A256130 -id:A256130 - cp's OEIS frontend

A246935 Number A(n,k) of partitions of n into k sorts of parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 3, 0, 1, 4, 12, 14, 5, 0, 1, 5, 20, 39, 34, 7, 0, 1, 6, 30, 84, 129, 74, 11, 0, 1, 7, 42, 155, 356, 399, 166, 15, 0, 1, 8, 56, 258, 805, 1444, 1245, 350, 22, 0, 1, 9, 72, 399, 1590, 4055, 5876, 3783, 746, 30, 0

Offset: 0

Alois P. Heinz, Sep 08 2014

In general, column k > 1 is asymptotic to c * k^n, where c = Product_{j>=1} 1/(1-1/k^j) = 1/QPochhammer[1/k,1/k]. - Vaclav Kotesovec, Mar 19 2015

When k is a prime power greater than 1, A(n,k) is the number of conjugacy classes of n X n matrices over a field of size k. - Geoffrey Critzer, Nov 11 2022

			A(2,2) = 6: [2a], [2b], [1a,1a], [1a,1b], [1b,1a], [1b,1b].
Square array A(n,k) begins:
  1,  1,   1,    1,     1,      1,      1,      1, ...
  0,  1,   2,    3,     4,      5,      6,      7, ...
  0,  2,   6,   12,    20,     30,     42,     56, ...
  0,  3,  14,   39,    84,    155,    258,    399, ...
  0,  5,  34,  129,   356,    805,   1590,   2849, ...
  0,  7,  74,  399,  1444,   4055,   9582,  19999, ...
  0, 11, 166, 1245,  5876,  20455,  57786, 140441, ...
  0, 15, 350, 3783, 23604, 102455, 347010, 983535, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Columns k=0-10 give: A000007, A000041, A070933, A242587, A246936, A246937, A246938, A246939, A246940, A246941, A246942.
Rows n=0-4 give: A000012, A001477, A002378, A027444, A186636.
Main diagonal gives A124577.
Cf. A144064, A255970, A256130.

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, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);

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_, k_] := b[n, n, k];  Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 03 2015, after Alois P. Heinz *)

G.f. of column k: Product_{i>=1} 1/(1-k*x^i).

T(n,k) = Sum_{i=0..k} C(k,i) * A255970(n,i).

A258466 Number of partitions of n into parts of sorts {1, 2, ... } which are introduced in ascending order.

Original entry on oeis.org

1, 1, 3, 8, 25, 82, 307, 1256, 5688, 28044, 149598, 855811, 5217604, 33711592, 229798958, 1646312694, 12355368849, 96861178984, 791258781708, 6720627124140, 59234364096426, 540812222095821, 5106663817156741, 49798678280227488, 500857393908312587

Offset: 0

Alois P. Heinz, May 30 2015

nonn

Also number of ways of partitioning a multiset with multiplicities some partition of n into disjoint blocks. Example: a(4) = 25: 1111; 111,2; 1112; 11,22; 1122; 11,2,3; 11,23; 112,3; 113,2; 1123; 1,2,3,4; 1,2,34; 1,23,4; 1,24,3; 1,234; 12,3,4; 12,34; 13,2,4; 13,24; 14,2,3; 14,23; 123,4; 124,3; 134,2; 1234. Formula: a(n) is the sum of Bell numbers of lengths of all integer partitions of n. - Gus Wiseman, Feb 17 2016

			a(3) = 8: 1a1a1a, 2a1a, 3a, 1a1a1b, 1a1b1a, 1a1b1b, 2a1b, 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 A256130.

Cf. A000110, A035310, A262496, A278644, A319731.

