A048574 -id:A048574 - cp's OEIS frontend

A122768 Number of combinations which can be taken from the integer partitions of n. Total number of cases in the (n,m)-fragmentation process.

0, 1, 3, 7, 15, 29, 54, 95, 163, 270, 439, 696, 1088, 1669, 2530, 3780, 5591, 8173, 11845, 17000, 24215, 34210, 48008, 66895, 92660, 127554, 174651, 237830, 322297, 434625, 583524, 779972, 1038356, 1376787, 1818755, 2393775, 3139812, 4104433, 5348375, 6947545, 8998201, 11620313, 14965126, 19220569

Offset: 0

Thomas Wieder, Sep 11 2006

nonn

Consider a fragmentation process of an n-object which consists of n unlabeled elements (= 1-parts). By definition the n-object can scatter into up to n m-parts where an m-part consists of 1 up to n elements. A 4-object can split up for example into 4 1-parts which corresponds to the integer partition [1,1,1,1], or it can, for example, rest unfragmented which corresponds to [4]. Since the number of integer partitions of n=4 equals 5, there are 5 n=4-fragmentation processes.
Now we ask for the probability of getting an m-part after an n-fragmentation. Think of a Greek statue which had been broken into n parts and covered by earth. We could find several m-parts, in the most lucky case we would find all m-parts which add up to m_1+m_2+...+m_n=n. Then the statue could be restored.
For example for n=4 we could ask for the probability prob(n=4,m=2) of just a single 2-part. We have 2 cases for a 2-part and we have 15 cases in total, thus prob(n=4,m=2)=2/15 (the 2 cases come from [1,1,2] and [2,2]). The chances to find the two 2-parts from the [2,2]-fragmentation are 1/15 only. The chances to find the n=4-object unsplitted are also 1/15 only.
This sequence is generated over the unordered partitions; for example, when n = 4 there are 1+3+2+5+4 = 15 cases. If we allow a null case for each of the five partitions then we have 15+5 = 20 which is A000712(4). - Alford Arnold, Dec 12 2006
Number of partitions into two kinds of parts with the first kind of parts used in each partition. - Joerg Arndt, Jun 21 2011

			a(n=4) = 15 because the possible combinations of all five integer partitions of n=4 are: [1], [1, 1], [1, 1, 1], [1, 1, 1, 1], [1], [2], [1, 1], [1, 2], [1, 1, 2], [2], [2, 2], [1], [3], [1, 3], [4].

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

Cf. A000041, A000079, A000262.
Cf. A000712, A074139, A074141.
Cf. A048574, A144064.

a122768 n = a122768_list !! n
a122768_list = 0 : f (tail a000041_list) [1] where
   f (p:ps) rs = (sum $ zipWith (*) rs $ tail a000041_list) : f ps (p : rs)
-- Reinhard Zumkeller, Nov 09 2015

A122768 := proc(n::integer) local i,j,prttnlst,prttn,ZahlTeile,H; prttnlst:=partition(n); H := NULL; for i from 1 to nops(prttnlst) do prttn := prttnlst[i]; ZahlTeile := nops(prttn); for j from 1 to ZahlTeile do H := H,op(choose(prttn,j)); od; od; print(n,H,nops([H])); end proc;
A000712 := proc(n) option remember ; add(combinat[numbpart](k)*combinat[numbpart](n-k),k=0..n) ; end: A000041 := proc(n) combinat[numbpart](n) ; end: A122768 := proc(n::integer) RETURN( A000712(n)-A000041(n)) ; end: for n from 0 to 80 do printf("%d,",A122768(n)) ; od: # R. J. Mathar, Aug 25 2008
# third Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, add(
      k*numtheory[sigma](j)*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n,2)-b(n,1):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 31 2017

