A035098 -id:A035098 - cp's OEIS frontend

A036035 Least integer of each prime signature, in graded (reflected or not) colexicographic order of exponents.

1, 2, 4, 6, 8, 12, 30, 16, 24, 36, 60, 210, 32, 48, 72, 120, 180, 420, 2310, 64, 96, 144, 216, 240, 360, 900, 840, 1260, 4620, 30030, 128, 192, 288, 432, 480, 720, 1080, 1800, 1680, 2520, 6300, 9240, 13860, 60060, 510510, 256, 384, 576, 864, 1296, 960, 1440, 2160

Offset: 0

N. J. A. Sloane

The exponents can be read off Abramowitz and Stegun, p. 831, column labeled "pi".
Here are the partitions in the order used by Abramowitz and Stegun (graded reflected colexicographic order): 0; 1; 2, 1+1; 3, 1+2, 1+1+1; 4, 1+3, 2+2, 1+1+2, 1+1+1+1; 5, 1+4, 2+3, 1+1+3, 1+2+2, 1+1+1+2, 1+1+1+1+1; ... (Cf. A036036)
Here are the partitions in graded colexicographic order: 0; 1; 2, 1+1; 3, 2+1, 1+1+1; 4, 3+1, 2+2, 2+1+1, 1+1+1+1; 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1; ... (Cf. A036037)
Since the prime signature is a partition of Omega(n), so to speak, the internal order is only a matter of convention and has no effect on the least integer with a given prime signature.
The graded colexicographic order has the advantage that the exponents are in the same order as the least integer with a given prime signature (also used on the wiki page, see links).
Embedded values include the primorial numbers 1, 2, 6, 30, 210, 2310, 30030 ... (A002110) with unordered factorizations counted by A000110 (Bell numbers) and ordered factorizations by A000670 (ordered Bell numbers).
When viewed as a table the n-th row has partition(n) (A000041(n)) terms. - Alford Arnold, Jul 31 2003
A closely related sequence, A096443(n), gives the number of partitions of the n-th multiset. - Alford Arnold, Sep 29 2005

			1;
2;
4, 6;
8, 12, 30;
16, 24, 36, 60, 210;
32, 48, 72, 120, 180, 420, 2310;
64, 96, 144, 216, 240, 360, 900, 840, 1260, 4620, 30030;
128, 192, 288, 432, 480, 720, 1080, 1800, 1680, 2520, 6300, 9240, 13860, 60060, 510510;

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings).

Peter Luschny, Rows n = 0..25, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
John Baez, What happens when a particle gets created?
OEIS Wiki, Prime signature.

A025487 in a different order. Cf. A035098, A002110, A000110 and A000670.

Cf. A025487, A059901, A096443.

with(combinat):
A036035_row := proc(n) local e, w; w := proc(e) local i, p;
p := [seq(ithprime(nops(e)-i+1), i=1..nops(e))];
mul(p[i]^e[i], i=1..nops(e)) end:
seq(w(conjpart(e)), e = partition(n)) end:
seq(A036035_row(i), i=0..10);  # Peter Luschny, Aug 01 2013

nmax = 52; primeSignature[n_] := Sort[ FactorInteger[n], #1[[2]] > #2[[2]] & ][[All, 2]]; ip[n_] := Reverse[ Sort[#]] & /@ Split[ Sort[ IntegerPartitions[n], Length[#1] < Length[#2] & ], Length[#1] == Length[#2] & ]; tip = Flatten[ Table[ip[n], {n, 0, 8}], 2]; a[n_] := (sig = tip[[n+1]]; k = 1; While[sig =!= primeSignature[k++]]; k-1); a[0] = 1; a[1] = 2; Table[an = a[n]; Print[an]; an, {n, 0, nmax}](* Jean-François Alcover, Nov 16 2011 *)

Row(n)={[prod(i=1, #p, prime(i)^p[#p+1-i]) | p<-partitions(n)]} \\ Andrew Howroyd, Oct 19 2020

More terms from Alford Arnold; corrected Sep 10 2002
More terms from Ray Chandler, Jul 13 2003
Definition corrected by Daniel Forgues, Jan 16 2011

A035310 Let f(n) = number of ways to factor n = A001055(n); a(n) = sum of f(k) over all terms k in A025487 that have n factors.