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):
a:= n-> add(T(n, k), k=0..n):
seq(a(n), n=0..25);

Table[Plus @@ BellB /@ Length /@ IntegerPartitions[n], {n, 0, 24}] (* Gus Wiseman, Feb 17 2016 *)
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}]; a[n_] := Sum[T[n, k], {k, 0, n}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Sep 01 2016, after Alois P. Heinz *)

a(n) = Sum_{k=0..n} A256130(n,k).
a(n) ~ Bell(n) = A000110(n). - Vaclav Kotesovec, Jun 01 2015
G.f.: Sum_{k>=0} Bell(k) * x^k / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Jan 28 2020

A262495 Number T(n,k) of partitions of n into parts of exactly k sorts which are introduced in ascending order such that sorts of adjacent parts are different; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 4, 1, 0, 1, 6, 12, 7, 1, 0, 1, 10, 31, 33, 11, 1, 0, 1, 14, 73, 130, 77, 16, 1, 0, 1, 21, 165, 464, 438, 157, 22, 1, 0, 1, 29, 357, 1558, 2216, 1223, 289, 29, 1, 0, 1, 41, 760, 5039, 10423, 8331, 2957, 492, 37, 1

Offset: 0

Alois P. Heinz, Sep 24 2015

			T(3,1) = 1: 3a.
T(3,2) = 2: 2a1b, 1a1b1a.
T(3,3) = 1: 1a1b1c.
T(5,3) = 12: 3a1b1c, 2a2b1c, 2a1b1a1c, 2a1b1c1a, 2a1b1c1b, 1a1b1a1b1c, 1a1b1a1c1a, 1a1b1a1c1b, 1a1b1c1a1b, 1a1b1c1a1c, 1a1b1c1b1a, 1a1b1c1b1c.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  2,   1;
  0, 1,  4,   4,    1;
  0, 1,  6,  12,    7,     1;
  0, 1, 10,  31,   33,    11,    1;
  0, 1, 14,  73,  130,    77,   16,    1;
  0, 1, 21, 165,  464,   438,  157,   22,   1;
  0, 1, 29, 357, 1558,  2216, 1223,  289,  29,  1;
  0, 1, 41, 760, 5039, 10423, 8331, 2957, 492, 37, 1;
  ...

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

Columns k=0-10 give: A000007, A057427, A000065, A262445/6 = A320545, A320546, A320547, A320548, A320549, A320550, A320551, A320552.
Main diagonal and lower diagonal give: A000012, A000124 (shifted).
Row sums give A262496.
T(2n,n) gives A262529.
Cf. A256130.

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):
seq(seq(T(n, k), k=0..n), n=0..12);

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}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 24 2017, translated from Maple *)

A300453 Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (x + 1)^n + x^2 - 1.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 2, 2, 0, 3, 4, 1, 0, 4, 7, 4, 1, 0, 5, 11, 10, 5, 1, 0, 6, 16, 20, 15, 6, 1, 0, 7, 22, 35, 35, 21, 7, 1, 0, 8, 29, 56, 70, 56, 28, 8, 1, 0, 9, 37, 84, 126, 126, 84, 36, 9, 1, 0, 10, 46, 120, 210, 252, 210, 120, 45, 10, 1, 0, 11, 56, 165

Offset: 0

Franck Maminirina Ramaharo, Mar 06 2018

This is essentially the usual Pascal triangle A007318, horizontally shifted and with the first three columns altered.
Let P(n;x) = (x + 1)^n + x^2 - 1. Then P(n;x) = P(n-1;x) + x*(x + 1)^(n - 1), with P(0;x) = x^2.
Let a (2,n)-torus knot be projected on the plane. The resulting projection is a planar diagram with n double points. Then, T(n,k) gives the number of state diagrams having k components that are obtained by deleting each double point, then pasting the edges in one of the two ways as shown below.
         \  /     \_/                   \  /     \   /
   (1)    \/  ==>                 (2)      \/  ==>  | |
          /\       _                     /\       | |
         /  \     /   \                   /  \     /   \
See example for the case n = 2.

