^(n+1)*euler(n+1, 1), 9); # _Peter Luschny_, Jan 26 2016
%t A111593 t[0, 0] = 1; t[n_, m_] := Sum[ Binomial[k+m-1, m-1]*(k+m)!*(-1)^(k)*2^(n-k-m)*StirlingS2[n, k+m], {k, 0, n-m}]/m!; Table[t[n, m], {n, 0, 11}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 05 2013, after _Vladimir Kruchinin_ *)
%t A111593 BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A111593 rows = 12;
%t A111593 M = BellMatrix[2^(#+1)*EulerE[#+1, 1]&, rows];
%t A111593 Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 23 2018, after _Peter Luschny_ *)
%o A111593 (Maxima)
%o A111593 T(n,m):=if n=0 and m=0 then 1 else sum(binomial(k+m-1,m-1)*(k+m)!*(-1)^(k)*2^(n-k-m)*stirling2(n,k+m),k,0,n-m)/m!; /* _Vladimir Kruchinin_, Jun 09 2011 */
%o A111593 (Sage) # uses[riordan_array from A256893]
%o A111593 riordan_array(1, tanh(x), 9, exp=true) # _Peter Luschny_, Apr 19 2015
%Y A111593 Row sums: A003723. Unsigned row sums: A006229.
%Y A111593 Cf. A059419, A060081, A111594.
%Y A111593 Cf. A002378.
%K A111593 sign,easy,tabl
%O A111593 0,8
%A A111593 _Wolfdieter Lang_, Aug 23 2005