A108470 Table read by antidiagonals: T(n,k) = number of labeled partitions of (n,k) into pairs (i,j).

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, 1, 1, 127, 727, 1351, 1351, 727, 127, 1, 1, 255, 2185, 5235, 6721, 5235, 2185, 255, 1, 1, 511, 6559, 20119, 31831, 31831, 20119, 6559, 511, 1, 1, 1023, 19681, 77379

Offset

1,5

Comments

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.

Links

Table of n, a(n) for n=1..59.

Formula

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

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

Example

1 1 1 1 1 ...
1 3 7 15 31 ...
1 7 25 79 241 ...
1 15 79 339 1351 ...
1 31 241 1351 6721 ...

Prog

(Maxima)
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 */
(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
(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

Crossrefs

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

Sequence in context: A063394 A344527 A193871 * A328300 A157152 A136126

Adjacent sequences: A108467 A108468 A108469 * A108471 A108472 A108473

Keyword

nonn,tabl

Author

Christian G. Bower, Jun 03 2005

Status

approved