A111593 - OEIS

%I #74 Jan 20 2025 20:24:06

%S 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 -272,0,616,0,-70,0,1,0,0,-3968,0,2016,0,-112,0,1,0,7936,0,-28160,0,

%U 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 Triangle of tanh numbers.

%C Sheffer triangle associated to Sheffer triangle A060081.

%C For Sheffer triangles (matrices) see the explanation and S. Roman reference given under A048854.

%C In the umbral calculus (see the S. Roman reference) this triangle would be called associated for (1,arctanh(y)).

%C Without the n=0 row and m=0 column and unsigned, this is the Jabotinsky triangle A059419.

%C The inverse matrix of A with elements a(n,m), n,m>=0, is A111594.

%C 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 The row polynomials p(n,x) (defined above) have e.g.f. exp(x*tanh(y)).

%C Exponential Riordan array [1, tanh(x)], inverse of [1, arctanh(x)] which is A111594. - _Paul Barry_, May 30 2010

%C Also the Bell transform of A155585(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 26 2016

%H W. Lang, <a href="/A111593/a111593.txt">First 10 rows.</a>

%F E.g.f. for column m>=0: ((tanh(x))^m)/m!.

%F a(n, m) = coefficient of x^n of ((tanh(x))^m)/m!, n>=m>=0, else 0.

%F 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 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 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 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 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 s(3,x)= -5*x+x^3; s(2,x)= -1+x^2, s(1,x)= x, s(0,x)= 1;

%e 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 From _Paul Barry_, May 30 2010: (Start)

%e Triangle begins:

%e   1;

%e   0,     1;

%e   0,     0,     1;

%e   0,    -2,     0,     1;

%e   0,     0,    -8,     0,     1;

%e   0,    16,     0,   -20,     0,     1;

%e   0,     0,   136,     0,   -40,     0,     1;

%e   0,  -272,     0,   616,     0,   -70,     0,     1;

%e   0,     0, -3968,     0,  2016,     0,  -112,     0,     1;

%e Production matrix begins:

%e   0,   1;

%e   0,   0,   1;

%e   0,  -2,   0,   1;

%e   0,   0,  -6,   0,   1;

%e   0,   0,   0, -12,   0,   1;

%e   0,   0,   0,   0, -20,   0,   1;

%e   0,   0,   0,   0,   0, -30,   0,   1;

%e   0,   0,   0,   0,   0,   0, -42,   0,   1;

%e   0,   0,   0,   0,   0,   0,   0, -56,   0,   1; (End)

%p # The function BellMatrix is defined in A264428.

%p BellMatrix(n -> 2^(n+1)*euler(n+1, 1), 9); # _Peter Luschny_, Jan 26 2016

%t 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 BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

%t rows = 12;

%t M = BellMatrix[2^(#+1)*EulerE[#+1, 1]&, rows];

%t Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 23 2018, after _Peter Luschny_ *)

%o (Maxima)

%o 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 (Sage) # uses[riordan_array from A256893]

%o riordan_array(1, tanh(x), 9, exp=true) # _Peter Luschny_, Apr 19 2015

%Y Row sums: A003723. Unsigned row sums: A006229.

%Y Cf. A059419, A060081, A111594.

%Y Cf. A002378.

%K sign,easy,tabl

%O 0,8

%A _Wolfdieter Lang_, Aug 23 2005