A028244 -id:A028244 - cp's OEIS frontend

A281779 Number of distinct topologies on an n-set that have exactly 11 open sets.

0, 0, 0, 0, 0, 500, 16980, 342160, 5486040, 77926380, 1031160060, 13047426920, 160124426880, 1921105846660, 22632779709540, 262513678889280, 3002768326532520, 33914184260797340, 378596540805849420, 4181330954328313240, 45727913513193402960, 495618273676457274420

Offset: 0

Geoffrey Critzer, Jan 29 2017

Colin Barker, Table of n, a(n) for n = 0..950
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
Index entries for linear recurrences with constant coefficients, signature (55,-1320,18150,-157773,902055,-3416930,8409500,-12753576,10628640,-3628800).

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A028244, A281774, A281775, A281776, A281777,A281778, A281779, A281780.

a(n) = 25*5!*stirling(n, 5, 2)/6 + 79*6!*stirling(n, 6, 2)/6 + 29*7!*stirling(n, 7, 2)/2 + 39*8!*stirling(n, 8, 2)/4 + 4*9!*stirling(n, 9, 2) + 10!*stirling(n, 10, 2) \\ Colin Barker, Jan 30 2017

concat(vector(4), Vec(20*x^5*(25 - 526*x + 3413*x^2 + 292*x^3 - 72756*x^4 + 226800*x^5) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)*(1 - 9*x)*(1 - 10*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

a(n) = 25/6*5! Stirling2(n, 5) + 79/6*6! Stirling2(n, 6) + 29/2*7! Stirling2(n, 7) + 39/4*8! Stirling2(n, 8) + 4*9! Stirling2(n, 9) + 10! Stirling2(n, 10).

G.f.: 20*x^5*(25 - 526*x + 3413*x^2 + 292*x^3 - 72756*x^4 + 226800*x^5) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)*(1 - 9*x)*(1 - 10*x)). - Colin Barker, Jan 30 2017

A281780 Number of distinct topologies on an n-set that have exactly 12 open sets.

Original entry on oeis.org

0, 0, 0, 0, 12, 660, 20400, 445620, 7977732, 126860580, 1873839000, 26381789940, 359484471852, 4784481401700, 62538498859200, 805447464281460, 10241415118476372, 128722997969290020, 1600670708273985000, 19705915838479512180, 240330009637668935292

Offset: 0

Geoffrey Critzer, Jan 29 2017

nonn

Ray Chandler, Table of n, a(n) for n = 0..960
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
Index entries for linear recurrences with constant coefficients, signature (66, -1925, 32670, -357423, 2637558, -13339535, 45995730, -105258076, 150917976, -120543840, 39916800).

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A028244, A281774, A281775, A281776, A281777, A281778, A281779, A281780.

a(n) = 1/2*4! Stirling2(n, 4) + 9/2*5! Stirling2(n, 5) + 16*6! Stirling2(n, 6) + 295/12*7! Stirling2(n, 7) + 85/4*8! Stirling2(n, 8) + 49/4*9! Stirling2(n, 9) + 9/2*10! Stirling2(n, 10) + 11!*Stirling2(n, 11).

A032180 Number of ways to partition n labeled elements into 6 pie slices.

Original entry on oeis.org

120, 2520, 31920, 317520, 2739240, 21538440, 158838240, 1118557440, 7612364760, 50483192760, 328191186960, 2100689987760, 13282470124680, 83169792213480, 516729467446080, 3190281535536480, 19596640721427000, 119876382958008600

Offset: 6

Christian G. Bower

For n>=6, a(n) is equal to the number of functions f: {1,2,...,n-1}->{1,2,3,4,5,6} such that Im(f) contains 5 fixed elements. - Aleksandar M. Janjic and Milan Janjic, Feb 27 2007

Vincenzo Librandi, Table of n, a(n) for n = 6..1000
C. G. Bower, Transforms (2)
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for sequences related to necklaces
Index entries for linear recurrences with constant coefficients, signature (21,-175,735,-1624,1764,-720).

