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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A346500 Number A(n,k) of partitions of the (n+k)-multiset {1,2,...,n,1,2,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 4, 4, 5, 15, 11, 9, 11, 15, 52, 36, 26, 26, 36, 52, 203, 135, 92, 66, 92, 135, 203, 877, 566, 371, 249, 249, 371, 566, 877, 4140, 2610, 1663, 1075, 712, 1075, 1663, 2610, 4140, 21147, 13082, 8155, 5133, 3274, 3274, 5133, 8155, 13082, 21147
Offset: 0

Views

Author

Alois P. Heinz, Jul 20 2021

Keywords

Comments

Also number A(n,k) of factorizations of Product_{i=1..n} prime(i) * Product_{i=1..k} prime(i); A(2,2) = 9: 2*2*3*3, 3*3*4, 6*6, 2*3*6, 4*9, 2*2*9, 3*12, 2*18, 36.

Examples

			A(2,2) = 9: 1122, 11|22, 12|12, 1|122, 112|2, 11|2|2, 1|1|22, 1|12|2, 1|1|2|2.
Square array A(n,k) begins:
    1,    1,    2,     5,    15,     52,     203,     877, ...
    1,    2,    4,    11,    36,    135,     566,    2610, ...
    2,    4,    9,    26,    92,    371,    1663,    8155, ...
    5,   11,   26,    66,   249,   1075,    5133,   26683, ...
   15,   36,   92,   249,   712,   3274,   16601,   91226, ...
   52,  135,  371,  1075,  3274,  10457,   56135,  325269, ...
  203,  566, 1663,  5133, 16601,  56135,  198091, 1207433, ...
  877, 2610, 8155, 26683, 91226, 325269, 1207433, 4659138, ...
  ...

Links

Crossrefs

Columns (or rows) k=0-10 give: A000110, A035098, A322764, A322768, A346881, A346882, A346883, A346884, A346885, A346886, A346887.
Main diagonal gives A020555.
First upper (or lower) diagonal gives A322766.
Second upper (or lower) diagonal gives A322767.
Antidiagonal sums give A346490.
A(2n,n) gives A322769.
Cf. A001055, A002110, A322765, A346426, A346517.

Programs

  • Maple
    g:= proc(n, k) option remember; uses numtheory; `if`(n>k, 0, 1)+
     `if`(isprime(n), 0, add(`if`(d>k or max(factorset(n/d))>d, 0,
        g(n/d, d)), d=divisors(n) minus {1, n}))
    end:
p:= proc(n) option remember; `if`(n=0, 1, p(n-1)*ithprime(n)) end:
A:= (n, k)-> g(p(n)*p(k)$2):
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
A:= proc(n, k) option remember; `if`(n
  • Mathematica
    b[n_] := b[n] = If[n == 0, 1, Sum[b[n-j] Binomial[n-1, j-1], {j, 1, n}]];
A[n_, k_] := A[n, k] = If[n < k, A[k, n],
     If[k == 0, b[n], (A[n + 1, k - 1] + Sum[A[n - k + j, j]*
     Binomial[k - 1, j], {j, 0, k - 1}] + A[n, k - 1])/2]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Aug 18 2021, after Alois P. Heinz's second program *)

Formula

A(n,k) = A001055(A002110(n)*A002110(k)).
A(n,k) = A(k,n).
A(n,k) = A322765(abs(n-k),min(n,k)).

A346428 Total number of partitions of all n-multisets {0,...,0,1,2,...,j} for 0 <= j <= n.

Original entry on oeis.org

1, 2, 6, 17, 53, 180, 683, 2866, 13219, 66307, 358532, 2074229, 12761831, 83086064, 570017222, 4106269668, 30965072776, 243778358992, 1998878586251, 17034471643814, 150591119435358, 1378657063570498, 13050460812585580, 127553991370245410, 1285578058726241427
Offset: 0

Views

Author

Alois P. Heinz, Jul 16 2021

Keywords

Comments

Also total number of factorizations of 2^(n-j) * Product_{i=1..j} prime(i+1) for 0 <= j <= n; a(2) = 6: 2*2, 4, 2*3, 6, 3*5, 15; a(3) = 17: 2*2*2, 2*4, 8, 2*2*3, 3*4, 2*6, 12, 2*3*5, 5*6, 3*10, 2*15, 30, 3*5*7, 7*15, 5*21, 3*35, 105.

Examples

			a(2) = 6: 00, 0|0, 01, 0|1, 12, 1|2.
a(3) = 17: 000, 0|00, 0|0|0, 001, 00|1, 0|01, 0|0|1, 012, 0|12, 02|1, 01|2, 0|1|2, 123, 1|23, 13|2, 12|3, 1|2|3.

Links

Crossrefs

Antidiagonal sums of A346426.
Cf. A346490, A346521.

Programs

  • 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(add(S(n-i, j)*b(i, j), j=0..n-i), i=0..n):
seq(a(n), n=0..25);
  • 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[Sum[S[n - i, j]*b[i, j], {j, 0, n - i}], {i, 0, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 12 2022, after Alois P. Heinz *)

Formula

a(n) = Sum_{j=0..n} A346426(n-j,j).

A346518 Total number of partitions of all n-multisets {1,2,...,n-j,1,2,...,j} into distinct multisets for 0 <= j <= n.

Original entry on oeis.org

1, 2, 5, 16, 53, 202, 826, 3724, 17939, 93390, 516125, 3042412, 18923139, 124368810, 857827458, 6208594458, 46937360868, 370335617694, 3039823038753, 25928519847988, 229285625745624, 2099543718917418, 19872430464012935, 194203934113959970, 1956736801957704866
Offset: 0

Views

Author

Alois P. Heinz, Jul 21 2021

Keywords

Comments

Also total number of factorizations of Product_{i=1..n-j} prime(i) * Product_{i=1..j} prime(i) into distinct factors for 0 <= j <= n.

Links

Crossrefs

Antidiagonal sums of A346517.
Cf. A002110, A045778, A346490.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
A:= proc(n, k) option remember; `if`(n add(A(n-j, j), j=0..n):
seq(a(n), n=0..24);
  • Mathematica
    (* Q is A322770 *)
Q[m_, n_] := Q[m, n] = If[n == 0, BellB[m], (1/2) (Q[m + 2, n - 1] +
     Q[m + 1, n - 1] - Sum[Binomial[n - 1, k] Q[m, k], {k, 0, n - 1}])];
A[n_, k_] := Q[Abs[n - k], Min[n, k]];
a[n_] := Sum[A[n - j, j], {j, 0, n}];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Apr 06 2022 *)

Formula

a(n) = Sum_{j=0..n} A045778(A002110(n-j)*A002110(j)).
a(n) = Sum_{j=0..n} A346517(n-j,j).
Showing 1-3 of 3 results.

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