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 A060081 #66 Apr 20 2025 02:25:55
%S A060081 1,0,1,-1,0,1,0,-5,0,1,5,0,-14,0,1,0,61,0,-30,0,1,-61,0,331,0,-55,0,1,
%T A060081 0,-1385,0,1211,0,-91,0,1,1385,0,-12284,0,3486,0,-140,0,1,0,50521,0,
%U A060081 -68060,0,8526,0,-204,0,1,-50521,0,663061
%N A060081 Exponential Riordan array (sech(x), tanh(x)).
%C A060081 Previous name was: "Triangle of coefficients (lower triangular matrix) of certain (binomial) convolution polynomials related to 1/cosh(x) and tanh(x). Use trigonometric functions for the unsigned version".
%C A060081 Row sums give A009265(n) (signed); A009244(n) (unsigned). Column sequences without interspersed zeros and unsigned: A000364 (Euler), A000364, A060075-8 for m=0,...,5.
%C A060081 a(n,m) = ((-1)^((n-m)/2))*ay(m+1,(n-m)/2) if n-m is even, else 0; where the rectangular array ay(n,m) is defined in A060058 Formula.
%C A060081 The row polynomials p(n,x) appear in a problem of thermo field dynamics (Bogoliubov transformation for the harmonic Bose oscillator). See the link to a .ps.gz file where they are called R_{n}(x).
%C A060081 The inverse of this Sheffer matrix with elements a(n,m) is the Sheffer matrix A060524. This Sheffer triangle appears in the Moyal star product of the harmonic Bose oscillator: x^{*n} = Sum_{m=0..n} a(n,m) x^m with x = 2 (bar a) a/hbar. See the Th. Spernat link, pp. 28, 29, where the unsigned version is used for y=-ix. - _Wolfdieter Lang_, Jul 22 2005
%C A060081 In the umbral calculus (see Roman reference under A048854) the p(n,x) are called Sheffer for (g(t)=1/cosh(arctanh(t)) = 1/sqrt(1-t^2), f(t)=arctanh(t)).
%C A060081 p(n,x) := Sum_{m=0..n} a(n,m)*x^m, n >= 0, are monic polynomials satisfying p(n,x+y) = Sum_{k=0..n} binomial(n,k)*p(k,x)*q(n-k,y) (binomial, also called exponential, convolution polynomials) with the row polynomials of the associated triangle q(n,x) := Sum_{m=0..n} A111593(n,m)*x^m. E.g.f. for p(n,x) is exp(x*tanh(z))*cosh(z)(signed). [Corrected by _Wolfdieter Lang_, Sep 12 2005]
%C A060081 Exponential Riordan array [sech(x), tanh(x)]. Unsigned triangle is [sec(x), tan(x)]. - _Paul Barry_, Jan 10 2011
%D A060081 W. Lang, Two normal ordering problems and certain Sheffer polynomials, in Difference Equations, Special Functions and Orthogonal Polynomials, edts. S. Elaydi et al., World Scientific, 2007, pages 354-368. [From _Wolfdieter Lang_, Feb 06 2009]
%H A060081 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry4/barry122.html">Exponential Riordan arrays and permutation enumeration</a>,Journal of Integer Sequences, Vol. 13 (2010)
%H A060081 Wolfdieter Lang, <a href="/A060081/a060081.ps">Thermo field dynamics, exercise 29. WS 2008/2009 (in German)</a>
%H A060081 Th. Spernat, <a href="/A060081/a060081.pdf">Diplomarbeit 2004 (in German)</a> (with permission)
%F A060081 E.g.f. for column m: (((tanh(x))^m)/m!)/cosh(x), m >= 0. Use trigonometric functions for unsigned case.
%F A060081 a(n, m) = a(n-1, m-1)-((m+1)^2)*a(n-1, m+1); a(0, 0)=1; a(n, -1) := 0, a(n, m)=0 if n < m. Use sum of the two recursion terms for unsigned case.
%F A060081 a(n, k) = (1/(k+1)!)*Sum_{q=0..n} C(n,q)*((-1)^(n-q)+1)*((-1)^(q-k)+1)*Sum_{j=0..q-k} C(j+k,k)*(j+k+1)!*2^(q-j-k-2)*(-1)^j*Stirling2(q+1,j+k+1). - _Vladimir Kruchinin_, Feb 12 2019
%e A060081 p(3,x) = -5*x + x^3.
