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(MAGMAMagma)
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approved
proposed
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editing
proposed
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proposed
The triangular sequence of symmetrical Lah numbers (A111596, A008297) : L(n, m)=If[m = = 0, KroneckerDelta[n, 0], (-1)^n* binomial(n!/m!,k)*Binomial[binomial(n - 1, m k- 1]] + If[ -m + )*( (n == 0, KroneckerDelta[-k)! + (n, 0], -k)*(k-1)^n* ! ), with L(0,0) = 2, L(n! Binomial[,0) = L(n - 1, (-m + ,n) = (- 1]/(-m + )^n)! ].
Row sums are: {2, -2, 6, -26, 146, -1002, 8102, -75266, 788706, -9193106, 117882182, ...} = signed version of 2*A000262;.
{2, -2, 6, -26, 146, -1002, 8102, -75266, 788706, -9193106, 117882182,...}.
This sequence uses Riordan's definition.
G. C. Greubel, <a href="/A156786/b156786.txt">Rows n = 0..100 of triangle, flattened</a>
L(n, m) =If[ if m == 0, then KroneckerDelta[(n, 0], ) otherwise (-1)^n*(n!/m!)*Binomial[ 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]]).
+ If[ -m + n == 0, KroneckerDelta[n, 0], (-1)^n* n! Binomial[n - 1, (-m + n) - 1]/(-m + n)! ].
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
{2},
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},;
{-1, -362952, -1453536, -1721664, -1058400, -1058400, -1721664, -1453536, -362952, -1},
{1, 3628890, 16332840, 21833280, 13335840, 7620480, 13335840, 21833280, 16332840, 3628890, 1}
L[n_, m_k_] := If[m n==0 && k== 0, KroneckerDelta2, If[n, k==0], || k==n, (-1)^n, (-1)^n*( Binomial[n!/m!), k]*Binomial[n - 1, m k-1]*( (n-k)! + (n-k)*(k- 1)! )]]; Table[L[n, k], {n, 0, 10}, {k, 0, n}] +//Flatten
If[ - m + n == 0, KroneckerDelta[n, 0], (-1)^n* n! Binomial[n - 1, (-m + n) - 1]/(-m + n)! ];
Table[Table[L[n, m], {m, 0, n}], {n, 0, 10}];
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
(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
(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
sign,tabl,uned
Edited by G. C. Greubel, May 20 2019
approved
editing
_Roger L. Bagula (rlbagulatftn(AT)yahoo.com), _, Feb 15 2009
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
0,1
Row sums are:signed version of 2*A000262;
{2, -2, 6, -26, 146, -1002, 8102, -75266, 788706, -9193106, 117882182,...}.
This sequence uses Riordan's definition.
J. Riordan, Combinatorial Identities, Wiley, 1968, p.48
L(n,m)=If[m == 0, KroneckerDelta[n, 0], (-1)^n*(n!/m!)*Binomial[n - 1, m - 1]]
+ If[ -m + n == 0, KroneckerDelta[n, 0], (-1)^n* n! Binomial[n - 1, (-m + n) - 1]/(-m + 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, -1721664, -1058400, -1058400, -1721664, -1453536, -362952, -1},
{1, 3628890, 16332840, 21833280, 13335840, 7620480, 13335840, 21833280, 16332840, 3628890, 1}
L[n_, m_] = If[m == 0, KroneckerDelta[n, 0], (-1)^n*(n!/m!)*Binomial[n - 1, m - 1]] +
If[ - m + n == 0, KroneckerDelta[n, 0], (-1)^n* n! Binomial[n - 1, (-m + n) - 1]/(-m + n)! ];
Table[Table[L[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]
sign,tabl,uned,new
Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 15 2009
approved