Original entry on oeis.org

1, 4, 12, 47, 170, 750, 3255, 16010, 81199, 448156, 2579626, 15913058, 102488024, 698976419, 4976098729, 37195337408, 289517846210, 2352125666883, 19841666995265, 173888579505200, 1577888354510786, 14820132616197925, 143746389756336173, 1438846957477988926

Offset: 1

Alford Arnold

Ways of partitioning an n-multiset with multiplicities some partition of n.

Number of multiset partitions of strongly normal multisets of size n, where a finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities. The (weakly) normal version is A255906. - Gus Wiseman, Dec 31 2019

			a(3) = 12 because there are 3 terms in A025487 with 3 factors, namely 8, 12, 30; and f(8)=3, f(12)=4, f(30)=5 and 3+4+5 = 12.
From _Gus Wiseman_, Dec 31 2019: (Start)
The a(1) = 1 through a(3) = 12 multiset partitions of strongly normal multisets:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1,2}}    {{1,1,2}}
         {{1},{1}}  {{1,2,3}}
         {{1},{2}}  {{1},{1,1}}
                    {{1},{1,2}}
                    {{1},{2,3}}
                    {{2},{1,1}}
                    {{2},{1,3}}
                    {{3},{1,2}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{3}}
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A025487, A000041, A000110, A035098, A080688.
Sequence A035341 counts the ordered cases. Tables A093936 and A095705 distribute the values; e.g. 81199 = 30 + 536 + 3036 + 6181 + 10726 + 11913 + 14548 + 13082 + 21147.
Cf. A035341, A093936, A095705.
Row sums of A317449.
The uniform case is A317584.
The case with empty intersection is A317755.
The strict case is A317775.
The constant case is A047968.
The set-system case is A318402.
The case of strict parts is A330783.
Multiset partitions of integer partitions are A001970.
Unlabeled multiset partitions are A007716.
Cf. A001055, A255906, A269134, A316652, A317654, A318284, A330467, A330471, A330475, A330675.

with(numtheory):
g:= proc(n, k) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, g(n/d, d)), d=divisors(n) minus {1, n}))
    end:
b:= proc(n, i, l)
      `if`(n=0, g(mul(ithprime(t)^l[t], t=1..nops(l))$2),
      `if`(i<1, 0, add(b(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> b(n$2, []):
seq(a(n), n=1..10);  # Alois P. Heinz, May 26 2013

g[n_, k_] := g[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d > k, 0, g[n/d, d]], {d, Divisors[n] ~Complement~ {1, n}}]]; b[n_, i_, l_] := If[n == 0, g[p = Product[Prime[t]^l[[t]], {t, 1, Length[l]}], p], If[i < 1, 0, Sum[b[n - i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; a[n_] := b[n, n, {}]; Table[Print[an = a[n]]; an, {n, 1, 13}] (* Jean-François Alcover, Dec 12 2013, after Alois P. Heinz *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n))-1, -n)/prod(i=1, #v, i^v[i]*v[i]!)}
seq(n)={my(s=0); forpart(p=n, s+=D(p,n)); s} \\ Andrew Howroyd, Dec 30 2020

from sympy.core.cache import cacheit
from sympy import divisors, isprime, prime
from operator import mul
@cacheit
def g(n, k):
    return (0 if n > k else 1) + (0 if isprime(n) else sum(g(n//d, d) for d in divisors(n)[1:-1] if d <= k))
@cacheit
def b(n, i, l):
    if n==0:
        p = reduce(mul, (prime(t + 1)**l[t] for t in range(len(l))))
        return g(p, p)
    else:
        return 0 if i<1 else sum([b(n - i*j, i - 1, l + [i]*j) for j in range(n//i + 1)])
def a(n):
    return b(n, n, [])
for n in range(1, 11): print(a(n)) # Indranil Ghosh, Aug 19 2017, after Maple code

More terms from Erich Friedman.
81199 from Alford Arnold, Mar 04 2008
a(10) from Alford Arnold, Mar 31 2008
a(10) corrected by Alford Arnold, Aug 07 2008
a(11)-a(13) from Alois P. Heinz, May 26 2013
a(14) from Alois P. Heinz, Sep 27 2014
a(15) from Alois P. Heinz, Jan 10 2015
Terms a(16) and beyond from Andrew Howroyd, Dec 30 2020

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

Alois P. Heinz, Jul 16 2021

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.

