A346520 -id:A346520 - cp's OEIS frontend

A036469 Partial sums of A000009 (partitions into distinct parts).

1, 2, 3, 5, 7, 10, 14, 19, 25, 33, 43, 55, 70, 88, 110, 137, 169, 207, 253, 307, 371, 447, 536, 640, 762, 904, 1069, 1261, 1483, 1739, 2035, 2375, 2765, 3213, 3725, 4310, 4978, 5738, 6602, 7584, 8697, 9957, 11383, 12993, 14809, 16857, 19161, 21751, 24661

Offset: 0

N. J. A. Sloane

nonn

Also number of 1's in all partitions of n+1 into odd parts. Example: a(4)=7 because the partitions of 5 into odd parts are [5], [3,1,1], [1,1,1,1,1], having a total number of 7 1's. - Emeric Deutsch, Mar 29 2006
Convolved with A035363 = A000070. - Gary W. Adamson, Jun 09 2009
Equals row sums of triangle A166240. - Gary W. Adamson, Oct 09 2009
a(n) = if n <= 1 then A201377(1,n) else A201377(n,1). - Reinhard Zumkeller, Dec 02 2011
a(n) equals the sum of the parts of the form 2^k (k >= 0) in all partitions of n + 1 into distinct parts. Example: a(6) = 14. The partitions of 7 into distinct parts are [7], [6,1], [5,2], [4,3] and [4,2,1] having sum over parts of the form 2^k equal to 1 + 2 + 4 + 4 + 2 + 1 = 14. - Peter Bala, Dec 01 2013
Number of partitions of the (n+1)-multiset {0,...,0,1} with n 0's into distinct multisets; a(3) = 5: 0|00|1, 00|01, 000|1, 0|001, 0001.  Also number of factorizations of 3*2^n into distinct factors; a(3) = 5: 2*3*4, 4*6, 3*8, 2*12, 24. - Alois P. Heinz, Jul 30 2021

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
A. V. Chekhonadskikh, Some classical number sequences in control system design, Siberian Electronic Mathematical Reports, Volume 14, p. 620-628. See Theorem 2.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 774

Cf. A000009, A265093.
Cf. A035363, A000070. - Gary W. Adamson, Jun 09 2009
Cf. A166240. - Gary W. Adamson, Oct 09 2009
Column k=1 of A346520.

g:=1/(1-x)/product(1-x^(2*j-1),j=1..30): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=0..46); # Emeric Deutsch, Mar 29 2006
# second Maple program:
b:= proc(n, i) b(n, i):= `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1)+`if`(i>n, 0, b(n-i, min(n-i, i-1)))))
    end:
a:= proc(n) option remember; b(n, n) +`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 21 2012

CoefficientList[ Series[Product[(1 + t^i), {i, 1, Infinity}]/(1 - t), {t, 0, 46}], t] (* Geoffrey Critzer, May 16 2010 *)
b[n_, i_] := If[n == 0, 1, If[i<1, 0, b[n, i-1]+If[i>n, 0, b[n-i, Min[n-i, i-1]]]]]; a[n_] := a[n] = b[n, n]+If[n>0, a[n-1], 0]; Table[a[n], {n, 0, 60}] // Flatten (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)
Accumulate[Table[PartitionsQ[n], {n, 0, 50}]] (* Vaclav Kotesovec, Oct 26 2016 *)

G.f.: 1/[(1-x)*product(1-x^(2j-1), j=1..infinity)]. - Emeric Deutsch, Mar 29 2006
a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (2*Pi*n^(1/4)) * (1 + (18+13*Pi^2) / (48*Pi*sqrt(3*n)) + (2916 - 1404*Pi^2 + 121*Pi^4)/(13824*Pi^2*n)). - Vaclav Kotesovec, Feb 26 2015, updated Oct 26 2016
For n > 0, a(n) = A026906(n) + 1. - Vaclav Kotesovec, Oct 26 2016
Faster converging g.f.: A(x) = (1/(1 - x))*Sum_{n >= 0} x^(n*(2*n-1))/Product_{k = 1..2*n} (1 - x^k). - Peter Bala, Feb 02 2021

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).

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 *)

A346519 Number of partitions of the 2n-multiset {0,...,0,1,2,...,n} into distinct multisets.

Original entry on oeis.org

1, 2, 9, 59, 442, 3799, 36332, 379831, 4288933, 51867573, 667168482, 9076862555, 130018298663, 1953284957029, 30675458303547, 502166867458649, 8547908294767932, 150965367603029126, 2760941474553823577, 52196915577464262360, 1018499212583077293854

Offset: 0

Alois P. Heinz, Jul 21 2021

nonn

Also number of factorizations of 2^n * Product_{i=1..n} prime(i+1) into distinct factors; a(2) = 9: 3*4*5, 2*5*6, 6*10, 2*3*10, 5*12, 4*15, 3*20, 2*30, 60.

			a(0) = 1: {}.
a(1) = 2: 01, 0|1.
a(2) = 9: 00|1|2, 001|2, 1|002, 0|01|2, 0|1|02, 01|02, 00|12, 0|012, 0012.

Alois P. Heinz, Table of n, a(n) for n = 0..510

Main diagonal of A346520.

Cf. A000009, A000040, A000079, A045778, A048993, A070826, A346424.