Cf. A000770, A008277, A000225, A028243, A028244, A028245.

[5*2^(n-1)-10*3^(n-1)+10*4^(n-1)-5^n+6^(n-1)-1: n in [6..30]]; // Vincenzo Librandi, Oct 19 2013

with (combstruct):ZL:=[S, {S=Sequence(U, card=r), U=Set(Z, card>=1)}, labeled]: seq(count(subs(r=6, ZL), size=m)/6, m=6..21); # Zerinvary Lajos, Mar 08 2008

CoefficientList[Series[120/((x - 1) (2 x - 1) (3 x - 1) (4 x - 1) (5 x - 1) (6 x - 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 19 2013 *)
Table[120*StirlingS2[n,6], {n,6,30}] (* G. C. Greubel, Nov 19 2017 *)

for(n=6,30, print1(120*stirling(n,6,2), ", ")) \\ G. C. Greubel, Nov 19 2017

"CIJ[ 6 ]" (necklace, indistinct, labeled, 6 parts) transform of 1, 1, 1, 1...
a(n) = 120*S(n, 6).
From Emeric Deutsch, May 02 2004: (Start)
a(n) = 5*2^(n-1) - 10*3^(n-1) + 10*4^(n-1) - 5^n + 6^(n-1) - 1.
a(n) = 120*A000770(n). (End)
G.f.: 120*x^6/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)). - Colin Barker, Sep 03 2012
E.g.f.: (Sum_{k=0..6} (-1)^(6-k)*binomial(6,k)*exp(k*x))/6 with a(n) = 0 for n = 0..5. - Wolfdieter Lang, May 03 2017

More terms from Vincenzo Librandi Oct 19 2013

A293617 Array of triangles read by ascending antidiagonals, T(m, n, k) = Pochhammer(m, k) * Stirling2(n + m, k + m) with m >= 0, n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 2, 1, 0, 1, 10, 3, 7, 3, 0, 1, 15, 4, 25, 12, 2, 0, 1, 21, 5, 65, 30, 6, 1, 0, 1, 28, 6, 140, 60, 12, 15, 7, 0, 1, 36, 7, 266, 105, 20, 90, 50, 12, 0, 1, 45, 8, 462, 168, 30, 350, 195, 60, 6, 0, 1, 55, 9, 750, 252, 42, 1050, 560, 180, 24, 1, 0

Offset: 0

Peter Luschny, Oct 20 2017

			Array starts:
m\j| 0   1  2     3       4       5       6       7       8       9      10
---|-----------------------------------------------------------------------
m=0| 1,  0, 0,    0,      0,      0,      0,      0,      0,      0,      0
m=1| 1,  1, 1,    1,      3,      2,      1,      7,     12,      6,      1
m=2| 1,  3, 2,    7,     12,      6,     15,     50,     60,     24,     31
m=3| 1,  6, 3,   25,     30,     12,     90,    195,    180,     60,    301
m=4| 1, 10, 4,   65,     60,     20,    350,    560,    420,    120,   1701
m=5| 1, 15, 5,  140,    105,     30,   1050,   1330,    840,    210,   6951
m=6| 1, 21, 6,  266,    168,     42,   2646,   2772,   1512,    336,  22827
m=7| 1, 28, 7,  462,    252,     56,   5880,   5250,   2520,    504,  63987
m=8| 1, 36, 8,  750,    360,     72,  11880,   9240,   3960,    720, 159027
m=9| 1, 45, 9, 1155,    495,     90,  22275,  15345,   5940,    990, 359502
   A000217, A001296,A027480,A002378,A001297,A293475,A033486,A007531,A001298
.
m\j| ...      11      12      13      14
---|-----------------------------------------
m=0| ...,      0,      0,      0,      0, ... [A000007]
m=1| ...,     15,     50,     60,     24, ... [A028246]
m=2| ...,    180,    390,    360,    120, ... [A053440]
m=3| ...,   1050,   1680,   1260,    360, ... [A294032]
m=4| ...,   4200,   5320,   3360,    840, ...
m=5| ...,  13230,  13860,   7560,   1680, ...
m=6| ...,  35280,  31500,  15120,   3024, ...
m=7| ...,  83160,  64680,  27720,   5040, ...
m=8| ..., 178200, 122760,  47520,   7920, ...
m=9| ..., 353925, 218790,  77220,  11880, ...
         A293476,A293608,A293615,A052762, ...
.
The parameter m runs over the triangles and j indexes the triangles by reading them by rows. Let T(m, n) denote the row [T(m, n, k) for 0 <= k <= n] and T(m) denote the triangle [T(m, n) for n >= 0]. Then for instance T(2) is the triangle A053440, T(3, 2) is row 2 of A294032 (which is [25, 30, 12]) and T(3, 2, 1) = 30.
.
Remark: To adapt the sequences A028246 and A053440 to our enumeration use the exponential generating functions exp(x)/(1 - y*(exp(x) - 1)) and exp(x)*(2*exp(x) - y*exp(2*x) + 2*y*exp(x) - 1 - y)/(1 - y*(exp(x) - 1))^2 instead of those indicated in their respective entries.

