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A108470 Table read by antidiagonals: T(n,k) = number of labeled partitions of (n,k) into pairs (i,j). 1
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 (list; table; graph; refs; listen; history; text; internal format)
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: A205497 A063394 A193871 * A157152 A136126 A046802

Adjacent sequences:  A108467 A108468 A108469 * A108471 A108472 A108473

KEYWORD

nonn,tabl

AUTHOR

Christian G. Bower, Jun 03 2005

STATUS

approved

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Last modified February 18 02:07 EST 2019. Contains 320237 sequences. (Running on oeis4.)