A346424 Number of partitions of the 2n-multiset {0,...,0,1,2,...,n}.

1, 2, 11, 74, 592, 5317, 52902, 572402, 6670707, 83025806, 1096662664, 15292076689, 224145880470, 3440981816071, 55153081768896, 920494136057715, 15959177281931953, 286834809549486462, 5334308665713522860, 102476857445135062727, 2030589375575413246579

Offset

0,2

Comments

Also number of factorizations of 2^n * Product_{i=1..n} prime(i+1); a(2) = 11: 2*2*3*5, 3*4*5, 2*5*6, 6*10, 2*3*10, 5*12, 4*15, 2*2*15, 3*20, 2*30, 60.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..509

Formula

a(n) = A001055(A000079(n)*A070826(n+1)).

a(n) = Sum_{j=0..n} A048993(n,j)*A292508(n,j+1).

a(n) = A346426(n,n).

Example

a(0) = 1: {}.
a(1) = 2: 01, 0|1.
a(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.

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-> add(S(n, j)*b(n, j), j=0..n):
seq(a(n), n=0..21);

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_] := Sum[S[n, j]*b[n, j], {j, 0, n}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Apr 06 2022, after Alois P. Heinz *)

Crossrefs

Main diagonal of A346426.

Cf. A000040, A000041, A000079, A000110, A001055, A048993, A070826, A292508, A346519.

Sequence in context: A231556 A207397 A365153 * A319743 A166992 A058789

Adjacent sequences: A346421 A346422 A346423 * A346425 A346426 A346427

Keyword

nonn

Author

Alois P. Heinz, Jul 16 2021

Status

approved