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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.

%I A156786 #9 Sep 08 2022 08:45:41
%S A156786 2,-1,-1,1,4,1,-1,-12,-12,-1,1,36,72,36,1,-1,-140,-360,-360,-140,-1,1,
%T A156786 750,2100,2400,2100,750,1,-1,-5082,-15750,-16800,-16800,-15750,-5082,
%U A156786 -1,1,40376,142296,152880,117600,152880,142296,40376,1,-1,-362952,-1453536
%N 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.
%C A156786 Row sums are: {2, -2, 6, -26, 146, -1002, 8102, -75266, 788706, -9193106, 117882182, ...} = signed version of 2*A000262.
%D A156786 J. Riordan, Combinatorial Identities, Wiley, 1968, p.48
%H A156786 G. C. Greubel, <a href="/A156786/b156786.txt">Rows n = 0..100 of triangle, flattened</a>
%F A156786 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).
%F A156786 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
%e A156786 Triangle begins as:
%e A156786    2;
%e A156786   -1,    -1;
%e A156786    1,     4,      1;
%e A156786   -1,   -12,    -12,     -1;
%e A156786    1,    36,     72,     36,      1;
%e A156786   -1,  -140,   -360,   -360,   -140,     -1;
%e A156786    1,   750,   2100,   2400,   2100,    750,      1;
%e A156786   -1, -5082, -15750, -16800, -16800, -15750,  -5082,    -1;
%e A156786    1, 40376, 142296, 152880, 117600, 152880, 142296, 40376, 1;
%t A156786 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
%o A156786 (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
%o A156786 (Magma)
%o A156786 [[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
%o A156786 (Sage)
%o A156786 def L(n, k):
%o A156786     if (k==0 and n==0): return 2
%o A156786     elif (k==0 or k==n): return (-1)^n
%o A156786     else: return (-1)^n*binomial(n,k)*binomial(n-1, k-1)*( factorial(n-k) + (n-k)*factorial(k-1) )
%o A156786 [[L(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, May 20 2019
%Y A156786 Cf. A111596, A008297.
%K A156786 sign,tabl
%O A156786 0,1
%A A156786 _Roger L. Bagula_, Feb 15 2009
%E A156786 Edited by _G. C. Greubel_, May 20 2019