			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, ...
  ...

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

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.

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);

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 *)

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).

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

Alois P. Heinz, Jul 20 2021

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.

			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, ...
  ...

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

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.

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

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 *)

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

A096443 Number of partitions of a multiset whose signature is the n-th partition (in Mathematica order).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 5, 7, 9, 11, 15, 7, 12, 16, 21, 26, 36, 52, 11, 19, 29, 38, 31, 52, 74, 66, 92, 135, 203, 15, 30, 47, 64, 57, 98, 141, 109, 137, 198, 296, 249, 371, 566, 877, 22, 45, 77, 105, 97, 171, 250, 109, 212, 269, 392, 592, 300, 444, 560, 850, 1315, 712, 1075

Offset: 0

Jon Wild, Aug 11 2004

nonn

The signature of a multiset is the partition consisting of the multiplicities of its elements; e.g., {a,a,a,b,c} is represented by [3,1,1]. The Mathematica order for partitions orders by ascending number of total elements, then by descending numerical order of its representation. The list begins:
n.....#elements.....n-th partition
....0 elements:....[]
....1 element:.....[1]
....2 elements:....[2]
...................[1,1]
....3 elements:....[3]
...................[2,1]
...................[1,1,1]
....4 elements:....[4]
...................[3,1]
...................[2,2]
..................[2,1,1]
..................[1,1,1,1]
...5 elements:....[5]
..................[4,1]
A000041 and A000110 are subsequences for conjugate partitions. A000070 and A035098 are also subsequences for conjugate partitions. - Alford Arnold, Dec 31 2005
A002774 and A020555 is another pair of subsequences for conjugate partitions. - Franklin T. Adams-Watters, May 16 2006

			The 10th partition is [2,1,1]. The partitions of a multiset whose elements have multiplicities 2,1,1 - for example, {a,a,b,c} - are:
{{a,a,b,c}}
{{a,a,b},{c}}
{{a,a,c},{b}}
{{a,b,c},{a}}
{{a,a},{b,c}}
{{a,b},{a,c}}
{{a,a},{b},{c}}
{{a,b},{a},{c}}
{{a,c},{a},{b}}
{{b,c},{a},{a}}
{{a},{a},{b},{c}}
We see there are 11 partitions of this multiset, so a(10)=11.
Also, a(n) is the number of distinct factorizations of A063008(n). For example, A063008(10) = 60 and 60 has 11 factorizations: 60, 30*2, 20*3, 15*4, 15*2*2, 12*5, 10*6, 10*3*2, 6*5*2, 5*4*3, 5*3*2*2 which confirms that a(10) = 11.

Jun Kyo Kim and Sang Guen Hahn, Recursive Formulae for the Multiplicative Partition Function, Internat. J. Math. & Math. Sci., 22(1) (1999), 213-216.
A. Knopfmacher, M. E. Mays, A survey of factorization counting functions, International Journal of Number Theory, 1(4):563-581,(2005). See P(n) page 3.

Cf. A035098, A035310.

MultiPartiteP[n : {___Integer?NonNegative}] :=
Block[{p, $RecursionLimit = 1024, firstPositive},
  firstPositive =
   Compile[{{vv, _Integer, 1}},
    Module[{k = 1}, Do[If[el == 0, k++, Break[]], {el, vv}]; k]];
  p[{0 ...}] := 1;
  p[v_] :=
   p[v] = Module[{len = Length[v], it, k, zeros, sum, pos, gcd},
     it = Array[k, len];
     pos = firstPositive[v];
     zeros = ConstantArray[0, len];
     sum = 0;
     Do[If[it == zeros, Continue[]];
      gcd = GCD @@ it;
      sum += it[[pos]] DivisorSigma[-1, gcd] p[v - it];,
      Evaluate[Sequence @@ Thread[{it, 0, v}]]];
     sum/v[[pos]]];
  p[n]];
ParallelMap[MultiPartiteP,
Flatten[Table[IntegerPartitions[k], {k, 0, 8}], 1]]
(* Oleksandr Pavlyk, Jan 23 2011 *)

Edited by Franklin T. Adams-Watters, May 16 2006

A035341 Sum of ordered factorizations over all prime signatures with n factors.