Eric Weisstein's World of Mathematics, Nørlund Polynomial.

A000217(n) = T(n, 1, 0), A001296(n) = T(n, 2, 0), A027480(n) = T(n, 2, 1),
A002378(n) = T(n, 2, 2), A001297(n) = T(n, 3, 0), A293475(n) = T(n, 3, 1),
A033486(n) = T(n, 3, 2), A007531(n) = T(n, 3, 3), A001298(n) = T(n, 4, 0),
A293476(n) = T(n, 4, 1), A293608(n) = T(n, 4, 2), A293615(n) = T(n, 4, 3),
A052762(n) = T(n, 4, 4), A052787(n) = T(n, 5, 5), A000225(n) = T(1, n, 1),
A028243(n) = T(1, n, 2), A028244(n) = T(1, n, 3), A028245(n) = T(1, n, 4),
A032180(n) = T(1, n, 5), A228909(n) = T(1, n, 6), A228910(n) = T(1, n, 7),
A000225(n) = T(2, n, 0), A007820(n) = T(n, n, 0).
A028246(n,k) = T(1, n, k), A053440(n,k) = T(2, n, k), A294032(n,k) = T(3, n, k),
A293926(n,k) = T(n, n, k), A124320(n,k) = T(n, k, k), A156991(n,k) = T(k, n, n).
Cf. A293616.

A293617 := proc(m, n, k) option remember:
if m = 0 then 0^n elif k < 0 or k > n then 0 elif n = 0 then 1 else
(k+m)*A293617(m,n-1,k) + k*A293617(m,n-1,k-1) + A293617(m-1,n,k) fi end:
for m in [$0..4] do for n in [$0..6] do print(seq(A293617(m, n, k), k=0..n)) od od;
# Sample uses:
A027480 := n -> A293617(n, 2, 1): A293608 := n -> A293617(n, 4, 2):
# Flatten:
a := proc(n) local w; w := proc(k) local t, s; t := 1; s := 1;
while t <= k do s := s + 1; t := t + s od; [s - 1, s - t + k] end:
seq(A293617(n - k, w(k)[1], w(k)[2]), k=0..n) end: seq(a(n), n = 0..11);

T[m_, n_, k_] := Pochhammer[m, k] StirlingS2[n + m, k + m];
For[m = 0, m < 7, m++, Print[Table[T[m, n, k], {n,0,6}, {k,0,n}]]]
A293617Row[m_, n_] := Table[T[m, n, k], {k,0,n}];
(* Sample use: *)
A293926Row[n_] := A293617Row[n, n];

T(m,n,k) = (k + m)*T(m, n-1, k) + k*T(m, n-1, k-1) + T(m-1, n, k) with boundary conditions T(0, n, k) = 0^n; T(m, n, k) = 0 if k<0 or k>n; and T(m, 0, k) = 0^k.

