A262495 -id:A262495 - cp's OEIS frontend

A256130 Number T(n,k) of partitions of n into parts of exactly k sorts which are introduced in ascending order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 5, 12, 7, 1, 0, 7, 30, 33, 11, 1, 0, 11, 72, 130, 77, 16, 1, 0, 15, 160, 463, 438, 157, 22, 1, 0, 22, 351, 1557, 2216, 1223, 289, 29, 1, 0, 30, 743, 5031, 10422, 8331, 2957, 492, 37, 1, 0, 42, 1561, 15877, 46731, 52078, 26073, 6401, 788, 46, 1

Offset: 0

Alois P. Heinz, Mar 15 2015

In general, column k>1 is asymptotic to c*k^n, where c = 1/(k!*Product_{n>=1} (1-1/k^n)) = 1/(k!*QPochhammer[1/k, 1/k]). - Vaclav Kotesovec, Jun 01 2015

			T(3,1) = 3: 1a1a1a, 2a1a, 3a.
T(3,2) = 4: 1a1a1b, 1a1b1a, 1a1b1b, 2a1b.
T(3,3) = 1: 1a1b1c.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  2,    1;
  0,  3,    4,     1;
  0,  5,   12,     7,     1;
  0,  7,   30,    33,    11,     1;
  0, 11,   72,   130,    77,    16,     1;
  0, 15,  160,   463,   438,   157,    22,    1;
  0, 22,  351,  1557,  2216,  1223,   289,   29,   1;
  0, 30,  743,  5031, 10422,  8331,  2957,  492,  37,  1;
  0, 42, 1561, 15877, 46731, 52078, 26073, 6401, 788, 46,  1;
  ...

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

Columns k=0-10 give: A000007, A000041 (for n>0), A258457, A258458, A258459, A258460, A258461, A258462, A258463, A258464, A258465.
Row sums give A258466.
T(2n,n) give A258467.
Cf. A000142, A246935, A255970, A262495, A319730.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; T[n_, k_] := Sum[b[n, n, k-i]*(-1)^i/(i!*(k-i)!), {i, 0, k}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 21 2016, after Alois P. Heinz *)

T(n,k) = A255970(n,k)/k! = (Sum_{i=0..k} (-1)^i * C(k,i) * A246935(n,k-i)) / A000142(k).

A320545 Number of partitions of n into parts of exactly three sorts which are introduced in ascending order such that sorts of adjacent parts are different.

Original entry on oeis.org

1, 4, 12, 31, 73, 165, 357, 760, 1582, 3270, 6678, 13589, 27482, 55468, 111588, 224259, 449908, 902106, 1807173, 3619162, 7244557, 14499238, 29011551, 58044194, 116115782, 232275383, 464607730, 929306306, 1858730674, 3717648658, 7435541392, 14871467784

Offset: 3

Alois P. Heinz, Oct 15 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column k=3 of A262495.

Cf. A258458, A262445.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):
a:= n-> (k-> add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k))(3):
seq(a(n), n=3..40);

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, k^(n - 1), b[n, i - 1, k] + If[i > n, 0, k b[n - i, i, k]]];
A[n_, k_] := If[n == 0, 1, If[k < 2, k, k b[n, n, k - 1]]];
a[n_] := With[{k = 3}, Sum[A[n, k - i] (-1)^i/(i! (k - i)!), {i, 0, k}]];
a /@ Range[3, 40] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)

a(n) ~ 2^(n-2) / QPochhammer[1/2]. - Vaclav Kotesovec, Oct 25 2018

A262445 Number of exact 3-colored partitions such that no adjacent parts have the same color.

Original entry on oeis.org

0, 0, 0, 6, 24, 72, 186, 438, 990, 2142, 4560, 9492, 19620, 40068, 81534, 164892, 332808, 669528, 1345554, 2699448, 5412636, 10843038, 21714972, 43467342, 86995428, 174069306, 348265164, 696694692, 1393652298, 2787646380, 5575837836, 11152384044, 22305891948, 44613248352, 89228806704, 178460625402, 356925987924

Offset: 0

Ran Pan, Sep 23 2015

nonn

a(1) = a(2) = 0 because we need to use exactly three colors, which means the number of parts should be greater than two.

All terms are multiples of 6.

			a(3)=6 because there are three partitions of 3 and there are no ways to color [3] or [2,1] but there are six ways to color [1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Ran Pan, A note on enumerating colored integer partitions, arXiv:1509.06107 [math.CO], 2015.
Ran Pan, Exercise S, Project P.