Original entry on oeis.org

1, 1, 5, 25, 173, 1297, 12225, 124997, 1492765, 19452389, 284145077, 4500039733, 78159312233, 1460072616929, 29459406350773, 634783708448137, 14613962109584749, 356957383060502945, 9241222160142506097, 252390723655315856437, 7260629936987794508973

Offset: 0

Alford Arnold

Let f(n) = number of ordered factorizations of n (A074206(n)); a(n) = sum of f(k) over all terms k in A025487 that have n factors.
When the unordered spectrum A035310 is so ordered the sequences A000041 A000070 ...A035098 A000110 yield A000079 A001792 ... A005649 A000670 respectively.
Row sums of A095705. - David Wasserman, Feb 22 2008
From Ludovic Schwob, Sep 23 2023: (Start)
a(n) is the number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums. The a(3) = 25 matrices:
  [1 1 1] [1 2] [2 1] [3]
  .
  [1 1] [1 1] [1 1 0] [1 0 1] [0 1 1] [2] [0 2] [2 0]
  [1 0] [0 1] [0 0 1] [0 1 0] [1 0 0] [1] [1 0] [0 1]
  .
  [1] [1 0] [0 1] [1 0] [0 1] [1 0 0] [1 0 0] [0 1] [1 0]
  [1] [1 0] [0 1] [0 1] [1 0] [0 1 0] [0 0 1] [1 0] [0 1]
  [1] [0 1] [1 0] [1 0] [0 1] [0 0 1] [0 1 0] [1 0] [0 1]
  .
  [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [1 0 0] [0 1 0] [1 0 0] (End)

			a(3) = 25 because there are 3 terms in A025487 with 3 factors, namely 8, 12, 30; and f(8)=4, f(12)=8, f(30)=13 and 4+8+13 = 25.

Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 36 terms from David Wasserman)
Eric Weisstein's World of Mathematics, Perfect Partition

Cf. A005651, A025487, A035310, A255906, A365961.

Row sums of A261719.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..n/i)))
    end:
a:= n->add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 29 2015

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, k]*If[j == 0, 1, Binomial[i + k - 1, k - 1]^j], {j, 0, n/i}]]];
a[n_] := Sum[Sum[b[n, n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}], {k, 0, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 26 2015, after Alois P. Heinz, updated Dec 15 2020 *)

R(n,k)=Vec(-1 + 1/prod(j=1, n, 1 - binomial(k+j-1,j)*x^j + O(x*x^n)))
seq(n) = {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Sep 23 2023

a(n) ~ c * n! / log(2)^n, where c = 1/(2*log(2)) * Product_{k>=2} 1/(1-1/k!) = A247551 / (2*log(2)) = 1.8246323... . - Vaclav Kotesovec, Jan 21 2017

More terms from Erich Friedman.

More terms from David Wasserman, Feb 22 2008

A126442 Triangular array t read by rows: t(0,k) is p(k), the number of partitions of the k-multiset {0,0,...,0} with k zeros. For 0 <= n < k, t(n, k) is the number of partitions of the k-multiset {0, 0, ..., 0, 1, 2, 3, ..., k-n} with n zeros.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 5, 7, 11, 15, 7, 12, 21, 36, 52, 11, 19, 38, 74, 135, 203, 15, 30, 64, 141, 296, 566, 877, 22, 45, 105, 250, 592, 1315, 2610, 4140, 30, 67, 165, 426, 1098, 2752, 6393, 13082, 21147, 42, 97, 254, 696, 1940, 5317, 13960, 33645, 70631, 115975

Offset: 1

Alford Arnold, Jan 28 2007

First in a series of triangular arrays which comprise subsequences of A096443(n).
The second array begins 9 16 26 29 52 92 47 98 198 371 and when the arrays are aligned as illustrated in triangle A126441 with p(n) values they sum to A035310 which counts unordered multisets.
Let t(n, k) be the number of ways to partition the k-multiset {0,0,...,0,1,2,3,4,...,k-n} with n zeros, 0 <= n < k. Then t(n, k) = sum_i = 0..k j = 0..n S(n, j) C(i,  j) p(k - n - i), where S(n, j) are Stirling numbers of the second kind, C(i, j) are the number of compositions of i distinct objects into j parts, and p is the integer partition function.
To see this, partition [n] into j blocks; there are S(n, j) partitions. For such a partition x and for each i, there are C(i, j) ways to distribute i zeros into x, because the blocks of x are all distinct. There are p(k-n-i) ways to partition the remaining k-n-i zeros. Multiplying and summing gives the result. - George Beck, Jan 10 2011
Values are also part of A096443, A129306 and A249620. Columns are also columns of the last one of these irregular triangles. See "Partitions_of_multisets" link. - Tilman Piesk, Nov 09 2014