1/QPochhammer[x]^2 - 1/QPochhammer[x] + O[x]^50 // CoefficientList[#, x]& (* Jean-François Alcover, Feb 05 2017, after Joerg Arndt *)

x='x+O('x^66); /* that many terms */
Vec(1/eta(x)^2-1/eta(x)) /* show terms (omitting initial zero) */
/* Joerg Arndt, Jun 21 2011 */

from sympy import npartitions
def A122768(n): return (sum(npartitions(k)*npartitions(n-k) for k in range(1,n+1>>1))<<1) + (0 if n&1 else npartitions(n>>1)**2) + npartitions(n) if n else 0 # Chai Wah Wu, Sep 25 2023

G.f.: 1/P(x)^2 - 1/P(x) where P(x)=prod(k>=1, 1-x^k ). - Joerg Arndt, Jun 21 2011
With sum_i^P(n) = the sum over all P(n) integer partitions of n, sum_j^p(i) = the sum over all p(i) parts of the i-th integer partition, prttn(i) = the i-th partition whereat prttn(i) is a list, choose(L,k) = construct the list LC of combinations of a list L (see Maple), |LC| = number of elements of list LC (=Maple's nops command) we have a(n) = sum_i^P(n) sum_j^p(i) |choose(prttn,j)|
a(n) = A000712(n) - A000041(n). - Alford Arnold, Dec 12 2006
a(n) = A144064(n,2)-A144064(n,1). - Alois P. Heinz, Mar 31 2017
a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*3^(3/4)*n^(5/4)) * (1 - (Pi/12 + 45/(16*Pi))/sqrt(3*n)). - Vaclav Kotesovec, Mar 31 2017

Extended by R. J. Mathar, Aug 25 2008

A060642 Triangle read by rows: row n lists number of ordered partitions into k parts of partitions of n.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 5, 10, 6, 1, 7, 22, 21, 8, 1, 11, 43, 59, 36, 10, 1, 15, 80, 144, 124, 55, 12, 1, 22, 141, 321, 362, 225, 78, 14, 1, 30, 240, 669, 944, 765, 370, 105, 16, 1, 42, 397, 1323, 2266, 2287, 1437, 567, 136, 18, 1, 56, 640, 2511, 5100, 6215, 4848, 2478, 824, 171, 20, 1

Offset: 1

Alford Arnold, Apr 16 2001

Also the convolution triangle of A000041. - Peter Luschny, Oct 07 2022

			Table begins:
   1;
   2,   1;
   3,   4,    1;
   5,  10,    6,    1;
   7,  22,   21,    8,    1;
  11,  43,   59,   36,   10,    1;
  15,  80,  144,  124,   55,   12,   1;
  22, 141,  321,  362,  225,   78,  14,   1;
  30, 240,  669,  944,  765,  370, 105,  16,  1;
  42, 397, 1323, 2266, 2287, 1437, 567, 136, 18, 1;
  ...
For n=4 there are 5 partitions of 4, namely 4, 31, 22, 211, 11111. There are 5 ways to pick 1 of them; 10 ways to partition one of them into 2 ordered parts: 3,1; 1,3; 2,2; 21,1; 1,21; 2,11; 11,2; 111,1; 1,111; 11,11; 6 ways to partition one of them into 3 ordered parts: 2,1,1; 1,2,1; 1,1,2; 11,1,1; 1,11,1; 1,1,11; and one way to partition one of them into 4 ordered parts: 1,1,1,1. So row 4 is 5,10,6,1.

Alois P. Heinz, Rows n = 1..141, flattened

Columns k=1-10 give: A000041, A048574, A341221, A341222, A341223, A341225, A341226, A341227, A341228, A341236.
Row sums give A055887.
Cf. A097805, A010815, A055888, A144064, A261719, A326346.
T(2n,n) gives A340987.

A:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      A(n-j, k)*numtheory[sigma](j), j=1..n)/n)
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Mar 12 2015
# Uses function PMatrix from A357368. Adds row and column for n, k = 0.
PMatrix(10, combinat:-numbpart); # Peter Luschny, Oct 07 2022

A[n_, k_] := A[n, k] = If[n==0, 1, k*Sum[A[n-j, k]*DivisorSigma[1, j], {j, 1, n}]/n]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[ Table[ T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)

G.f. A(n;x) for n-th row satisfies A(n;x) = Sum_{k=0..n-1} A000041(n-k)*A(k;x)*x, A(0;x) = 1. - Vladeta Jovovic, Jan 02 2004
T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A144064(n,k-i). - Alois P. Heinz, Mar 12 2015
Sum_{k=1..n} k * T(n,k) = A326346(n). - Alois P. Heinz, Sep 11 2019
Sum_{k=0..n} (-1)^k * T(n,k) = A010815(n). - Alois P. Heinz, Feb 07 2021
G.f. of column k: (-1 + Product_{j>=1} 1 / (1 - x^j))^k. - Ilya Gutkovskiy, Feb 13 2021

More terms from Vladeta Jovovic, Jan 02 2004

A338463 Expansion of g.f.: (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^2.

Original entry on oeis.org

1, 0, 2, 2, 3, 4, 5, 8, 9, 12, 15, 20, 23, 28, 36, 44, 52, 62, 76, 90, 106, 124, 149, 176, 203, 236, 279, 324, 372, 430, 499, 576, 657, 752, 867, 992, 1124, 1280, 1463, 1662, 1876, 2124, 2410, 2722, 3061, 3446, 3889, 4374, 4896, 5490, 6166, 6900, 7700, 8600

Offset: 2

Ilya Gutkovskiy, Jan 31 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 2..1002

Cf. A000700, A022597, A047654, A048574, A073252, A327380, A338223.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( (-1 + (&*[1+x^(2*j+1): j in [0..m+2]]) )^2 )); // G. C. Greubel, Sep 07 2023

nmax = 55; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^2, {x, 0, nmax}], x] // Drop[#, 2] &
With[{k=2}, Drop[CoefficientList[Series[(2/QPochhammer[-1,-x] -1)^k, {x,0,80}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

m=80
def f(x): return (-1 + product(1+x^(2*j-1) for j in range(1,m+3)) )^2
def A338463_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
a=A338463_list(m); a[2:] # G. C. Greubel, Sep 07 2023

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^2.
a(n) = Sum_{k=1..n-1} A000700(k) * A000700(n-k).
a(n) = A073252(n) - 2 * A000700(n) for n > 0.
a(n) = [x^n]( (2/QPochhammer(-1,-x) - 1)^2 ). - G. C. Greubel, Sep 07 2023

A341221 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^3.

Original entry on oeis.org

1, 6, 21, 59, 144, 321, 669, 1323, 2511, 4604, 8202, 14253, 24241, 40449, 66363, 107234, 170910, 269004, 418566, 644436, 982536, 1484482, 2223942, 3305484, 4876620, 7144455, 10398123, 15039564, 21624678, 30919323, 43973708, 62222844, 87619212, 122810585

Offset: 3

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..10000

Column k=3 of A060642.

Cf. A000041, A000716, A048574, A327381, A341222, A341223, A341225, A341226, A341227, A341228.

b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 3):
seq(a(n), n=3..36);  # Alois P. Heinz, Feb 07 2021

nmax = 36; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^3, {x, 0, nmax}], x] // Drop[#, 3] &

a(n) ~ A000716(n). - Vaclav Kotesovec, Feb 20 2021

A341222 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^4.

Original entry on oeis.org

1, 8, 36, 124, 362, 944, 2266, 5100, 10903, 22340, 44168, 84692, 158137, 288452, 515344, 903740, 1558465, 2646820, 4432964, 7329916, 11977507, 19358524, 30970444, 49077936, 77081679, 120054268, 185514428, 284540060, 433360308, 655622392, 985604644, 1472751228

Offset: 4

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..10000

Column k=4 of A060642.

Cf. A000041, A023003, A048574, A327382, A341221, A341223, A341225, A341226, A341227, A341228.

b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..35);  # Alois P. Heinz, Feb 07 2021

nmax = 35; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^4, {x, 0, nmax}], x] // Drop[#, 4] &

a(n) ~ A023003(n). - Vaclav Kotesovec, Feb 20 2021

A341223 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^5.

