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%I #41 Aug 18 2021 15:28:05
%S 1,1,1,2,2,2,5,5,4,3,15,15,11,7,5,52,52,36,21,12,7,203,203,135,74,38,
%T 19,11,877,877,566,296,141,64,30,15,4140,4140,2610,1315,592,250,105,
%U 45,22,21147,21147,13082,6393,2752,1098,426,165,67,30,115975,115975,70631,33645,13960,5317,1940,696,254,97,42
%N Number A(n,k) of partitions of the (n+k)-multiset {0,...,0,1,2,...,k} with n 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C Also number A(n,k) of factorizations of 2^n * Product_{i=1..k} prime(i+1); A(3,1) = 7: 2*2*2*3, 2*3*4, 4*6, 2*2*6, 3*8, 2*12, 24; A(1,2) = 5: 2*3*5, 5*6, 3*10, 2*15, 30.
%H Alois P. Heinz, <a href="/A346426/b346426.txt">Antidiagonals n = 0..140, flattened</a>
%F A(n,k) = A001055(A000079(n)*A070826(k+1)).
%F A(n,k) = Sum_{j=0..k} A048993(k,j)*A292508(n,j+1).
%F A(n,k) = Sum_{j=0..k} Stirling2(k,j)*Sum_{i=0..n} binomial(j+i-1,i)*A000041(n-i).
%e A(2,2) = 11: 00|1|2, 001|2, 1|002, 0|0|1|2, 0|01|2, 0|1|02, 01|02, 00|12, 0|0|12, 0|012, 0012.
%e Square array A(n,k) begins:
%e 1, 1, 2, 5, 15, 52, 203, 877, 4140, ...
%e 1, 2, 5, 15, 52, 203, 877, 4140, 21147, ...
%e 2, 4, 11, 36, 135, 566, 2610, 13082, 70631, ...
%e 3, 7, 21, 74, 296, 1315, 6393, 33645, 190085, ...
%e 5, 12, 38, 141, 592, 2752, 13960, 76464, 448603, ...
%e 7, 19, 64, 250, 1098, 5317, 28009, 158926, 963913, ...
%e 11, 30, 105, 426, 1940, 9722, 52902, 309546, 1933171, ...
%e 15, 45, 165, 696, 3281, 16972, 95129, 572402, 3670878, ...
%e 22, 67, 254, 1106, 5372, 28582, 164528, 1015356, 6670707, ...
%e ...
%p s:= proc(n) option remember; expand(`if`(n=0, 1,
%p x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
%p end:
%p S:= proc(n, k) option remember; coeff(s(n), x, k) end:
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=0,
%p combinat[numbpart](n), add(b(n-j, i-1), j=0..n)))
%p end:
%p A:= (n, k)-> add(S(k, j)*b(n, j), j=0..k):
%p seq(seq(A(n, d-n), n=0..d), d=0..10);
%t s[n_] := s[n] = Expand[If[n == 0, 1, x Sum[s[n - j] Binomial[n - 1, j - 1], {j, 1, n}]]];
%t S[n_, k_] := S[n, k] = Coefficient[s[n], x, k];
%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, PartitionsP[n], Sum[b[n - j, i - 1], {j, 0, n}]]];
%t A[n_, k_] := Sum[S[k, j] b[n, j], {j, 0, k}];
%t Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* _Jean-François Alcover_, Aug 18 2021, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A000041, A000070, A082775, A093802, A346857, A346858, A346859, A346860, A346861, A346862, A346863.
%Y Rows n=0+1, 2-10 give: A000110, A035098, A169587, A169588, A346851, A346852, A346853, A346854, A346855, A346856.
%Y Main diagonal gives A346424.
%Y Antidiagonal sums give A346428.
%Y Cf. A000079, A001055, A008277, A048993, A070826, A126442, A129306, A219727, A255903, A292508, A346520.
%K nonn,tabl
%O 0,4
%A _Alois P. Heinz_, Jul 16 2021
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