A108470 - OEIS

%I #9 Apr 14 2015 10:39:38

%S 1,1,1,1,3,1,1,7,7,1,1,15,25,15,1,1,31,79,79,31,1,1,63,241,339,241,63,

%T 1,1,127,727,1351,1351,727,127,1,1,255,2185,5235,6721,5235,2185,255,1,

%U 1,511,6559,20119,31831,31831,20119,6559,511,1,1,1023,19681,77379

%N Table read by antidiagonals: T(n,k) = number of labeled partitions of (n,k) into pairs (i,j).

%C Partitions of n black objects labeled 1..n and n white objects labeled 1..n. Each partition must have at least one black object and at least one white object.

%F Double e.g.f.: exp((exp(x)-1)*(exp(y)-1)).

%F T(n,k) = Sum{m=1..min(k,n-k+1)} m!*stirling2(k,m)*stirling2(n-k+1,m). - _Vladimir Kruchinin_, Apr 11 2015

%e 1 1 1 1 1 ...

%e 1 3 7 15 31 ...

%e 1 7 25 79 241 ...

%e 1 15 79 339 1351 ...

%e 1 31 241 1351 6721 ...

%o (Maxima)

%o T(n,k):=sum(m!*stirling2(k,m)*stirling2(n-k+1,m),m,1,min(k,n-k+1)); /* _Vladimir Kruchinin_, Apr 11 2015 */

%o (PARI) antidiag(nn) = {for (n=1, nn, for (k=1, n, print1(sum(m=1, min(k, n-k+1), m!*stirling(k, m, 2)*stirling(n-k+1, m, 2)), ", "); ); print(););} \\ _Michel Marcus_, Apr 11 2015

%o (PARI) tabl(nn) = {default(seriesprecision, nn); for (n=1, nn, for (k=1, nn, print1(k!*polcoeff(polcoeff(n!*exp((exp(x)-1)*(exp(y)-1))+O(x^(n+1)), n, x), k, y), ", "); ); print(););} \\ _Michel Marcus_, Apr 11 2015

%Y Cf. A108461. Columns 1-3: A000012, A000225, A058481. Main diagonal: A023997.

%K nonn,tabl

%O 1,5

%A _Christian G. Bower_, Jun 03 2005