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A322770 Array read by upwards antidiagonals: T(m,n) = number of set partitions into distinct parts 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. 8
1, 1, 1, 2, 3, 5, 5, 9, 18, 40, 15, 31, 70, 172, 457, 52, 120, 299, 801, 2295, 6995, 203, 514, 1393, 4025, 12347, 40043, 136771, 877, 2407, 7023, 21709, 70843, 243235, 875936, 3299218, 4140, 12205, 38043, 124997, 431636, 1562071, 5908978, 23308546 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

REFERENCES

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

LINKS

Table of n, a(n) for n=0..43.

D. E. Knuth, Partitioning a multiset into submultisets, Email to N. J. A. Sloane, Dec 29 2018.

FORMULA

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

EXAMPLE

The array begins:

     1,    1,     5,     40,      457,      6995,      136771, ...

     1,    3,    18,    172,     2295,     40043,      875936, ...

     2,    9,    70,    801,    12347,    243235,     5908978, ...

     5,   31,   299,   4025,    70843,   1562071,    41862462, ...

    15,  120,  1393,  21709,   431636,  10569612,   310606617, ...

    52,  514,  7023, 124997,  2781372,  75114998,  2407527172, ...

   203, 2407, 38043, 764538, 18885177, 559057663, 19449364539, ...

   ...

MAPLE

B := n -> combinat[bell](n):

Q := proc(m, n) local k; global B; option remember;

if n = 0 then B(m)  else

(1/2)*( Q(m+2, n-1) + Q(m+1, n-1) - add( binomial(n-1, k)*Q(m, k), k=0..n-1) ); fi; end;  # Q(m, n) (which is Knuth's notation) is T(m, n)

MATHEMATICA

Q[m_, n_] := Q[m, n] = If[n == 0, BellB[m], (1/2)(Q[m+2, n-1] + Q[m+1, n-1] - Sum[Binomial[n-1, k] Q[m, k], {k, 0, n-1}])];

Table[Q[m-n, n], {m, 0, 8}, {n, 0, m}] // Flatten (* Jean-Fran├žois Alcover, Jan 02 2019, from Maple *)

CROSSREFS

Rows include A094574, A322771, A322772.

Columns include A000110, A087648, A322773, A322774.

Main diagonal is A322775.

Sequence in context: A222705 A241381 A237365 * A257008 A265822 A257012

Adjacent sequences:  A322767 A322768 A322769 * A322771 A322772 A322773

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Dec 30 2018

STATUS

approved

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Last modified March 29 15:16 EDT 2020. Contains 333107 sequences. (Running on oeis4.)