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A123160 Triangle read by rows: T(0,0)=1; T(n,k) = n!(n+k-1)!/((n-k)!(n-1)!(k!)^2) for 0 <= k <= n. 5
1, 1, 1, 1, 4, 3, 1, 9, 18, 10, 1, 16, 60, 80, 35, 1, 25, 150, 350, 350, 126, 1, 36, 315, 1120, 1890, 1512, 462, 1, 49, 588, 2940, 7350, 9702, 6468, 1716, 1, 64, 1008, 6720, 23100, 44352, 48048, 27456, 6435, 1, 81, 1620, 13860, 62370, 162162, 252252, 231660 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

T(n,k) is also the number of order-preserving partial transformations (of an n-element chain) of width k (width(alpha) = |Dom(alpha)|). - Abdullahi Umar, Aug 25 2008

REFERENCES

Frederick T. Wall, Chemical Thermodynamics, W. H. Freeman, San Francisco, 1965 pages 296 and 305

LINKS

Table of n, a(n) for n=0..52.

A. Laradji and A. Umar, Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.

A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8.

FORMULA

T(n,m) = If [n == m == 0, 1, n!*(n + m - 1)!/((n - m)!*(n - 1)!(m!)^2)].

T(n,k) = binomial(n,k)*binomial(n+k-1,k). The row polynomials (except the first) are (1+x)*P(n,0,1,2x+1), where P(n,a,b,x) denotes the Jacobi polynomial. The columns of this triangle give the diagonals of A122899. - Peter Bala, Jan 24 2008

Or, T(n,k) = binomial(n,k)*(n+k-1)!/((n-1)!*k!.

T(n,k)= binomial(n,k)*binomial(n+k-1,n-1). - Abdullahi Umar, Aug 25 2008

G.f.: (x+1)/(2*sqrt((1-x)^2-4*y)) + 1/2. - Vladimir Kruchinin, Jun 16 2015

From _Peter Bala, Jul 20 2015: (Start)

O.g.f. (1 + x)/( 2*sqrt((1 - x)^2 - 4*x*y) ) + 1/2 = 1 + (1 + y)*x + (1 + 4*y + 3*y^2)*x^2 + ....

For n >= 1, the n-th row polynomial R(n,y) = (1 + y)*r(n-1,y), where r(n,y) is the n-th row polynomial of A178301.

exp( Sum_{n >= 1} R(n,y)*x^n/n ) = 1 + (1 + y)*x + (1 + 3*y + 2*y^2)*x^2 + ... is the o.g.f for A088617. (End)

EXAMPLE

Triangle begins:

  1;

  1,  1;

  1,  4,   3;

  1,  9,  18,  10;

  1, 16,  60,  80,  35;

  1, 25, 150, 350, 350, 126;

  ...

MAPLE

T:=proc(n, k) if k=0 and n=0 then 1 elif k<=n then n!*(n+k-1)!/(n-k)!/(n-1)!/(k!)^2 else 0 fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

MATHEMATICA

t[n_, m_] = If [n == m == 0, 1, n!*(n + m - 1)!/((n - m)!*(n - 1)!(m!)^2)]; a = Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[a]

max = 9; s = (x+1)/(2*Sqrt[(1-x)^2-4*y])+1/2 + O[x]^(max+2) + O[y]^(max+2); T[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {y, 0, k}]; Table[T[n-k, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 18 2015, after Vladimir Kruchinin *)

CROSSREFS

Cf. A059481, A122899, A088617, A178301.

Sequence in context: A197698 A193011 A214859 * A109692 A039758 A157894

Adjacent sequences:  A123157 A123158 A123159 * A123161 A123162 A123163

KEYWORD

nonn,easy,tabl

AUTHOR

Roger L. Bagula, Oct 02 2006

EXTENSIONS

Edited by N. J. A. Sloane, Oct 26 2006 and Jul 03 2008

STATUS

approved

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Last modified December 3 02:10 EST 2020. Contains 338898 sequences. (Running on oeis4.)