T(m,n,k) = Pochhammer(m, k)*binomial(n + m, k + m)*NorlundPolynomial(n - k, -k - m).

A298668 Number T(n,k) of set partitions of [n] into k blocks such that the absolute difference between least elements of consecutive blocks is always > 1; triangle T(n,k), n>=0, 0<=k<=ceiling(n/2), read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 3, 0, 1, 7, 2, 0, 1, 15, 12, 0, 1, 31, 50, 6, 0, 1, 63, 180, 60, 0, 1, 127, 602, 390, 24, 0, 1, 255, 1932, 2100, 360, 0, 1, 511, 6050, 10206, 3360, 120, 0, 1, 1023, 18660, 46620, 25200, 2520, 0, 1, 2047, 57002, 204630, 166824, 31920, 720

Offset: 0

Alois P. Heinz, Jan 24 2018

			T(5,1) = 1: 12345.
T(5,2) = 7: 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345.
T(5,3) = 2: 124|3|5, 12|34|5.
T(7,4) = 6: 1246|3|5|7, 124|36|5|7, 124|3|56|7, 126|34|5|7, 12|346|5|7, 12|34|56|7.
T(9,5) = 24: 12468|3|5|7|9, 1246|38|5|7|9, 1246|3|58|7|9, 1246|3|5|78|9, 1248|36|5|7|9, 124|368|5|7|9, 124|36|58|7|9, 124|36|5|78|9, 1248|3|56|7|9, 124|38|56|7|9, 124|3|568|7|9, 124|3|56|78|9, 1268|34|5|7|9, 126|348|5|7|9, 126|34|58|7|9, 126|34|5|78|9, 128|346|5|7|9, 12|3468|5|7|9, 12|346|58|7|9, 12|346|5|78|9, 128|34|56|7|9, 12|348|56|7|9, 12|34|568|7|9, 12|34|56|78|9.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1;
  0, 1,    1;
  0, 1,    3;
  0, 1,    7,     2;
  0, 1,   15,    12;
  0, 1,   31,    50,     6;
  0, 1,   63,   180,    60;
  0, 1,  127,   602,   390,    24;
  0, 1,  255,  1932,  2100,   360;
  0, 1,  511,  6050, 10206,  3360,  120;
  0, 1, 1023, 18660, 46620, 25200, 2520;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Wenqin Cao, Emma Yu Jin, and Zhicong Lin, Enumeration of inversion sequences avoiding triples of relations, Discrete Applied Mathematics (2019); see also author's copy

Columns k=0-11 give (offsets may differ): A000007, A057427, A168604, A028243, A028244, A028245, A032180, A228909, A228910, A228911, A228912, A228913.
Row sums give A229046(n-1) for n>0.
T(2n+1,n+1) gives A000142.
T(2n,n) gives A001710(n+1).
Cf. A000629, A000670, A008277, A028246, A048993, A076726, A110654, A136011, A173018.

b:= proc(n, m, t) option remember; `if`(n=0, x^m, add(
      b(n-1, max(m, j), `if`(j>m, 1, 0)), j=1..m+1-t))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..14);
# second Maple program:
T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), (k-1)!*Stirling2(n-k+1, k)):
seq(seq(T(n, k), k=0..ceil(n/2)), n=0..14);
# third Maple program:
T:= proc(n, k) option remember; `if`(k<2, `if`(n=0 xor k=0, 0, 1),
      `if`(k>ceil(n/2), 0, add((k-j)*T(n-1-j, k-j), j=0..1)))
    end:
seq(seq(T(n, k), k=0..ceil(n/2)), n=0..14);

T[n_, k_] := T[n, k] = If[k < 2, If[Xor[n == 0, k == 0], 0, 1],
     If[k > Ceiling[n/2], 0, Sum[(k-j) T[n-1-j, k-j], {j, 0, 1}]]];
Table[Table[T[n, k], {k, 0, Ceiling[n/2]}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Mar 08 2021, after third Maple program *)

