A094314 -id:A094314 - cp's OEIS frontend

A058087 Triangle read by rows, giving coefficients of the ménage hit polynomials ordered by descending powers. T(n, k) for 0 <= k <= n.

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

Offset: 0

N. J. A. Sloane, Dec 02 2000

Riordan's book (page 197) notes that an alternative convention is to put 2 in the first row of the triangle. - William P. Orrick, Aug 09 2020

			The triangle begins:
  1;
  2, -1;
  2,  0,   0;
  2,  3,   0,    1;
  2,  8,   4,    8,     2;
  2, 15,  20,   40,    30,    13;
  2, 24,  60,  152,   210,   192,    80;
  2, 35, 140,  469,   994,  1477,  1344,    579;
  2, 48, 280, 1232,  3660,  7888, 11672,  10800,  4738;
  2, 63, 504, 2856, 11268, 32958, 70152, 104256, 97434, 43387;
The polynomials start:
  [0] 1;
  [1] 2*x - 1;
  [2] 2*x^2;
  [3] 2*x^3 + 3*x^2 + 1;
  [4] 2*x^4 + 8*x^3 + 4*x^2 + 8*x + 2;
  [5] 2*x^5 + 15*x^4 + 20*x^3 + 40*x^2 + 30*x + 13.

I. Kaplansky and J. Riordan, The probleme des menages, Scripta Mathematica, 1946, 12 (2), 113-124.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 198.
Tolman, L. Kirk, "Extensions of derangements", Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing, Humboldt State University, Arcata, California, September 5-7, 1979. Vol. 26. Utilitas Mathematica Pub., 1980. See Table I. - N. J. A. Sloane, Jul 06 2014

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.

Diagonals give A000179, A000425, A000033, A000159, A000181, A000185, A058089, A058090.

Essentially a mirror image of A094314.

U := proc(n) if n = 0 then return 1 fi;
add((2*n/(2*n-k))*binomial(2*n-k, k)*(n-k)!*(x-1)^k, k=0..n) end:
W := proc(r, s) coeff(U(r), x, s ) end:
T := (n, k) -> W(n, n-k): seq(seq(T(n, k), k=0..n), n=0..9);

u[n_] := Sum[ 2*n/(2*n-k)*Binomial[2*n-k, k]*(n-k)!*(x-1)^k, {k, 0, n}]; w[r_, s_] := Coefficient[u[r], x, s]; a[n_, k_] := w[n, n-k]; a[0, 0]=1; Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 10 2012, translated from Maple *)
T[n_, k_]:= If[n==0, 1, Sum[(-1)^j*(2*n*(k-j)!/(n+k-j))*Binomial[j+n-k, n - k]*Binomial[n+k-j, n-k+j], {j, 0, k}]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 15 2021 *)

U(n,t)=sum(k=0,n, ((2*n/(2*n-k))*binomial(2*n-k,k)*(n-k)!*(t-1)^k));
print1(1,", "); for(n=1,9,forstep(k=n,0,-1,print1(polcoef(U(n,'x),k),", "))) \\ Hugo Pfoertner, Aug 30 2020

def A058087(n,k): return 1 if (n==0) else sum( (-1)^j*(2*n*factorial(k-j)/(n+k-j))*binomial(j+n-k, n-k)*binomial(n+k-j, n-k+j) for j in (0..k) )
flatten([[A058087(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 15 2021

a = [[1]]
for n in range(1, 10):
    g = expand(
        sum((x - 1)^ k * (2*n) * binomial(2*n-k, k) * factorial(n-k) / (2*n-k)
            for k in range(0, n + 1)
        )
    )
    coeffs = g.coefficients(sparse=False)
    coeffs.reverse()
    a.append(coeffs) # William P. Orrick, Aug 12 2020

G.f.: (1-x*(y-1))*Sum_{n>=0} ( n!*(x*y)^n/(1+x*(y-1))^(2*n+1) ). - Vladeta Jovovic, Dec 14 2009
Row n of the triangle lists the coefficients of the polynomial U_n(t) = Sum_{k=0..n} (2*n/(2*n-k))*binomial(2*n-k,k)*(n-k)!*(t-1)^k, with higher order terms first (Kaplansky and Riordan). - William P. Orrick, Aug 09 2020
T(n, k) = Sum_{j=0..k} (-1)^j*(2*n*(k-j)!/(n+k-j))*binomial(n-k+j, n-k)*binomial(n+k-j, n-k+j), with T(0, k) = 1. - G. C. Greubel, May 15 2021 [Corrected by Sean A. Irvine, Jul 23 2022]

T(1,1) set to -1 to accord with Riordan by William P. Orrick, Aug 09 2020

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

Original entry on oeis.org

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

Roger L. Bagula, Feb 20 2009

			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;

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

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.

Cf. A000179, A094314, A156995.

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

(* 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 *)

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

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)

Edited by G. C. Greubel, May 14 2021

cp's OEIS Frontend

A058087 Triangle read by rows, giving coefficients of the ménage hit polynomials ordered by descending powers. T(n, k) for 0 <= k <= n.

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

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Author

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Magma

Mathematica

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cp's OEIS Frontend

A058087 Triangle read by rows, giving coefficients of the ménage hit polynomials ordered by descending powers. T(n, k) for 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

SageMath

Formula

Extensions

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.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

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