cp's OEIS Frontend

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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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.

Original entry on oeis.org

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

Views

Author

Alois P. Heinz, Jul 16 2021

Keywords

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.

Examples

			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, ...
  ...

Links

Crossrefs

Columns k=0-10 give: A000041, A000070, A082775, A093802, A346857, A346858, A346859, A346860, A346861, A346862, A346863.
Rows n=0+1, 2-10 give: A000110, A035098, A169587, A169588, A346851, A346852, A346853, A346854, A346855, A346856.
Main diagonal gives A346424.
Antidiagonal sums give A346428.
Cf. A000079, A001055, A008277, A048993, A070826, A126442, A129306, A219727, A255903, A292508, A346520.

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, 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 *)

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).

A346490 Total number of partitions of all n-multisets {1,2,...,n-j,1,2,...,j} for 0 <= j <= n.

Original entry on oeis.org

1, 2, 6, 18, 61, 228, 926, 4126, 19688, 101582, 556763, 3258810, 20134527, 131591030, 902915694, 6506096000, 48986713992, 385159376478, 3151457714098, 26806601933838, 236457090358459, 2160451562170100, 20408176433186475, 199086685731569740, 2002713693735431017
Offset: 0

Views

Author

Alois P. Heinz, Jul 19 2021

Keywords

Comments

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

Links

Crossrefs

Antidiagonal sums of A346500.
Cf. A001055, A002110, A346428, A346518.

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
    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]];
a[n_] := Sum[A[n - j, j], {j, 0, n}];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Mar 12 2022, after Alois P. Heinz *)

Formula

a(n) = Sum_{j=0..n} A001055(A002110(n-j)*A002110(j)).
a(n) = Sum_{j=0..n} A346500(n-j,j).

A346521 Total number of partitions of all n-multisets {0,...,0,1,2,...,j} into distinct multisets for 0 <= j <= n.

Original entry on oeis.org

1, 2, 5, 15, 46, 161, 624, 2669, 12483, 63261, 344631, 2005058, 12390086, 80945545, 556896913, 4021109557, 30382294412, 239589006143, 1967343509525, 16786587081641, 148561276135546, 1361378815644787, 12897870827339021, 126158299918183469, 1272377007364596242
Offset: 0

Views

Author

Alois P. Heinz, Jul 21 2021

Keywords

Comments

Also total number of factorizations of 2^(n-j) * Product_{i=1..j} prime(i+1) into distinct factors for 0 <= j <= n; a(2) = 5: 4, 2*3, 6, 3*5, 15; a(3) = 15: 2*4, 8, 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) = 5: 00, 01, 0|1, 12, 1|2.
a(3) = 15: 000, 0|00, 001, 00|1, 0|01, 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 A346520.
Cf. A000009, A000040, A000079, A000110, A045778, A070826, A346428.

Programs

  • Maple
    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):
a:= n-> add(A(n-j, j), j=0..n):
seq(a(n), n=0..24);
  • Mathematica
    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}];
a[n_] := Sum[A[n - j, j], {j, 0, n}];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Jul 31 2021, after Alois P. Heinz *)

Formula

a(n) = Sum_{j=0..n} A346520(n-j,j).
a(n) = Sum_{j=0..n} A045778(A000079(n-j)*A070826(j+1)).
Showing 1-3 of 3 results.

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