T(n,k) = (k-1)! * Stirling2(n-k+1,k) for k>0, T(n,0) = A000007(n).
T(n,k) = Sum_{j=0..k-1} (-1)^j*C(k-1,j)*(k-j)^(n-k) for k>0, T(n,0) = A000007(n).
T(n,k) = (k-1)! * A136011(n,k) for n, k >= 1.
Sum_{j>=0} T(n+j,j) = A076726(n) = 2*A000670(n) = A000629(n) + A000007(n).

A245602 Triangle read by rows: the negative terms of A163626.

Original entry on oeis.org

-1, -3, -7, -6, -15, -60, -31, -390, -120, -63, -2100, -2520, -127, -10206, -31920, -5040, -255, -46620, -317520, -181440, -511, -204630, -2739240, -3780000, -362880, -1023, -874500, -21538440, -59875200, -19958400, -2047

Offset: 0

Paul Curtz, Dec 17 2014

These numbers a(n) are the companion of A249163(n).
Consider the Worpitzky fractions A163626(n)/A002260(n) yielding the second Bernoulli numbers A164555(n)/A027642(n):
1,
1, -1/2,
1, -3/2,  +2/3,
1, -7/2, +12/3, -6/4,
etc.
From the second row on, the sum of the numerators is 0.
The absolute values of every row of the numerators triangle A163626 are 1, 2, 6, 26, ... = A000629(n).
a(n) triangle is shifted. It starts from second row and second column of triangle above.
-1,
-3,
-7,       -6,
-15,     -60,
-31,    -390,    -120,
-63,   -2100,   -2520,
-127, -10206,  -31920,   -5040,
-255, -46620, -317520, -181440,
etc.
Sum of successive rows: -1, -3, -13, -75, ... = -A000670(n+1).
Successive columns: A000225, A028244, from the Stirling numbers of second kind S(n,2), S(n,4), S(n,6), S(n,8), S(n,10), ... . See A000770, A032180, A049434, A228910, A049435, A228912, A008277.
Thanks to Jean-François Alcover.

Cf. A000225, A000629, A000670, A000770, A002260, A008277, A009445, A027642, A028244, A032180, A049434, A049435, A163626, A164555, A228910, A228912, A249163.

Select[ Table[ (-1)^k*k!*StirlingS2[n+1, k+1], {n, 0, 12}, {k, 0, n}] // Flatten, Negative] (* Jean-François Alcover, Dec 26 2014 *)

A285867 Triangle T(n, k) read by rows: T(n, k) = S2(n, k)k! + S2(n, k-1)(k-1)! with the Stirling2 triangle S2 = A048993.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 7, 12, 0, 1, 15, 50, 60, 0, 1, 31, 180, 390, 360, 0, 1, 63, 602, 2100, 3360, 2520, 0, 1, 127, 1932, 10206, 25200, 31920, 20160, 0, 1, 255, 6050, 46620, 166824, 317520, 332640, 181440, 0, 1, 511, 18660, 204630, 1020600, 2739240, 4233600, 3780000, 1814400, 0, 1, 1023, 57002, 874500, 5921520, 21538440, 46070640, 59875200, 46569600, 19958400

Offset: 0

Wolfdieter Lang, May 03 2017

This triangle T(n, k) appears in the e.g.f. of the sum of powers SP(n, m) = Sum_{j=0..m} j^n, n >= 0, m >= 0 with 0^0:=1 as ESP(n, t) = exp(t)*(Sum_{k=0..n} T(n, k)*t^k/k! + t^(n+1)/(n+1)), n >= 0.
The sub-triangle T(n, k) for 1 <= k <=n, see A028246(n+1,k) (diagonal not needed).
For S2(n, m)*m! see A131689.
The columns (starting sometimes with n=k) are A000007, A000012, A000225, A028243(n-1), A028244(n-1), A028245(n-1), A032180(n-1), A228909, A228910, A228911, A228912, A228913. See below for the e.g.f.s and o.g.f.s.
The row sums are 1 for n=1 and A000629(n) - n! for n >= 1, See A285868.

