A156786 The triangular sequence of symmetrical Lah numbers (A111596, A008297) : L(n, m) = (-1)^n* binomial(n,k)*binomial(n-1, k-1)*( (n-k)! + (n-k)*(k-1)! ), with L(0,0) = 2, L(n,0) = L(n,n) = (-1)^n.

2, -1, -1, 1, 4, 1, -1, -12, -12, -1, 1, 36, 72, 36, 1, -1, -140, -360, -360, -140, -1, 1, 750, 2100, 2400, 2100, 750, 1, -1, -5082, -15750, -16800, -16800, -15750, -5082, -1, 1, 40376, 142296, 152880, 117600, 152880, 142296, 40376, 1, -1, -362952, -1453536

Offset: 0

Author

Roger L. Bagula, Feb 15 2009

Keywords

• sign
• tabl

Comments

Row sums are: {2, -2, 6, -26, 146, -1002, 8102, -75266, 788706, -9193106, 117882182, ...} = signed version of 2*A000262.

Examples

			Triangle begins as:
   2;
  -1,    -1;
   1,     4,      1;
  -1,   -12,    -12,     -1;
   1,    36,     72,     36,      1;
  -1,  -140,   -360,   -360,   -140,     -1;
   1,   750,   2100,   2400,   2100,    750,      1;
  -1, -5082, -15750, -16800, -16800, -15750,  -5082,    -1;
   1, 40376, 142296, 152880, 117600, 152880, 142296, 40376, 1;
		

References

• J. Riordan, Combinatorial Identities, Wiley, 1968, p.48

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A111596, A008297.

Programs

Magma

[[n eq 0 and k eq 0 select 2 else k eq 0 or k eq n select (-1)^n else (-1)^n*Binomial(n,k)*Binomial(n-1, k-1)*( Factorial(n-k) + (n-k)* Factorial(k-1) ): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 20 2019

Mathematica

L[n_, k_]:= If[n==0 && k==0, 2, If[k==0 || k==n, (-1)^n, (-1)^n* Binomial[n,k]*Binomial[n-1,k-1]*( (n-k)! + (n-k)*(k-1)! )]]; Table[L[n, k], {n, 0, 10}, {k, 0, n}]//Flatten

PARI

{ L(n, k) = if(n==0 && k==0, 2, if(k==0 || k==n, (-1)^n, (-1)^n* binomial(n,k)*binomial(n-1, k-1)*( (n-k)! + (n-k)*(k-1)! ) )) }; \\ G. C. Greubel, May 20 2019

Sage

def L(n, k):
    if (k==0 and n==0): return 2
    elif (k==0 or k==n): return (-1)^n
    else: return (-1)^n*binomial(n,k)*binomial(n-1, k-1)*( factorial(n-k) + (n-k)*factorial(k-1) )
[[L(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 20 2019

Formula

L(n, m) = if m = 0 then KroneckerDelta(n, 0) otherwise (-1)^n*(n!/m!)* binomial(n-1, m-1) + if m = n then KroneckerDelta(n, 0) otherwise (-1)^n* n! *binomial(n,m)* binomial(n-1, n-m-1).

L(n, m) = (-1)^n* binomial(n,k)*binomial(n-1, k-1)*( (n-k)! + (n-k)*(k-1)! ), with L(0,0) = 2, L(n,0) = L(n,n) = (-1)^n. - G. C. Greubel, May 20 2019

Extensions

Edited by G. C. Greubel, May 20 2019