			The triangle T(n,k) begins
n\k  0   1   2    3     4     5     6     7     8     9    10   11  12  13 14
0:   0   0   1
1:   0   1   1
2:   0   2   2
3:   0   3   4    1
4:   0   4   7    4     1
5:   0   5  11   10     5     1
6:   0   6  16   20    15     6     1
7:   0   7  22   35    35    21     7     1
8:   0   8  29   56    70    56    28     8     1
9:   0   9  37   84   126   126    84    36     9     1
10:  0  10  46  120   210   252   210   120    45    10     1
11:  0  11  56  165   330   462   462   330   165    55    11    1
12:  0  12  67  220   495   792   924   792   495   220    66   12   1
13:  0  13  79  286   715  1287  1716  1716  1287   715   286   78  13   1
14:  0  14  92  364  1001  2002  3003  3432  3003  2002  1001  364  91  14  1
...
The states of the (2,2)-torus knot (Hopf Link) are the last four diagrams:
                                    ____  ____
                                   /    \/    \
                                  /     /\     \
                                 |     |  |     |
                                 |     |  |     |
                                  \     \/     /
                                   \____/\____/
                           ___    ____         __________
                         /    \  /    \       /    __    \
                        /     /  \     \     /    /  \    \
                       |      |  |      |   |     |  |     |
                       |      |  |      |   |     |  |     |
                        \      \/      /     \     \/     /
                         \_____/\_____/       \____/\____/
      ____    ____        ____    ____        ____________        __________
     /    \  /    \      /    \  /    \      /     __     \      /    __    \
    /     /  \     \    /     /  \     \    /     /  \     \    /    /  \    \
   |     |    |     |  |     |    |     |  |     |    |     |  |    |    |    |
   |     |    |     |  |     |    |     |  |     |    |     |  |    |    |    |
    \     \  /     /    \     \__/     /    \     \  /     /    \    \__/    /
     \____/  \____/      \____________/      \____/  \____/      \__________/
There are 2 diagrams that consist of two components, and 2 diagrams that consist of one component.

Colin Adams, The Knot Book, W. H. Freeman and Company, 1994.
Louis H. Kauffman, Knots and Physics, World Scientific Publishers, 1991.

Michael De Vlieger, Table of n, a(n) for n = 0..11327 (rows 0 <= n <= 150, flattened).
Agnijo Banerjee, Knot theory [Foil knot family].
Allison Henrich, Rebecca Hoberg, Slavik Jablan, Lee Johnson, Elizabeth Minten and Ljiljana Radovic, The Theory of Pseudoknots, arXiv preprint arXiv:1210.6934 [math.GT], 2012.
Abdullah Kopuzlu, Abdulgani Şahin and Tamer Ugur, On polynomials of K(2,n) torus knots, Applied Mathematical Sciences, Vol. 3 (2009), 2899-2910.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Franck Ramaharo, Note on sequences A123192, A137396 and A300453, arXiv:1911.04528 [math.CO], 2019.
Franck Ramaharo, A bracket polynomial for 2-tangle shadows, arXiv:2002.06672 [math.CO], 2020.
Wikipedia, Torus knot.
Xinfei Li, Xin Liu and Yong-Chang Huang, Tackling tangledness of cosmic strings by knot polynomial topological invariants, arxiv preprint arXiv:1602.08804 [hep-th], 2016.

Row sums: A000079 (powers of 2).
Triangles related to the regular projection of some knots: A299989 (connected summed trefoils); A300184 (chain links); A300454 (twist knot).
When n = 3 (trefoil), the corresponding 4-tuplet (0,3,4,1) appears in several triangles: A030528 (row 5), A256130 (row 3), A128908 (row 3), A186084 (row 6), A284938 (row 7), A101603 (row 3), A155112 (row 3), A257566 (row 3).
Cf. A001477, A007318, A007318, A152947.

f[n_] := CoefficientList[ Expand[(x + 1)^n + x^2 - 1], x]; Array[f, 12, 0] // Flatten (* or *)
CoefficientList[ CoefficientList[ Series[(x^2 + y*x/(1 - y*(x + 1)))/(1 - y), {y, 0, 11}, {x, 0, 11}], y], x] // Flatten (* Robert G. Wilson v, Mar 08 2018 *)

P(n, x) := (x + 1)^n + x^2 - 1$
T : []$
for i:0 thru 20 do
  T : append(T, makelist(ratcoef(P(i, x), x, n), n, 0, max(2, i)))$
T;

row(n) = Vecrev((x + 1)^n + x^2 - 1);
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Mar 12 2018

T(n,1) = A001477(n).
T(n,2) = A152947(n).
T(n,k) = A007318(n,k-1), k >= 1.
T(n,0) = 0, T(0,1) = 1, T(0,2) = 1 and T(n,k) = T(n-1,k) + A007318(n-1,k-1).
G.f.: (x^2 + y*x/(1 - y*(x + 1)))/(1 - y).

