cp's OEIS Frontend

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.

%I A322765 #42 Jul 21 2021 14:41:17
%S A322765 1,1,2,2,4,9,5,11,26,66,15,36,92,249,712,52,135,371,1075,3274,10457,
%T A322765 203,566,1663,5133,16601,56135,198091,877,2610,8155,26683,91226,
%U A322765 325269,1207433,4659138,4140,13082,43263,149410,537813,2014321,7837862,31638625,132315780
%N 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.
%D A322765 D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778.
%H A322765 Seiichi Manyama, <a href="/A322765/b322765.txt">Antidiagonals n = 0..139, flattened</a>
%F A322765 Knuth p. 779 gives a recurrence using the Bell numbers A000110 (see Maple program).
%F A322765 From _Alois P. Heinz_, Jul 21 2021: (Start)
%F A322765 A(n,k) = A001055(A002110(n+k)*A002110(k)).
%F A322765 A(n,k) = A346500(n+k,k). (End)
%e A322765 The array begins:
%e A322765     1,    2,     9,     66,      712,     10457,      198091, ...
%e A322765     1,    4,    26,    249,     3274,     56135,     1207433, ...
%e A322765     2,   11,    92,   1075,    16601,    325269,     7837862, ...
%e A322765     5,   36,   371,   5133,    91226,   2014321,    53840640, ...
%e A322765    15,  135,  1663,  26683,   537813,  13241402,   389498179, ...
%e A322765    52,  566,  8155, 149410,  3376696,  91914202,  2955909119, ...
%e A322765   203, 2610, 43263, 894124, 22451030, 670867539, 23456071495, ...
%e A322765   ...
%p A322765 B := n -> combinat[bell](n):
%p A322765 P := proc(m,n) local k; global B; option remember;
%p A322765 if n = 0 then B(m)  else
%p A322765 (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)
%t A322765 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}])];
%t A322765 Table[P[m-n, n], {m, 0, 8}, {n, 0, m}] // Flatten (* _Jean-François Alcover_, Jan 02 2019, from Maple *)
%o A322765 (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
%Y A322765 Rows include A020555, A322766, A322767.
%Y A322765 Columns include A000110, A035098, A322764, A322768.
%Y A322765 Main diagonal is A322769.
%Y A322765 See A322770 for partitions into distinct parts.
%Y A322765 Cf. A001055, A002110, A346500.
%K A322765 nonn,tabl
%O A322765 0,3
%A A322765 _N. J. A. Sloane_, Dec 30 2018