A322765 Array read by upwards antidiagonals: T(m,n) = number of set partitions of the multiset consisting of one copy each of x_1, x_2, ..., x_m, and two copies each of y_1, y_2, ..., y_n, for m >= 0, n >= 0.

1, 1, 2, 2, 4, 9, 5, 11, 26, 66, 15, 36, 92, 249, 712, 52, 135, 371, 1075, 3274, 10457, 203, 566, 1663, 5133, 16601, 56135, 198091, 877, 2610, 8155, 26683, 91226, 325269, 1207433, 4659138, 4140, 13082, 43263, 149410, 537813, 2014321, 7837862, 31638625, 132315780

Offset

0,3

References

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778.

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Formula

Knuth p. 779 gives a recurrence using the Bell numbers A000110 (see Maple program).

From Alois P. Heinz, Jul 21 2021: (Start)

A(n,k) = A001055(A002110(n+k)*A002110(k)).

A(n,k) = A346500(n+k,k). (End)

Example

The array begins:
    1,    2,     9,     66,      712,     10457,      198091, ...
    1,    4,    26,    249,     3274,     56135,     1207433, ...
    2,   11,    92,   1075,    16601,    325269,     7837862, ...
    5,   36,   371,   5133,    91226,   2014321,    53840640, ...
   15,  135,  1663,  26683,   537813,  13241402,   389498179, ...
   52,  566,  8155, 149410,  3376696,  91914202,  2955909119, ...
  203, 2610, 43263, 894124, 22451030, 670867539, 23456071495, ...
  ...

Maple

B := n -> combinat[bell](n):
P := proc(m, n) local k; global B; option remember;
if n = 0 then B(m)  else
(1/2)*( P(m+2, n-1) + P(m+1, n-1) + add( binomial(n-1, k)*P(m, k), k=0..n-1) ); fi; end; # P(m, n) (which is Knuth's notation) is T(m, n)

Mathematica

P[m_, n_] := P[m, n] = If[n == 0, BellB[m], (1/2)(P[m+2, n-1] + P[m+1, n-1] + Sum[Binomial[n-1, k] P[m, k], {k, 0, n-1}])];
Table[P[m-n, n], {m, 0, 8}, {n, 0, m}] // Flatten (* Jean-François Alcover, Jan 02 2019, from Maple *)

Prog

(PARI) {T(n, k) = if(k==0, sum(j=0, n, stirling(n, j, 2)), (T(n+2, k-1)+T(n+1, k-1)+sum(j=0, k-1, binomial(k-1, j)*T(n, j)))/2)} \\ Seiichi Manyama, Nov 21 2020

Crossrefs

Rows include A020555, A322766, A322767.

Columns include A000110, A035098, A322764, A322768.

Main diagonal is A322769.

See A322770 for partitions into distinct parts.

Cf. A001055, A002110, A346500.

Sequence in context: A280362 A241130 A019822 * A281605 A199499 A351351

Adjacent sequences: A322762 A322763 A322764 * A322766 A322767 A322768

Keyword

nonn,tabl

Author

N. J. A. Sloane, Dec 30 2018

Status

approved