A028243 -id:A028243 - cp's OEIS frontend

A228910 a(n) = 8^n - 77^n + 216^n - 355^n + 354^n - 213^n + 72^n - 1.

0, 0, 0, 0, 0, 0, 0, 5040, 181440, 3780000, 59875200, 801496080, 9574044480, 105398092800, 1091804313600, 10794490827120, 102896614941120, 952741767650400, 8617145057539200, 76461500619902160, 667855517349303360, 5757691363157764800, 49099453300298016000, 414884142077935345200

Offset: 0

Richard V. Scholtz, III, Sep 07 2013

Calculates the eighth column of coefficients with respect to the derivatives, d^n/dx^n(y), of the logistic equation when written as y=1/[1+exp(-x)].

Seiichi Manyama, Table of n, a(n) for n = 0..1107
Index entries for linear recurrences with constant coefficients, signature (36,-546,4536,-22449,67284,-118124,109584,-40320).

Cf. A008277, A049434.
The eighth column of results of A163626.
Cf. A000225, A028243, A028244, A028245, A032180, A228909.

[8^(n)-7*7^(n)+21*6^(n)-35*5^(n)+35*4^(n)-21*3^(n)+7*2^(n)-1: n in [0..30]]; // G. C. Greubel, Nov 19 2017

Derivative[0][y][x] = y[x]; Derivative[1][y][x] = y[x]*(1 - y[x]); Derivative[n_][y][x] := Derivative[n][y][x] = D[Derivative[n - 1][y][x], x]; row[n_] := CoefficientList[ Derivative[n][y][x], y[x]] // Rest; Join[{0, 0, 0, 0, 0, 0, 0}, Table[ -row[n], {n, 7, 23}] [[All, 8]]] (* Jean-François Alcover, Dec 16 2014 *)
Table[7!*StirlingS2[n + 1, 8], {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)
Table[8^n - 7*7^n + 21*6^n - 35*5^n + 35*4^n - 21*3^n + 7*2^n - 1, {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)
CoefficientList[Series[5040*x^7 / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 16 2014 after Colin Barker *)

a(n)=8^(n)-7*7^(n)+21*6^(n)-35*5^(n)+35*4^(n)-21*3^(n)+7*2^(n)-1.

for(n=0,30, print1(5040*stirling(n+1,8,2), ", ")) \\ G. C. Greubel, Nov 19 2017

a(n) = 5040 * S2(n+1,8), n>=0.
G.f.: 5040*x^7 / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)). - Colin Barker, Sep 20 2013
E.g.f.: Sum_{k=1..8} (-1)^(8-k)*binomial(8-1,k-1)*exp(k*x). - Wolfdieter Lang, May 03 2017

Offset corrected by Vaclav Kotesovec, Dec 16 2014

A028245 a(n) = 5^(n-1) - 44^(n-1) + 63^(n-1) - 4*2^(n-1) + 1 (essentially Stirling numbers of second kind).

Original entry on oeis.org

0, 0, 0, 0, 24, 360, 3360, 25200, 166824, 1020600, 5921520, 33105600, 180204024, 961800840, 5058406080, 26308573200, 135666039624, 694994293080, 3542142833040, 17980946172000, 90990301641624

Offset: 1

N. J. A. Sloane, Doug McKenzie mckfam4(AT)aol.com

For n>=2, a(n) is equal to the number of functions f: {1,2,...,n-1}->{1,2,3,4,5} such that Im(f) contains 4 fixed elements. - Aleksandar M. Janjic and Milan Janjic, Mar 08 2007

Seiichi Manyama, Table of n, a(n) for n = 1..1431
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (15,-85,225,-274,120).

Cf. A000481, A008277, A163626, A000225, A028243, A028244.

[5^(n-1) - 4*4^(n-1) + 6*3^(n-1) - 4*2^(n-1) + 1: n in [1..30]]; // G. C. Greubel, Nov 19 2017

24StirlingS2[Range[30],5] (* Harvey P. Dale, Jun 18 2013 *)
Table[5^(n - 1) - 4*4^(n - 1) + 6*3^(n - 1) - 4*2^(n - 1) + 1, {n, 21}] (* or *)
Rest@ CoefficientList[Series[-24 x^5/((x - 1) (4 x - 1) (3 x - 1) (2 x - 1) (5 x - 1)), {x, 0, 21}], x] (* Michael De Vlieger, Sep 24 2016 *)

for(n=1,30, print1(24*stirling(n,5,2), ", ")) \\ G. C. Greubel, Nov 19 2017

a(n) = 24*S(n, 5) = 24*A000481(n). - Emeric Deutsch, May 02 2004
G.f.: -24*x^5/((x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009; checked and corrected by R. J. Mathar, Sep 16 2009
E.g.f.: (Sum_{k=0..5} (-1)^(5-k)*binomial(5,k)*exp(k*x))/5. with a(0) = 0. - Wolfdieter Lang, May 03 2017

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

A053154 Number of 2-element intersecting families (with not necessarily distinct sets) of an n-element set.