			This first array includes only the hook cases. A096443(9,14,16) correspond to partitions [2,2], [3,2] and [2,2,1] so these values do not appear in A126442.
The array begins:
1
2 2
3 4 5
5 7 11 15
7 12 21 36 52

Tilman Piesk, Partitions of multisets (Wikiversity)

Cf. A000041, A000070, A082775, A093802, A000291, A002763, A000412, A054225, A035310, A000110, A035098.

(* The triangle is flattened to a sequence. *)
t[n_, k_] := Sum[StirlingS2[n, j] * Binomial[-1 + i + j, i] * PartitionsP[k - n - i], {j, 0, n}, {i, 0, k - n}]; Table[ t[n, k], {k, 10}, {n, 0, k - 1}] // Flatten (* George Beck, Jan 10 2011 *)

Definition clarified by George Beck, Jan 11 2011

A322765 Array read by upwards antidiagonals: T(m,n) = number of set partitions of the multiset consisting of one copy each of x_1, x_2, ..., x_m, and two copies each of y_1, y_2, ..., y_n, for m >= 0, n >= 0.

Original entry on oeis.org

1, 1, 2, 2, 4, 9, 5, 11, 26, 66, 15, 36, 92, 249, 712, 52, 135, 371, 1075, 3274, 10457, 203, 566, 1663, 5133, 16601, 56135, 198091, 877, 2610, 8155, 26683, 91226, 325269, 1207433, 4659138, 4140, 13082, 43263, 149410, 537813, 2014321, 7837862, 31638625, 132315780

Offset: 0

N. J. A. Sloane, Dec 30 2018

			The array begins:
    1,    2,     9,     66,      712,     10457,      198091, ...
    1,    4,    26,    249,     3274,     56135,     1207433, ...
    2,   11,    92,   1075,    16601,    325269,     7837862, ...
    5,   36,   371,   5133,    91226,   2014321,    53840640, ...
   15,  135,  1663,  26683,   537813,  13241402,   389498179, ...
   52,  566,  8155, 149410,  3376696,  91914202,  2955909119, ...
  203, 2610, 43263, 894124, 22451030, 670867539, 23456071495, ...
  ...

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778.

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Rows include A020555, A322766, A322767.
Columns include A000110, A035098, A322764, A322768.
Main diagonal is A322769.
See A322770 for partitions into distinct parts.
Cf. A001055, A002110, A346500.

B := n -> combinat[bell](n):
P := proc(m,n) local k; global B; option remember;
if n = 0 then B(m)  else
(1/2)*( P(m+2,n-1) + P(m+1,n-1) + add( binomial(n-1,k)*P(m,k), k=0..n-1) ); fi; end; # P(m,n) (which is Knuth's notation) is T(m,n)

P[m_, n_] := P[m, n] = If[n == 0, BellB[m], (1/2)(P[m+2, n-1] + P[m+1, n-1] + Sum[Binomial[n-1, k] P[m, k], {k, 0, n-1}])];
Table[P[m-n, n], {m, 0, 8}, {n, 0, m}] // Flatten (* Jean-François Alcover, Jan 02 2019, from Maple *)

{T(n, k) = if(k==0, sum(j=0, n, stirling(n, j, 2)), (T(n+2, k-1)+T(n+1, k-1)+sum(j=0, k-1, binomial(k-1, j)*T(n, j)))/2)} \\ Seiichi Manyama, Nov 21 2020

Knuth p. 779 gives a recurrence using the Bell numbers A000110 (see Maple program).
From Alois P. Heinz, Jul 21 2021: (Start)
A(n,k) = A001055(A002110(n+k)*A002110(k)).
A(n,k) = A346500(n+k,k). (End)

A059606 Expansion of (1/2)(exp(2x)-1)*exp(exp(x)-1).