A255970 Number T(n,k) of partitions of n into parts of exactly k sorts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 3, 8, 6, 0, 5, 24, 42, 24, 0, 7, 60, 198, 264, 120, 0, 11, 144, 780, 1848, 1920, 720, 0, 15, 320, 2778, 10512, 18840, 15840, 5040, 0, 22, 702, 9342, 53184, 146760, 208080, 146160, 40320, 0, 30, 1486, 30186, 250128, 999720, 2129040, 2479680, 1491840, 362880

Offset: 0

Alois P. Heinz, Mar 12 2015

			T(3,1) = 3: 1a1a1a, 2a1a, 1a.
T(3,2) = 8: 1a1a1b, 1a1b1a, 1b1a1a, 1b1b1a, 1b1a1b, 1a1b1b, 2a1b, 2b1a.
T(3,3) = 6: 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  2,   2;
  0,  3,   8,    6;
  0,  5,  24,   42,    24;
  0,  7,  60,  198,   264,    120;
  0, 11, 144,  780,  1848,   1920,    720;
  0, 15, 320, 2778, 10512,  18840,  15840,   5040;
  0, 22, 702, 9342, 53184, 146760, 208080, 146160, 40320;
  ...

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

Columns k=0-1 give: A000007, A000041 (for n>0).
Main diagonal gives A000142.
Row sums give A278644.
Cf. A246935, A256130, A319600.

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*binomial(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* Binomial[k, i], {i, 0, k}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

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

T(n,k) = k! * A256130(n,k).

A319730 Number T(n,k) of plane 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.

Original entry on oeis.org

1, 0, 1, 0, 3, 2, 0, 6, 11, 3, 0, 13, 48, 33, 5, 0, 24, 165, 212, 75, 7, 0, 48, 573, 1253, 798, 172, 11, 0, 86, 1759, 6114, 6175, 2284, 326, 15, 0, 160, 5473, 29573, 45040, 25697, 6198, 631, 22, 0, 282, 16051, 131488, 289685, 238516, 86189, 14519, 1102, 30

Offset: 0

Alois P. Heinz, Sep 26 2018

			Triangle T(n,k) begins:
  1;
  0,   1;
  0,   3,     2;
  0,   6,    11,      3;
  0,  13,    48,     33,      5;
  0,  24,   165,    212,     75,      7;
  0,  48,   573,   1253,    798,    172,    11;
  0,  86,  1759,   6114,   6175,   2284,   326,    15;
  0, 160,  5473,  29573,  45040,  25697,  6198,   631,   22;
  0, 282, 16051, 131488, 289685, 238516, 86189, 14519, 1102, 30;
  ...

Alois P. Heinz, Rows n = 0..50, flattened
Eric Weisstein's World of Mathematics, Plane partition
Wikipedia, Plane partition

Columns k=0-1 give: A000007, A000219 (for n>0).
Main diagonal gives A000041.
Row sums give A319731.
T(2n,n) gives A319732.
Cf. A256130, A319600.

T(n,k) = 1/k! * A319600(n,k).

A258458 Number of partitions of n into parts of exactly 3 sorts which are introduced in ascending order.

Original entry on oeis.org

1, 7, 33, 130, 463, 1557, 5031, 15877, 49240, 151116, 460173, 1394645, 4212071, 12693724, 38195286, 114817389, 344911117, 1035659955, 3108817911, 9330152740, 27997803871, 84008165515, 252053831034, 756220333901, 2268778132337, 6806569134920, 20420175154486

Offset: 3

Alois P. Heinz, May 30 2015

nonn

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

Column k=3 of A256130.

