OFFSET
0,5
COMMENTS
Rows are associated with the expansions of (x^k/k!)exp(x)cosh(x) (leading zeros dropped). Rows include A011782, A057711, A080929, A082138, A080951, A082139, A082140, A082141. Columns are of the form 2^(k-1)C(n+k, k). Diagonals include A069723, A082143, A082144, A082145, A069720.
T(n, k) is also the number of idempotent order-preserving and order-decreasing partial transformations (of an n-chain) of width k (width(alpha)= |Dom(alpha)|). - Abdullahi Umar, Oct 02 2008
Read as a triangle this is A119468 with rows reversed. A119468 has e.g.f. exp(z*x)/(1-tanh(x)). - Peter Luschny, Aug 01 2012
Read as a triangle this is a subtriangle of A198793. - Philippe Deléham, Nov 10 2013
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
Laradji, A. and Umar, A. Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8.
FORMULA
Square array defined by T(n, k)=(2^(n-1)+0^n/2)C(n + k, n)= Sum{k=0..n, C(n+k, k+j)C(k+j, k)(1+(-1)^j)/2 }.
O.g.f. for array read as a triangle: (1-x*(1+t))/((1-x)*(1-x*(1+2*t))) = 1 + x*(1+t) + x^2*(1+2*t+2*t^2) + x^3*(1+3*t+6*t^2+4*t^3) + .... - Peter Bala, Apr 26 2012
For array read as a triangle: T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) -2*T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 10 2013
EXAMPLE
Rows begin
1 1 2 4 8 ...
1 2 6 16 40 ...
1 3 12 40 120 ...
1 4 20 80 280 ...
1 5 30 140 560 ...
Read as a triangle, this begins:
1
1, 1
1, 2, 2
1, 3, 6, 4
1, 4, 12, 16, 8
1, 5, 20, 40, 40, 16
1, 6, 30, 80, 120, 96, 32
... - Philippe Deléham, Nov 10 2013
MAPLE
# As a triangular array:
T := (n, k) -> 2^(k+0^k-1)*binomial(n, k):
for n from 0 to 9 do seq(T(n, k), k=0..n) od; # Peter Luschny, Nov 10 2017
MATHEMATICA
rows = 11; t[n_, k_] := 2^(n-1)*(n+k)!/(n!*k!); t[0, _] = 1; tkn = Table[ t[n, k], {k, 0, rows}, {n, 0, rows}]; Flatten[ Table[ tkn[[ n-k+1, k ]], {n, 1, rows}, {k, 1, n}]] (* Jean-François Alcover, Jan 20 2012 *)
PROG
(Sage)
def A082137_row(n) : # as a triangular array
var('z')
s = (exp(z*x)/(1-tanh(x))).series(x, n+2)
t = factorial(n)*s.coefficient(x, n)
return [t.coefficient(z, n-k) for k in (0..n)]
for n in (0..7) : print(A082137_row(n)) # Peter Luschny, Aug 01 2012
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Apr 06 2003
STATUS
approved