Original entry on oeis.org

1, 10, 55, 225, 765, 2287, 6215, 15680, 37265, 84300, 182933, 383070, 777705, 1536490, 2963120, 5592060, 10349465, 18817760, 33665870, 59341785, 103176877, 177131330, 300530125, 504318530, 837632700, 1377874861, 2246061540, 3630059510, 5819556060, 9258393655, 14622472250

Offset: 5

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..10000

Column k=5 of A060642.

Cf. A000041, A023004, A048574, A327383, A341221, A341222, A341225, A341226, A341227, A341228.

b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..35);  # Alois P. Heinz, Feb 07 2021

nmax = 35; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^5, {x, 0, nmax}], x] // Drop[#, 5] &

A341225 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^6.

Original entry on oeis.org

1, 12, 78, 370, 1437, 4848, 14719, 41148, 107610, 266296, 628941, 1427118, 3127369, 6646440, 13746081, 27744926, 54782271, 106029918, 201512970, 376630680, 693161334, 1257641676, 2251764699, 3982196910, 6961522279, 12038699766, 20607718317, 34938910360

Offset: 6

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..10000

Column k=6 of A060642.

Cf. A000041, A023005, A048574, A327384, A341221, A341222, A341223, A341226, A341227, A341228.

b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 6):
seq(a(n), n=6..33);  # Alois P. Heinz, Feb 07 2021

nmax = 33; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^6, {x, 0, nmax}], x] // Drop[#, 6] &

A341226 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^7.

Original entry on oeis.org

1, 14, 105, 567, 2478, 9317, 31269, 95965, 273896, 735966, 1879059, 4591342, 10797290, 24549924, 54171729, 116368308, 243991034, 500446135, 1006039762, 1985480063, 3852429483, 7358212272, 13850448185, 25718189483, 47150564517, 85417834621, 153015826880

Offset: 7

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..10000

Column k=7 of A060642.

Cf. A000041, A023006, A048574, A327385, A341221, A341222, A341223, A341225, A341227, A341228.

b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 7):
seq(a(n), n=7..33);  # Alois P. Heinz, Feb 07 2021

nmax = 33; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^7, {x, 0, nmax}], x] // Drop[#, 7] &

A341227 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^8.

Original entry on oeis.org

1, 16, 136, 824, 4004, 16608, 61076, 204200, 631714, 1831752, 5027312, 13159104, 33049090, 80030808, 187613348, 427201176, 947520103, 2051989360, 4347996772, 9030416704, 18412343832, 36905322248, 72807201940, 141525042736, 271321432489, 513454659312

Offset: 8

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..10000

Column k=8 of A060642.

Cf. A000041, A023007, A048574, A327386, A341221, A341222, A341223, A341225, A341226, A341228.

b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 8):
seq(a(n), n=8..33);  # Alois P. Heinz, Feb 07 2021

nmax = 33; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^8, {x, 0, nmax}], x] // Drop[#, 8] &

A341228 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^9.

Original entry on oeis.org

1, 18, 171, 1149, 6147, 27891, 111567, 403722, 1345896, 4189334, 12300174, 34337403, 91721385, 235645425, 584759880, 1406588073, 3289489002, 7498465029, 16697615817, 36391839264, 77758115283, 163123713621, 336420277812, 682877289213, 1365674365197, 2693384989056

Offset: 9

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..10000

Column k=9 of A060642.

Cf. A000041, A023008, A048574, A327387, A341221, A341222, A341223, A341225, A341226, A341227.

b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 9):
seq(a(n), n=9..34);  # Alois P. Heinz, Feb 07 2021

nmax = 34; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^9, {x, 0, nmax}], x] // Drop[#, 9] &

cp's OEIS Frontend

A122768 Number of combinations which can be taken from the integer partitions of n. Total number of cases in the (n,m)-fragmentation process.

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A060642 Triangle read by rows: row n lists number of ordered partitions into k parts of partitions of n.

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A338463 Expansion of g.f.: (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^2.

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A341221 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^3.

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A341222 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^4.

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A341223 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^5.

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A341225 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^6.

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A341226 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^7.