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A082137
Square array of transforms of binomial coefficients, read by antidiagonals.
14
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, 1, 7, 42, 140, 280, 336, 224, 64, 1, 8, 56, 224, 560, 896, 896, 512, 128, 1, 9, 72, 336, 1008, 2016, 2688, 2304, 1152, 256, 1, 10, 90, 480, 1680, 4032, 6720, 7680, 5760, 2560, 512
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
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 }.
As an infinite lower triangular matrix, equals A007318 * A134309. - Gary W. Adamson, Oct 19 2007
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
easy,nonn,tabl
AUTHOR
Paul Barry, Apr 06 2003
STATUS
approved