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-> add(S(n, j)*b(n, j), j=0..n):
seq(a(n), n=0..20);

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_] := Sum[S[n, j]*b[n, j], {j, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 06 2022, after Alois P. Heinz *)

a(n) = A045778(A000079(n)*A070826(n+1)).
a(n) = Sum_{j=0..n} Stirling2(n,j)*Sum_{i=0..n} binomial(j+i-1,i)*A000009(n-i).
a(n) = A346520(n,n).

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

Alois P. Heinz, Jul 21 2021

nonn

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.

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..576

Antidiagonal sums of A346520.

Cf. A000009, A000040, A000079, A000110, A045778, A070826, A346428.

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);

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 *)

a(n) = Sum_{j=0..n} A346520(n-j,j).

a(n) = Sum_{j=0..n} A045778(A000079(n-j)*A070826(j+1)).

A346813 Number of partitions of the (n+3)-multiset {0,0,0,1,2,...,n} into distinct multisets.

Original entry on oeis.org

2, 5, 16, 59, 244, 1112, 5516, 29505, 168938, 1029271, 6639768, 45162696, 322739878, 2415567889, 18884022724, 153826198419, 1302844673004, 11451006297332, 104262283711688, 981877752182053, 9550121928456718, 95809651792320751, 990221394289370920

Offset: 0

Alois P. Heinz, Aug 05 2021

nonn

Also number of factorizations of 8 * Product_{i=1..n} prime(i+1) into distinct factors.

Alois P. Heinz, Table of n, a(n) for n = 0..573

Row n=3 of A346520.

Cf. A169587.

A346814 Number of partitions of the (n+4)-multiset {0,0,0,0,1,2,...,n} into distinct multisets.

Original entry on oeis.org

2, 7, 25, 100, 442, 2134, 11147, 62505, 373832, 2372061, 15896786, 112087787, 828764232, 6407210143, 51661121427, 433442672918, 3776529680114, 34108004916520, 318791092171365, 3078872979139965, 30684482469719642, 315168093997478339, 3332424923926530990

Offset: 0

Alois P. Heinz, Aug 05 2021

nonn

Also number of factorizations of 16 * Product_{i=1..n} prime(i+1) into distinct factors.

Alois P. Heinz, Table of n, a(n) for n = 0..572

Row n=4 of A346520.

Cf. A169588.

A346815 Number of partitions of the (n+5)-multiset {0,...,0,1,2,...,n} with five 0's into distinct multisets.

Original entry on oeis.org

3, 10, 38, 161, 750, 3799, 20739, 121141, 752681, 4949969, 34312195, 249798408, 1904026799, 15153517600, 125621420076, 1082409498473, 9675240317432, 89561442387221, 857209445230483, 8471058440432651, 86318636142516675, 905868823242467245, 9779990840117785293

Offset: 0

Alois P. Heinz, Aug 05 2021

nonn

Also number of factorizations of 32 * Product_{i=1..n} prime(i+1) into distinct factors.

Alois P. Heinz, Table of n, a(n) for n = 0..572

Row n=5 of A346520.

Cf. A346851.

A346816 Number of partitions of the (n+6)-multiset {0,...,0,1,2,...,n} with six 0's into distinct multisets.

Original entry on oeis.org

4, 14, 56, 249, 1213, 6404, 36332, 220000, 1413937, 9600364, 68592684, 513952699, 4026607292, 32900195138, 279706376058, 2469224186643, 22592983270903, 213904411529522, 2092395705495048, 21117781135309376, 219627058223237983, 2351003452644642314

Offset: 0

Alois P. Heinz, Aug 05 2021

nonn

Also number of factorizations of 2^6 * Product_{i=1..n} prime(i+1) into distinct factors.

Alois P. Heinz, Table of n, a(n) for n = 0..571

Row n=6 of A346520.

Cf. A346852.

A346817 Number of partitions of the (n+7)-multiset {0,...,0,1,2,...,n} with seven 0's into distinct multisets.

Original entry on oeis.org

5, 19, 80, 372, 1887, 10340, 60727, 379831, 2516880, 17590607, 129185101, 993686245, 7982964121, 66815548603, 581342199992, 5247733142922, 49060074341483, 474248427057490, 4733396660516715, 48713824251499499, 516315132698019888, 5629552607340558361

Offset: 0

Alois P. Heinz, Aug 05 2021

nonn

Also number of factorizations of 2^7 * Product_{i=1..n} prime(i+1) into distinct factors.

Alois P. Heinz, Table of n, a(n) for n = 0..571

Row n=7 of A346520.

Cf. A346853.

cp's OEIS Frontend

A036469 Partial sums of A000009 (partitions into distinct parts).

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

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A346519 Number of partitions of the 2n-multiset {0,...,0,1,2,...,n} into distinct multisets.

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

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A346813 Number of partitions of the (n+3)-multiset {0,0,0,1,2,...,n} into distinct multisets.

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A346814 Number of partitions of the (n+4)-multiset {0,0,0,0,1,2,...,n} into distinct multisets.

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A346815 Number of partitions of the (n+5)-multiset {0,...,0,1,2,...,n} with five 0's into distinct multisets.

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A346816 Number of partitions of the (n+6)-multiset {0,...,0,1,2,...,n} with six 0's into distinct multisets.