A156996 Triangle T(n, k) = coefficients of p(n,x), where p(n,x) = Sum_{j=0..n} (2*n*(n-j)!/(2*n-j)) * binomial(2*n-j, j) * (x-1)^j and p(0,x) = 1, read by rows.

1, -1, 2, 0, 0, 2, 1, 0, 3, 2, 2, 8, 4, 8, 2, 13, 30, 40, 20, 15, 2, 80, 192, 210, 152, 60, 24, 2, 579, 1344, 1477, 994, 469, 140, 35, 2, 4738, 10800, 11672, 7888, 3660, 1232, 280, 48, 2, 43387, 97434, 104256, 70152, 32958, 11268, 2856, 504, 63, 2, 439792, 976000, 1036050, 695760, 328920, 115056, 30300, 6000, 840, 80, 2

Offset

0,3

References

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 197-199

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

I. Kaplansky and J. Riordan, The problème des ménages, Scripta Math. 12, (1946), 113-124. [Scan of annotated copy]

Anthony C. Robin, 90.72 Circular Wife Swapping, The Mathematical Gazette, Vol. 90, No. 519 (Nov., 2006), pp. 471-478.

L. Takacs, On the probleme des menages, Discr. Math. 36 (3) (1981) 289-297, Table 1.

Formula

T(n, k) = coefficients of p(n,x), where p(n,x) = Sum_{j=0..n} (2*n*(n-j)!/(2*n-j)) * binomial(2*n-j, j) * (x-1)^j and p(0,x) = 1.

Sum_{k=0..n} T(n, k) = n!.

From G. C. Greubel, May 14 2021: (Start)

T(n, 0) = A000179(n).

T(n, k) = Sum_{j=k..n} (-1)^(j+k)*(2*n*(n-j)!/(2*n-j))*binomial(j, k)*binomial(2*n-j, j), with T(0, k) = 1. (End)

Example

Triangle begins as:
       1;
      -1,      2;
       0,      0,       2;
       1,      0,       3,      2;
       2,      8,       4,      8,      2;
      13,     30,      40,     20,     15,      2;
      80,    192,     210,    152,     60,     24,     2;
     579,   1344,    1477,    994,    469,    140,    35,    2;
    4738,  10800,   11672,   7888,   3660,   1232,   280,   48,   2;
   43387,  97434,  104256,  70152,  32958,  11268,  2856,  504,  63,  2;
  439792, 976000, 1036050, 695760, 328920, 115056, 30300, 6000, 840, 80, 2;

Mathematica

(* first program *)
Table[CoefficientList[If[n==0, 1, Sum[Binomial[2*n-k, k]*(n-k)!*(2*n/(2*n-k))*(x- 1)^k, {k, 0, n}]], x], {n, 0, 12}]//Flatten
(* Second program *)
T[n_, k_]:= If[n==0, 1, Sum[(-1)^(j-k)*(2*n*(n-j)!/(2*n-j))*Binomial[j, k]*Binomial[2*n-j, j], {j, k, n}]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 14 2021 *)

Prog

(Magma)
A156996:= func< n, k | n eq 0 select 1 else (&+[(-1)^(j-k)*(2*n*Factorial(n-j)/(2*n-j))*Binomial(j, k)*Binomial(2*n-j, j): j in [k..n]]) >;
[A156996(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 14 2021
(Sage)
def A156996(n, k): return 1 if (n==0) else sum( (-1)^(j-k)*(2*n*factorial(n-j)/(2*n-j))*binomial(j, k)*binomial(2*n-j, j) for j in (k..n) )
flatten([[A156996(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 14 2021

Crossrefs

Cf. A000179, A094314, A156995.

Sequence in context: A178580 A035437 A339815 * A029304 A030202 A159818

Adjacent sequences: A156993 A156994 A156995 * A156997 A156998 A156999

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 20 2009

Extensions

Edited by G. C. Greubel, May 14 2021

Status

approved