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A346426
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.
24
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, 19, 11, 877, 877, 566, 296, 141, 64, 30, 15, 4140, 4140, 2610, 1315, 592, 250, 105, 45, 22, 21147, 21147, 13082, 6393, 2752, 1098, 426, 165, 67, 30, 115975, 115975, 70631, 33645, 13960, 5317, 1940, 696, 254, 97, 42
OFFSET
0,4
COMMENTS
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.
LINKS
FORMULA
A(n,k) = A001055(A000079(n)*A070826(k+1)).
A(n,k) = Sum_{j=0..k} A048993(k,j)*A292508(n,j+1).
A(n,k) = Sum_{j=0..k} Stirling2(k,j)*Sum_{i=0..n} binomial(j+i-1,i)*A000041(n-i).
EXAMPLE
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.
Square array A(n,k) begins:
1, 1, 2, 5, 15, 52, 203, 877, 4140, ...
1, 2, 5, 15, 52, 203, 877, 4140, 21147, ...
2, 4, 11, 36, 135, 566, 2610, 13082, 70631, ...
3, 7, 21, 74, 296, 1315, 6393, 33645, 190085, ...
5, 12, 38, 141, 592, 2752, 13960, 76464, 448603, ...
7, 19, 64, 250, 1098, 5317, 28009, 158926, 963913, ...
11, 30, 105, 426, 1940, 9722, 52902, 309546, 1933171, ...
15, 45, 165, 696, 3281, 16972, 95129, 572402, 3670878, ...
22, 67, 254, 1106, 5372, 28582, 164528, 1015356, 6670707, ...
...
MAPLE
s:= proc(n) option remember; expand(`if`(n=0, 1,
x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
end:
S:= proc(n, k) option remember; coeff(s(n), x, k) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=0,
combinat[numbpart](n), add(b(n-j, i-1), j=0..n)))
end:
A:= (n, k)-> add(S(k, j)*b(n, j), j=0..k):
seq(seq(A(n, d-n), n=0..d), d=0..10);
MATHEMATICA
s[n_] := s[n] = Expand[If[n == 0, 1, x Sum[s[n - j] Binomial[n - 1, j - 1], {j, 1, n}]]];
S[n_, k_] := S[n, k] = Coefficient[s[n], x, k];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, PartitionsP[n], Sum[b[n - j, i - 1], {j, 0, n}]]];
A[n_, k_] := Sum[S[k, j] b[n, j], {j, 0, k}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Aug 18 2021, after Alois P. Heinz *)
CROSSREFS
Main diagonal gives A346424.
Antidiagonal sums give A346428.
Sequence in context: A316660 A098101 A257670 * A105960 A081290 A168256
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 16 2021
STATUS
approved