%e A060081 Exponential convolution together with A111593 for row polynomials q(n,x), case n=2: -1+(x+y)^2 = p(2,x+y) = 1*p(0,x)*q(2,y) + 2*p(1,x)*q(1,y) + 1*p(2,x)*q(0,y) = 1*1*y^2 + 2*x*y + 1*(-1+x^2)*1.
%e A060081 Triangle begins:
%e A060081 1,
%e A060081 0, 1,
%e A060081 -1, 0, 1,
%e A060081 0, -5, 0, 1,
%e A060081 5, 0, -14, 0, 1,
%e A060081 0, 61, 0, -30, 0, 1,
%e A060081 -61, 0, 331, 0, -55, 0, 1,
%e A060081 0, -1385, 0, 1211, 0, -91, 0, 1,
%e A060081 1385, 0, -12284, 0, 3486, 0, -140, 0, 1,
%e A060081 0, 50521, 0, -68060, 0, 8526, 0, -204, 0, 1,
%e A060081 -50521, 0, 663061, 0, -281210, 0, 18522, 0, -285, 0, 1,
%e A060081 ...
%e A060081 As a right-aligned triangle:
%e A060081 1;
%e A060081 0, 1;
%e A060081 -1, 0, 1;
%e A060081 0, -5, 0, 1;
%e A060081 5, 0, -14, 0, 1;
%e A060081 0, 61, 0, -30, 0, 1;
%e A060081 -61, 0, 331, 0, -55, 0, 1;
%e A060081 0, -1385, 0, 1211, 0, -91, 0, 1;
%e A060081 1385, 0, -12284, 0, 3486, 0, -140, 0, 1;
%e A060081 0, 50521, 0, -68060, 0, 8526, 0, -204, 0, 1;
%e A060081 -50521, 0, 663061, 0, -281210, 0, 18522, 0, -285, 0, 1;
%e A060081 ...
%e A060081 Production matrix begins
%e A060081 0, 1;
%e A060081 -1, 0, 1;
%e A060081 0, -4, 0, 1;
%e A060081 0, 0, -9, 0, 1;
%e A060081 0, 0, 0, -16, 0, 1;
%e A060081 0, 0, 0, 0, -25, 0, 1;
%e A060081 0, 0, 0, 0, 0, -36, 0, 1;
%e A060081 0, 0, 0, 0, 0, 0, -49, 0, 1;
%e A060081 0, 0, 0, 0, 0, 0, 0, -64, 0, 1;
%e A060081 - _Paul Barry_, Jan 10 2011
%p A060081 riordan := (d,h,n,k) -> coeftayl(d*h^k,x=0,n)*n!/k!:
%p A060081 A060081 := (n,k) -> riordan(sech(x),tanh(x),n,k):
%p A060081 seq(print(seq(A060081(n,k),k=0..n)),n=0..5); # _Peter Luschny_, Apr 15 2015
%t A060081 max = 12; t = Transpose[ Table[ PadRight[ CoefficientList[ Series[ Tanh[x]^m/m!/Cosh[x], {x, 0, max}], x], max + 1, 0]*Table[k!, {k, 0, max}], {m, 0, max}]]; Flatten[ Table[t[[n, k]], {n, 1, max}, {k, 1, n}]] (* _Jean-François Alcover_, Sep 29 2011 *)
%o A060081 (Sage)
%o A060081 def A060081_triangle(dim): # computes unsigned T(n, k).
%o A060081 M = matrix(ZZ,dim,dim)
%o A060081 for n in (0..dim-1): M[n,n] = 1
%o A060081 for n in (1..dim-1):
%o A060081 for k in (0..n-1):
%o A060081 M[n,k] = M[n-1,k-1]+(k+1)^2*M[n-1,k+1]
%o A060081 return M
%o A060081 A060081_triangle(9) # _Peter Luschny_, Sep 19 2012
%K A060081 sign,easy,tabl
%O A060081 0,8
%A A060081 _Wolfdieter Lang_, Mar 29 2001
%E A060081 New name (using a comment from _Paul Barry_) from _Peter Luschny_, Apr 15 2015