Cf. A000041, A262444, A139582, A262495, A320545.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
a:= n-> `if`(n=0, 0, b(n$2, 2)/2*3-6*b(n$2, 1)+3):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 23 2015

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k*b[n - i, i, k]]]]; a[n_] := If[n == 0, 0, b[n, n, 2]/2*3 - 6*b[n, n, 1] + 3]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)

G.f.: 3/2*Product_{k>=1} (1/(1-2*x^k)) - 6*Product_{k>=1} (1/(1-x^k)) + 3/(1-x) + 3/2.
a(n) = A262444(n) - 6*A000041(n) + 3, for n >= 1.
a(n) = 6 * A262495(n,3). - Alois P. Heinz, Sep 24 2015

A262496 Number of partitions of n into parts of sorts {1, 2, ... } which are introduced in ascending order such that sorts of adjacent parts are different.

Original entry on oeis.org

1, 1, 2, 4, 10, 27, 87, 312, 1269, 5703, 28082, 149643, 855938, 5217753, 33712046, 229799508, 1646314498, 12355371024, 96861186897, 791258791159, 6720627161903, 59234364141343, 540812222291531, 5106663817387466, 49798678281320763, 500857393909589995

Offset: 0

Alois P. Heinz, Sep 24 2015

nonn

			a(3) = 4: 3a, 2a1b, 1a1b1a, 1a1b1c (in this example the sorts are labeled a, b, c).

Alois P. Heinz, Table of n, a(n) for n = 0..400

Row sums of A262495.

Cf. A258466.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):
T:= (n, k)->  add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> add(T(n, k), k=0..n):
seq(a(n), n=0..30);

b[n_, i_, k_] := b[n, i, k] = If[n==0 || i==1, k^(n-1), b[n, i-1, k] + If[i > n, 0, k*b[n-i, i, k]]]; A[n_, k_] := If[n==0, 1, If[k<2, k, k*b[n, n, k - 1]]]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i/(i!*(k-i)!), {i, 0, k}]; a[n_] := Sum[T[n, k], {k, 0, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 05 2017, translated from Maple *)

A320546 Number of partitions of n into parts of exactly four sorts which are introduced in ascending order such that sorts of adjacent parts are different.

Original entry on oeis.org

1, 7, 33, 130, 464, 1558, 5039, 15886, 49282, 151165, 460352, 1394863, 4212752, 12694566, 38197710, 114820403, 344919283, 1035670246, 3108844526, 9330186438, 27997888759, 84008273161, 252054096569, 756220672185, 2268778953179, 6806570182252, 20420177671614

Offset: 4

Alois P. Heinz, Oct 15 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..1000

Column k=4 of A262495.

Cf. A258459.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):
a:= n-> (k-> add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k))(4):
seq(a(n), n=4..40);

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, k^(n - 1), b[n, i - 1, k] + If[i > n, 0, k b[n - i, i, k]]];
A[n_, k_] := If[n == 0, 1, If[k < 2, k, k b[n, n, k - 1]]];
a[n_] := With[{k = 4}, Sum[A[n, k - i] (-1)^i/(i! (k - i)!), {i, 0, k}]];
a /@ Range[4, 40] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)

a(n) ~ 3^(n-1) / (3! * QPochhammer[1/3]). - Vaclav Kotesovec, Oct 25 2018

A320547 Number of partitions of n into parts of exactly five sorts which are introduced in ascending order such that sorts of adjacent parts are different.

Original entry on oeis.org

1, 11, 77, 438, 2216, 10423, 46732, 202826, 860599, 3593651, 14835058, 60735635, 247155920, 1001321100, 4043485479, 16288776186, 65500040622, 263035896496, 1055252507399, 4230340498375, 16949360224358, 67881450386237, 271777857121332, 1087867654290457

Offset: 5

Alois P. Heinz, Oct 15 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..1000

Column k=5 of A262495.

