A089225
Triangle T(n,k) read by rows, defined by T(n,k) = (n-k)*T(n-1,k)+Sum(k=1..n, T(n-1,k)); T(1,1) = 1, T(1,k)= 0 if k >1.
Original entry on oeis.org
1, 2, 1, 7, 4, 3, 35, 22, 17, 14, 228, 154, 122, 102, 88, 1834, 1310, 1060, 898, 782, 694, 17582, 13128, 10818, 9272, 8142, 7272, 6578, 195866, 151560, 126882, 109880, 97218, 87336, 79370, 72792, 2487832, 1981824, 1682196, 1470304, 1309776
Offset: 1
n=4: M = |1,1,1,1|1, 2,1, 1|1, 1, 3, 1|1, 1, 1, 4|
T(4, 1) = permanent of |2, 1, 1|1, 3, 1|1, 1, 4| = 26+5+4 = 35
T(4, 2) = permanent of |1, 1, 1|1, 3, 1|1, 1, 4| = 13+5+4 = 22
T(4, 3) = permanent of |1, 1, 1|1, 2, 1|1, 1, 4| = 9+5+3 = 17
T(4, 4) = permanent of |1, 1, 1|1, 2, 1|1, 1, 3| = 7+4+3 = 14
A091739
Third column (k=7) sequence of array A090216 ((5,5)-Stirling2) divided by 600.
Original entry on oeis.org
1, 4440, 12715200, 33158592000, 84365452800000, 213181366579200000, 537634980016128000000, 1355141067314135040000000, 3415172150786516582400000000, 8606389816065144913920000000000
Offset: 0
Cf.
A091553 (third column of array (4, 4)-Stirling2 divided by 72).
A248028
a(n) = Sum_{k=0..n} |Stirling1(n, k)|*(n-k)! for n>=0.
Original entry on oeis.org
1, 1, 2, 8, 65, 957, 22512, 773838, 36561289, 2271696241, 179538792358, 17584290721868, 2090031277816649, 296326507395472205, 49400463740287289892, 9566059122999739401954, 2129221864475839211318769, 539805407803681202368358785, 154636541536285163968515043306, 49702496963149041682740769491568
Offset: 0
For example, the (5-1) X (5-1) matrix of the form indicated above is equal to
[2 1 1 1]
[2 3 2 2]
[3 3 4 3]
[4 4 4 5]
and the permanent of the above matrix is equal to 957 = a(5). - _John M. Campbell_, Jan 21 2018
-
Table[Sum[Abs[StirlingS1[n,k]]*(n-k)!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Sep 30 2014 *)
-
{Stirling1(n, k)=if(n==0, 1, n!*polcoeff(binomial(x, n), k))}
{a(n)=sum(k=0, n, (-1)^(n-k)*Stirling1(n, k)*(n-k)!)}
for(n=0,20,print1(a(n),", "))
A382805
a(n) = Sum_{k=0..n} (-1)^(n-k) * (Stirling1(n,k) * k!)^2.
Original entry on oeis.org
1, 1, 3, 4, -272, -8524, -96596, 9634752, 983055168, 36429411456, -4303305703296, -1051644384152064, -89651253435644160, 10632887072757561600, 5599203549778990667520, 914684633796830925275136, -89559567563652079025946624, -104514775371103880549281775616
Offset: 0
-
Table[Sum[(-1)^(n - k) (StirlingS1[n, k] k!)^2, {k, 0, n}], {n, 0, 17}]
Table[(n!)^2 SeriesCoefficient[1/(1 + Log[1 + x] Log[1 - y]), {x, 0, n}, {y, 0, n}], {n, 0, 17}]
Comments