A082137 Square array of transforms of binomial coefficients, read by antidiagonals.

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

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 }.

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

Cf. A119468, A007318, A134309.

Sequence in context: A159830 A293472 A046726 * A091187 A318607 A340106

Adjacent sequences: A082134 A082135 A082136 * A082138 A082139 A082140

Keyword

easy,nonn,tabl

Author

Paul Barry, Apr 06 2003

Status

approved