Cf. 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:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> T(n,3):
seq(a(n), n=3..35);

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}];
a[n_] := T[n, 3];
Table[a[n], {n, 3, 35}] (* Jean-François Alcover, May 22 2018, translated from Maple *)

a(n) ~ c * 3^n, where c = 1/(6*Product_{n>=1} (1-1/3^n)) = 1/(6*QPochhammer[1/3, 1/3]) = 1/(6*A100220) = 0.297552056999755698394581... . - Vaclav Kotesovec, Jun 01 2015

A258459 Number of partitions of n into parts of exactly 4 sorts which are introduced in ascending order.

Original entry on oeis.org

1, 11, 77, 438, 2216, 10422, 46731, 202814, 860586, 3593561, 14834956, 60735095, 247155292, 1001318246, 4043482110, 16288762319, 65500024027, 263035832734, 1055252430510, 4230340216034, 16949359882259, 67881449170593, 271777855641517, 1087867649157513

Offset: 4

Alois P. Heinz, May 30 2015

nonn

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

Column k=4 of A256130.

Cf. A320546.

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):
a:= n-> T(n,4):
seq(a(n), n=4..35);

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[T[n, 4], {n, 4, 35}] (* Jean-François Alcover, May 25 2018, translated from Maple *)

a(n) ~ c * 4^n, where c = 1/(24*Product_{n>=1} (1-1/4^n)) = 1/(24*QPochhammer[1/4, 1/4]) = 1/(24*A100221) = 0.060514735102066542326446... . - Vaclav Kotesovec, Jun 01 2015

A258460 Number of partitions of n into parts of exactly 5 sorts which are introduced in ascending order.

Original entry on oeis.org

1, 16, 157, 1223, 8331, 52078, 307122, 1738441, 9552809, 51357781, 271624053, 1418856775, 7341440755, 37708531955, 192586153199, 979219591861, 4961598056587, 25071026497266, 126410385360189, 636282269208285, 3198360708483673, 16059685003763157

Offset: 5

Alois P. Heinz, May 30 2015

nonn

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

Column k=5 of A256130.

Cf. A320547.

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):
a:= n-> T(n,5):
seq(a(n), n=5..30);

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}];
a[n_] := T[n, 5];
Table[a[n], {n, 5, 30}] (* Jean-François Alcover, May 22 2018, translated from Maple *)

a(n) ~ c * 5^n, where c = 1/(5!*Product_{n>=1} (1-1/5^n)) = 1/(5!*QPochhammer[1/5, 1/5]) = 1/(5!*A100222) = 0.0109601129644612101609007882... . - Vaclav Kotesovec, Jun 01 2015

A258467 Number of partitions of 2n into parts of exactly n sorts which are introduced in ascending order.

Original entry on oeis.org

1, 2, 12, 130, 2216, 52078, 1558219, 56524414, 2406802476, 117575627562, 6478447651345, 397345158550386, 26842747368209994, 1980156804133210116, 158365138356099680582, 13647670818304698139989, 1260732993182758276252088, 124273946254095006307105363

Offset: 0

Alois P. Heinz, May 30 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A256130, A187655, A187657, A217899, A217900, A245109, A319732.

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):
a:= n-> T(2*n, n):
seq(a(n), n=0..20);

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}]; a[n_] := T[2n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 06 2017, translated from Maple *)

a(n) = A256130(2n,n).
a(n) ~ 2^(2*n-1/2) * n^(n-1/2) / (sqrt(Pi*(1-c)) * exp(n) * c^n * (2-c)^n), where c = -A226775 = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 31 2015
a(n) ~ Stirling2(2*n, n) = A007820(n). - Vaclav Kotesovec, Jun 01 2015

cp's OEIS Frontend

A246935 Number A(n,k) of partitions of n into k sorts of parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A258466 Number of partitions of n into parts of sorts {1, 2, ... } which are introduced in ascending order.

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A262495 Number T(n,k) of partitions of n into parts of exactly k sorts which are introduced in ascending order such that sorts of adjacent parts are different; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A300453 Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (x + 1)^n + x^2 - 1.

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A255970 Number T(n,k) of partitions of n into parts of exactly k sorts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A319730 Number T(n,k) of plane 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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A258458 Number of partitions of n into parts of exactly 3 sorts which are introduced in ascending order.

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A258459 Number of partitions of n into parts of exactly 4 sorts which are introduced in ascending order.

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