A346823 -id:A346823 - cp's OEIS frontend

A346520 Number A(n,k) of partitions of the (n+k)-multiset {0,...,0,1,2,...,k} with n 0's into distinct multisets; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 2, 2, 1, 5, 5, 3, 2, 15, 15, 9, 5, 2, 52, 52, 31, 16, 7, 3, 203, 203, 120, 59, 25, 10, 4, 877, 877, 514, 244, 100, 38, 14, 5, 4140, 4140, 2407, 1112, 442, 161, 56, 19, 6, 21147, 21147, 12205, 5516, 2134, 750, 249, 80, 25, 8, 115975, 115975, 66491, 29505, 11147, 3799, 1213, 372, 111, 33, 10

Offset: 0

Alois P. Heinz, Jul 21 2021

Also number A(n,k) of factorizations of 2^n * Product_{i=1..k} prime(i+1) into distinct factors; A(3,1) = 5: 2*3*4, 4*6, 3*8, 2*12, 24; A(1,2) = 5: 2*3*5, 5*6, 3*10, 2*15, 30.

			A(2,2) = 9: 00|1|2, 001|2, 1|002, 0|01|2, 0|1|02, 01|02, 00|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, ...
  1,  3,   9,  31,  120,   514,  2407,  12205,   66491, ...
  2,  5,  16,  59,  244,  1112,  5516,  29505,  168938, ...
  2,  7,  25, 100,  442,  2134, 11147,  62505,  373832, ...
  3, 10,  38, 161,  750,  3799, 20739, 121141,  752681, ...
  4, 14,  56, 249, 1213,  6404, 36332, 220000, 1413937, ...
  5, 19,  80, 372, 1887, 10340, 60727, 379831, 2516880, ...
  6, 25, 111, 539, 2840, 16108, 97666, 629346, 4288933, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000009, A036469, A346822, A346823, A346824, A346825, A346826, A346827, A346828, A346829, A346830.
Rows n=0+1,2-10 give: A000110, A087648, A346813, A346814, A346815, A346816, A346817, A346818, A346819, A346820.
Main diagonal gives A346519.
Antidiagonal sums give A346521.
Cf. A000040, A000079, A045778, A048993, A070826, A346426.

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
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, g(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..12);

g[n_] := g[n] = If[n == 0, 1, Sum[g[n - j]*Sum[If[OddQ[d], d, 0], {d, Divisors[j]}], {j, 1, n}]/n];
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, g[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, 12}] // Flatten (* Jean-François Alcover, Jul 31 2021, after Alois P. Heinz *)

A(n,k) = A045778(A000079(n)*A070826(k+1)).

A(n,k) = Sum_{j=0..k} Stirling2(k,j)*Sum_{i=0..n} binomial(j+i-1,i)*A000009(n-i).

A093802 Number of distinct factorizations of 105*2^n.

Original entry on oeis.org

5, 15, 36, 74, 141, 250, 426, 696, 1106, 1711, 2593, 3852, 5635, 8118, 11548, 16231, 22577, 31092, 42447, 57464, 77213, 103009, 136529, 179830, 235514, 306751, 397506, 512607, 658030, 841020, 1070490, 1357195, 1714274, 2157539, 2706174, 3383187, 4216358

Offset: 0

Alford Arnold, May 19 2004

			105*A000079 is 105, 210, 420, 840, 1680, 3360, ... and there are 15 distinct factorizations of 210 so a(1) = 15.
a(0) = 5: 105*2^0 = 105 = 3*5*7 = 3*35 = 5*21 = 7*15. - _Alois P. Heinz_, May 26 2013

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Similar sequences: 45*A000079 => A002763, [1, 3, 9, 27, 81, 243...]*A000079 => A054225, 1*A002110 => A000110, 2*A002110 => A035098, A000142 => A076716.
Cf. A001055, A126442, A129306, A346823.
Column k=3 of A346426.

with(numtheory):
b:= proc(n, k) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= n-> b((105*2^n)$2):
seq(a(n), n=0..50);  # Alois P. Heinz, May 26 2013

b[n_, k_] := b[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0,
     Sum[If[d > k, 0, b[n/d, d]], {d, Divisors[n][[2;;-2]]}]];
a[n_] := b[105*2^n, 105*2^n];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 15 2021, after Alois P. Heinz *)

2 more terms from Alford Arnold, Aug 29 2007

Corrected offset and extended beyond a(7) by Alois P. Heinz, May 26 2013

cp's OEIS Frontend

A346520 Number A(n,k) of partitions of the (n+k)-multiset {0,...,0,1,2,...,k} with n 0's into distinct multisets; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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