A111593 Triangle of tanh numbers.
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, -272, 0, 616, 0, -70, 0, 1, 0, 0, -3968, 0, 2016, 0, -112, 0, 1, 0, 7936, 0, -28160, 0, 5376, 0, -168, 0, 1, 0, 0, 176896, 0, -135680, 0, 12432, 0, -240, 0, 1, 0, -353792, 0, 1805056, 0, -508640, 0, 25872
Offset: 0
Examples
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: s(3,x)= -5*x+x^3; s(2,x)= -1+x^2, s(1,x)= x, s(0,x)= 1; 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). From _Paul Barry_, May 30 2010: (Start) Triangle begins: 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, -272, 0, 616, 0, -70, 0, 1; 0, 0, -3968, 0, 2016, 0, -112, 0, 1; Production matrix begins: 0, 1; 0, 0, 1; 0, -2, 0, 1; 0, 0, -6, 0, 1; 0, 0, 0, -12, 0, 1; 0, 0, 0, 0, -20, 0, 1; 0, 0, 0, 0, 0, -30, 0, 1; 0, 0, 0, 0, 0, 0, -42, 0, 1; 0, 0, 0, 0, 0, 0, 0, -56, 0, 1; (End)
Links
- W. Lang, First 10 rows.
Crossrefs
Programs
-
Maple
# The function BellMatrix is defined in A264428. BellMatrix(n -> 2^(n+1)*euler(n+1, 1), 9); # Peter Luschny, Jan 26 2016
-
Mathematica
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 *) BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]]; rows = 12; M = BellMatrix[2^(#+1)*EulerE[#+1, 1]&, rows]; Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *) -
Maxima
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 */
-
Sage
# uses[riordan_array from A256893] riordan_array(1, tanh(x), 9, exp=true) # Peter Luschny, Apr 19 2015
Formula
E.g.f. for column m>=0: ((tanh(x))^m)/m!.
a(n, m) = coefficient of x^n of ((tanh(x))^m)/m!, n>=m>=0, else 0.
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
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
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
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
A168402 E.g.f.: Sum_{n>=0} tanh(2^n*x)^n/n!.
1, 2, 16, 496, 63488, 32899584, 68048945152, 560546063546368, 18415227534711980032, 2416302301874471529480192, 1267360472364492854214927581184, 2658246833130441862918267501949222912
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 496*x^3/3! + 63488*x^4/4! +... A(x) = 1 + tanh(2*x) + tanh(4*x)^2/2! + tanh(8*x)^3/3! + tanh(16*x)^4/4! +...+ tanh(2^n*x)^n/n! +... a(n) = coefficient of x^n/n! in G(x)^(2^n) where G(x) = exp(tanh(x)): G(x) = 1 + x + x^2/2! - x^3/3! - 7*x^4/4! - 3*x^5/5! + 97*x^6/6! + 275*x^7/7! +...+ A003723(n)*x^n/n! +...
Programs
-
Mathematica
nmax = 15; CoefficientList[Series[Sum[Tanh[2^k*x]^k/k!, {k, 0, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 11 2020 *) -
PARI
{a(n)=n!*polcoeff(sum(k=0,n,tanh(2^k*x +x*O(x^n))^k/k!),n)} -
PARI
{a(n)=n!*polcoeff(exp(2^n*tanh(x +x*O(x^n))),n)}
Formula
a(n) = [x^n/n! ] exp(2^n*tanh(x)).
a(n) ~ 2^(n^2). - Vaclav Kotesovec, Oct 11 2020
Extensions
Example corrected by Paul D. Hanna, Nov 25 2009
A296835 Expansion of e.g.f. exp(x*tan(x/2)) (even powers only).
1, 1, 4, 33, 451, 9110, 253401, 9246881, 427272364, 24332740569, 1671761966755, 136185663849422, 12966840876896193, 1425738305622057713, 179172604156015950676, 25507107918052543195905, 4081610970381242583997171, 729135575105289450378655526
Offset: 0
Keywords
Examples
exp(x*tan(x/2)) = 1 + x^2/2! + 4*x^4/4! + 33*x^6/6! + 451*x^8/8! + ...
Programs
-
Mathematica
nmax = 17; Table[(CoefficientList[Series[Exp[x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
Formula
a(n) = (2*n)! * [x^(2*n)] exp(x*tan(x/2)).
A296836 Expansion of e.g.f. exp(x*tanh(x/2)) (even powers only).
1, 1, 2, 3, -3, 20, 105, -5271, 133826, -2714517, 25525845, 2131781300, -235250824479, 17527695547713, -1124258412169438, 58383380825728035, -975024061456732035, -398903577787777972396, 97649546210035758250281, -17069419358223320552890167
Offset: 0
Keywords
Examples
exp(x*tanh(x/2)) = 1 + x^2/2! + 2*x^4/4! + 3*x^6/6! - 3*x^8/8! + ...
Programs
-
Mathematica
nmax = 19; Table[(CoefficientList[Series[Exp[x Tanh[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
Formula
a(n) = (2*n)! * [x^(2*n)] exp(x*tanh(x/2)).
A009263 Expansion of e.g.f.: exp(tanh(x)).x.
0, 1, 2, 3, -4, -35, -18, 679, 2200, -18567, -150150, 587499, 11618004, -9780459, -1078005082, -2502478065, 118928710576, 742659420913, -15184771406478, -178572619407021, 2148536703814700, 44455465546363725
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..100
Programs
-
Mathematica
With[{nn=30},CoefficientList[Series[Exp[Tanh[x]]*x,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jun 24 2014 *)
Formula
a(n) = n * A003723(n-1). - Ralf Stephan, Sep 01 2003
Extensions
Extended with signs by Olivier Gérard, Mar 15 1997
Definition clarified and prior Mathematica program replaced by Harvey P. Dale, Jun 24 2014
Comments