			The triangle T(n, k) begins:
n\k 0  1    2     3      4       5        6        7        8        9  ...
0:  1
1:  0  1
2:  0  1    3
3:  0  1    7    12
4:  0  1   15    50     60
5:  0  1   31   180    390     360
6:  0  1   63   602   2100    3360     2520
7:  0  1  127  1932  10206   25200    31920    20160
8:  0  1  255  6050  46620  166824   317520   332640   181440
9:  0  1  511 18660 204630 1020600  2739240  4233600  3780000  1814400
...

Cf. A000629, A028246, A048993, A131689, A285868.

Table[If[k == 0, Boole[n == 0], StirlingS2[n, k] k! + StirlingS2[n, k - 1] (k - 1)!], {n, 0, 10}, {k, 0, n}] (* Michael De Vlieger, May 08 2017 *)

T(n, k) =  A131689(n, k) + A131689(n, k-1), 0 <= k <= n, with A131689(n, -1) = 0.
T(0, 0) = 1 and T(n, k) = Stirling2(n+1, k)*(k-1)! for n >= k >= 1. For Stirling2 see A048993. Stirling2(n, k)*(k-1)! = A028246(n, k) for n >= k >= 1.
Recurrence: T(0, 0) = 1, T(n, n) = (n+1)!/2, T(n, -1) = 0, T(n, k) = 0 if n < k, and T(n, k) = (k-1)*T(n-1, k-1) + k*T(n-1, k), for n > k >= 0.
E.g.f. for column k=0 is 1, and for  k >= 1: Sum_{j=1..k}((-1)^(k-j) * binomial(k-1, j-1) * exp(j*x)) - x^(k-1).
O.g.f. for column k = 0 is 1, and for  k >= 1: ((k-1)!*x^(k-1) / Product_{j=1..k} (1-j*x)) - (k-1)!*x^(k-1).

A130567 Expansion of x(2 - 7x + 2x^2)/((1-x)(1-4x)(1-2*x)).

Original entry on oeis.org

2, 7, 23, 79, 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, 16785407, 67125247, 268468223, 1073807359, 4295098367, 17180131327, 68720001023, 274878955519, 1099513724927, 4398050705407, 17592194433023, 70368760954879

Offset: 1

Roger L. Bagula, Aug 09 2007

Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A093069, A099393, A028244.

f[n_Integer?Positive] := f[n] = 2^(2*n - 1) + 2*f[n - 1] + 1; f[0] = 2; Table[f[n], {n, 0, 30}]
CoefficientList[Series[x*(2-7x+2x^2)/((1-x)(1-4x)(1-2x)),{x,0,30}],x] (* Harvey P. Dale, Sep 07 2015 *)

a(n) = 2^(2*n - 1) + 2*a(n - 1) + 1.
From R. J. Mathar, Jun 13 2008: (Start)
O.g.f.: x*(2 - 7*x + 2*x^2)/((1-x)*(1-4*x)*(1-2*x)).
a(n) = A093069(n-2), n>1. (End)

New name from Joerg Arndt, Feb 08 2015

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A285867 Triangle T(n, k) read by rows: T(n, k) = S2(n, k)*k! + S2(n, k-1)*(k-1)! with the Stirling2 triangle S2 = A048993.

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A130567 Expansion of x*(2 - 7*x + 2*x^2)/((1-x)*(1-4*x)*(1-2*x)).

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A285867 Triangle T(n, k) read by rows: T(n, k) = S2(n, k)k! + S2(n, k-1)(k-1)! with the Stirling2 triangle S2 = A048993.

A130567 Expansion of x(2 - 7x + 2x^2)/((1-x)(1-4x)(1-2*x)).