Original entry on oeis.org

0, 1, 4, 16, 68, 311, 1530, 8065, 45344, 270724, 1709526, 11376135, 79520644, 582207393, 4453142140, 35500884556, 294365897104, 2533900264547, 22604669612078, 208656457858161, 1990060882027600

Offset: 0

Vladeta Jovovic, Jan 29 2001

Starting (1, 4, 16, 68, 311, ...), = A008277 * A000217, i.e., the product of the Stirling2 triangle and triangular series. - Gary W. Adamson, Jan 31 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vladeta Jovovic, More information.

Cf. A000110, A005493, A059604, A059605.
Cf. A035098.
Cf. A008277, A000217.

s := series(1/2*(exp(2*x)-1)*exp(exp(x)-1), x, 21): for i from 0 to 20 do printf(`%d,`,i!*coeff(s,x,i)) od:

With[{nn=20},CoefficientList[Series[((Exp[2x]-1)Exp[Exp[x]-1])/2,{x,0,nn}] ,x] Range[0,nn]!] (* Harvey P. Dale, Nov 10 2011 *)

a(n) = Sum_{i=0..n} Stirling2(n, i)*binomial(i+1, 2).
a(n) = (1/2)*(Bell(n+2)-Bell(n+1)-Bell(n)). - Vladeta Jovovic, Sep 23 2003
G.f.: Sum_{k>=1} (k*(k + 1)/2)*x^k/Product_{j=1..k} (1 - j*x). - Ilya Gutkovskiy, Jun 19 2018
a(n) ~ n^2 * Bell(n) / (2*LambertW(n)^2) * (1 - LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021

A087648 a(n) = (1/2)*(Bell(n+2)+Bell(n+1)-Bell(n)).

Original entry on oeis.org

1, 3, 9, 31, 120, 514, 2407, 12205, 66491, 386699, 2388096, 15589732, 107165081, 773106715, 5836100685, 45981026703, 377230766908, 3215977070706, 28437411817135, 260380616093533, 2464930698184351, 24091925888687459, 242802079705721156, 2520198597834860148

Offset: 0

Vladeta Jovovic, Sep 23 2003

nonn

Sum of last number in all set partitions of n+1. E.g. The set partitions of 3 are {1,1,1}, {1,1,2}, {1,2,1}, {1,2,2} and {1,2,3}, so a(2) = 1+2+1+2+3 = 9. - Franklin T. Adams-Watters, Jun 07 2006

Number of partitions of the (n+2)-multiset {0,0,1,2,...,n} into distinct multisets. Also number of factorizations of 2 * Product_{i=1..n+1} prime(i) into distinct factors. - Alois P. Heinz, Jul 30 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000110, A035098, A059606.
Main diagonal of A120057, row sums of A120095.
Column 1 of array in A322770.
Row n=2 of A346520.

[(1/2)*(Bell(n+2)+Bell(n+1)-Bell(n)) : n in [0..30]]; // Vincenzo Librandi, Nov 13 2011

f[0]=1; f[n_] := Sum[ StirlingS2[n, k]*Binomial[k+2, k ], {k, 1, n}]; Table[ f[n], {n, 0, 20}] (* Zerinvary Lajos, Mar 31 2007 *)
(#[[3]]+#[[2]]-#[[1]])/2&/@Partition[BellB[Range[0,30]],3,1] (* Harvey P. Dale, Jul 20 2021 *)

cp's OEIS Frontend

A036035 Least integer of each prime signature, in graded (reflected or not) colexicographic order of exponents.

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A035310 Let f(n) = number of ways to factor n = A001055(n); a(n) = sum of f(k) over all terms k in A025487 that have n factors.

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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.

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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.

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A096443 Number of partitions of a multiset whose signature is the n-th partition (in Mathematica order).

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A035341 Sum of ordered factorizations over all prime signatures with n factors.

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A126442 Triangular array t read by rows: t(0,k) is p(k), the number of partitions of the k-multiset {0,0,...,0} with k zeros. For 0 <= n < k, t(n, k) is the number of partitions of the k-multiset {0, 0, ..., 0, 1, 2, 3, ..., k-n} with n zeros.

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A059606 Expansion of (1/2)(exp(2x)-1)*exp(exp(x)-1).