Cf. A258460.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):
a:= n-> (k-> add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k))(5):
seq(a(n), n=5..40);

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, k^(n - 1), b[n, i - 1, k] + If[i > n, 0, k b[n - i, i, k]]];
A[n_, k_] := If[n == 0, 1, If[k < 2, k, k b[n, n, k - 1]]];
a[n_] := With[{k = 5}, Sum[A[n, k - i] (-1)^i/(i! (k - i)!), {i, 0, k}]];
a /@ Range[5, 40] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)

a(n) ~ 4^(n-1) / (4! * QPochhammer[1/4]). - Vaclav Kotesovec, Oct 25 2018

A320548 Number of partitions of n into parts of exactly six sorts which are introduced in ascending order such that sorts of adjacent parts are different.

Original entry on oeis.org

1, 16, 157, 1223, 8331, 52078, 307123, 1738442, 9552826, 51357799, 271624228, 1418856967, 7341442171, 37708533544, 192586163135, 979219603193, 4961598120154, 25071026570558, 126410385741982, 636282269651863, 3198360710675384, 16059685006324807

Offset: 6

Alois P. Heinz, Oct 15 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..1000

Column k=6 of A262495.

Cf. A258461.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):
a:= n-> (k-> add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k))(6):
seq(a(n), n=6..40);

a(n) ~ 5^(n-1) / (5! * QPochhammer[1/5]). - Vaclav Kotesovec, Oct 25 2018

A320549 Number of partitions of n into parts of exactly seven sorts which are introduced in ascending order such that sorts of adjacent parts are different.

Original entry on oeis.org

1, 22, 289, 2957, 26073, 208516, 1558219, 11087757, 76079369, 507834036, 3318628468, 21330628088, 135325211035, 849659803048, 5290544985029, 32722489543068, 201296535411532, 1232850239281547, 7523511821702402, 45777353201698899, 277862479922939524

Offset: 7

Alois P. Heinz, Oct 15 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..1000

Column k=7 of A262495.

Cf. A258462.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):
a:= n-> (k-> add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k))(7):
seq(a(n), n=7..40);

a(n) ~ 6^(n-1) / (6! * QPochhammer[1/6]). - Vaclav Kotesovec, Oct 25 2018

A320550 Number of partitions of n into parts of exactly eight sorts which are introduced in ascending order such that sorts of adjacent parts are different.

Original entry on oeis.org

1, 29, 492, 6401, 70880, 704676, 6490951, 56524414, 471750268, 3810085913, 29989229889, 231255237342, 1754111872952, 13128442914265, 97189645391839, 713050007293418, 5192646586543845, 37581376345173772, 270593146238709314, 1939929376873532436

Offset: 8

Alois P. Heinz, Oct 15 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..1000

Column k=8 of A262495.

Cf. A258463.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):
a:= n-> (k-> add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k))(8):
seq(a(n), n=8..40);

a(n) ~ 7^(n-1) / (7! * QPochhammer[1/7]). - Vaclav Kotesovec, Oct 25 2018

A320551 Number of partitions of n into parts of exactly nine sorts which are introduced in ascending order such that sorts of adjacent parts are different.

Original entry on oeis.org

1, 37, 788, 12705, 172520, 2084836, 23169639, 241881526, 2406802476, 23064505722, 214505275666, 1947297442708, 17332491414655, 151788374232332, 1311496639251360, 11205023121317869, 94832831557101194, 796244028802170279, 6640545376656272033

Offset: 9

Alois P. Heinz, Oct 15 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..1000

Column k=9 of A262495.

Cf. A258464.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):
a:= n-> (k-> add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k))(9):
seq(a(n), n=9..40);

a(n) ~ 8^(n-1) / (8! * QPochhammer[1/8]). - Vaclav Kotesovec, Oct 25 2018

cp's OEIS Frontend

A256130 Number T(n,k) of partitions of n into parts of exactly k sorts which are introduced in ascending order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A320545 Number of partitions of n into parts of exactly three sorts which are introduced in ascending order such that sorts of adjacent parts are different.

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A262445 Number of exact 3-colored partitions such that no adjacent parts have the same color.

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A262496 Number of partitions of n into parts of sorts {1, 2, ... } which are introduced in ascending order such that sorts of adjacent parts are different.

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A320546 Number of partitions of n into parts of exactly four sorts which are introduced in ascending order such that sorts of adjacent parts are different.

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A320547 Number of partitions of n into parts of exactly five sorts which are introduced in ascending order such that sorts of adjacent parts are different.

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A320548 Number of partitions of n into parts of exactly six sorts which are introduced in ascending order such that sorts of adjacent parts are different.

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A320549 Number of partitions of n into parts of exactly seven sorts which are introduced in ascending order such that sorts of adjacent parts are different.

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