Original entry on oeis.org

0, 1, 5, 22, 95, 406, 1715, 7162, 29615, 121486, 495275, 2009602, 8124935, 32761366, 131834435, 529712842, 2125993055, 8525430046, 34166159195, 136858084882, 548012945975, 2193794127526, 8780404589555, 35137304693722

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Feb 28 2000

Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are intersecting but for which x is not a subset of y and y is not a subset of x, or 2) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x. - Ross La Haye, Jan 11 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, in Russian, Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, English translation, in Discrete Mathematics and Applications, 9, (1999), no. 6.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A036239, A083324.

Cf. A000225, A032263, A028243.

[(4^n-3^n+2^n-1)/2: n in [0..30]]; // Vincenzo Librandi, Oct 06 2017

Table[(4^n-3^n+2^n-1)/2, {n,1,30}] (* Clark Kimberling, Mar 12 2012 *)
CoefficientList[Series[x (1 - 5 x + 7 x^2) / ((1 - x) (1 - 4 x) (1 - 3 x) (1 - 2 x)), {x, 0, 33}], x] (* Vincenzo Librandi, Oct 06 2017 *)

a(n) = (4^n-3^n+2^n-1)/2; \\ Michel Marcus, Nov 30 2015

a(n) = (A083324(n) - 1)/2.
a(n) = (4^n - 3^n + 2^n - 1)/2.
a(n) = 3*StirlingS2(n+1,4) + 2*StirlingS2(n+1,3) + StirlingS2(n+1,2). - Ross La Haye, Jan 11 2008
From Wolfdieter Lang, Oct 28 2011 (Start)
E.g.f.: Sum_{j=1..4} ((-1)^j*exp(j*x))/2  = exp(x)*(exp(4*x)-1)/(exp(x)+1)/2.
O.g.f.: Sum_{j=1..4} (((-1)^j)/(1-j*x))/2 = x*(1-5*x+7*x^2)/product(1-j*x,j=1..4). See A196847.
(End)
G.f.: x*(1-5*x+7*x^2)/((1-x)*(1-4*x)*(1-3*x)*(1-2*x)). - Vincenzo Librandi, Oct 06 2017

A134169 a(n) = 2^(n-1)*(2^n - 1) + 1.

Original entry on oeis.org

1, 2, 7, 29, 121, 497, 2017, 8129, 32641, 130817, 523777, 2096129, 8386561, 33550337, 134209537, 536854529, 2147450881, 8589869057, 34359607297, 137438691329, 549755289601, 2199022206977, 8796090925057, 35184367894529

Offset: 0

Ross La Haye, Jan 12 2008

Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either (Case 0) x and y are disjoint, x is not a subset of y, and y is not a subset of x; or (Case 1) x and y are intersecting, but x is not a subset of y, and y is not a subset of x; or (Case 2) x and y are intersecting, and either x is a proper subset of y, or y is a proper subset of x; or (Case 3) x = y.

			a(2) = 7 because for P(A) = {{},{1},{2},{1,2}} we have for Case 0 {{1},{2}}; we have for Case 2 {{1},{1,2}}, {{2},{1,2}}; and we have for Case 3 {{},{}}, {{1},{1}}, {{2},{2}}, {{1,2},{1,2}}. There are 0 {x,y} of P(A) in this example that fall under Case 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A000392, A032263, A028243, A000079, A006516.

Table[EulerE[2,2^n],{n,0,60}]/2+1 (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)
LinearRecurrence[{7,-14,8},{1,2,7},30] (* Harvey P. Dale, Mar 12 2013 *)

a(n) = 2^(n-1)*(2^n - 1) + 1.
a(n) = StirlingS2(2^n,2^n - 1) + 1 = C(2^n,2) + 1 = A006516(n) + 1.
From R. J. Mathar, Feb 15 2010: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3).
G.f.: (1 - 5*x + 7*x^2)/((1-x) * (2*x-1) * (4*x-1)). (End)

More terms from Vladimir Joseph Stephan Orlovsky, Nov 03 2009

Edited by N. J. A. Sloane, Jan 25 2015 at the suggestion of Michel Marcus

A053156 Number of 2-element intersecting families (with not necessarily distinct sets) whose union is an n-element set.

Original entry on oeis.org

1, 3, 10, 33, 106, 333, 1030, 3153, 9586, 29013, 87550, 263673, 793066, 2383293, 7158070, 21490593, 64504546, 193579173, 580868590, 1742867913, 5229128026, 15688432653, 47067395110, 141206379633, 423627527506, 1270899359733

Offset: 1

Vladeta Jovovic and Goran Kilibarda, Feb 28 2000

Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 2) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x, or 3) x = y. - Ross La Haye, Jan 12 2008
From Paul Barry, Apr 27 2003: (Start)
With offset 0, this is a(n) = (3*3^n - 2*2^n + 1)/2.
G.f. (1-3*x+3*x^2)/((1-x)*(1-2*x)*(1-3*x)).
E.g.f. (3*exp(3*x) - 2*exp(2*x) + exp(x))/2.
Binomial transform of A083329.
Second binomial transform of A040001. (End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, in Russian, Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, English translation, in Discrete Mathematics and Applications, 9, (1999), no. 6.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A000225, A000392, A028243, A000079.
Cf. A036239.
Column k=2 of A288638.
Third column of A294201.

[(3^n-2^n+1)/2: n in [1..30]]; // G. C. Greubel, Oct 06 2017

A053156:=n->(3^n - 2^n + 1)/2: seq(A053156(n), n=1..40); # Wesley Ivan Hurt, Oct 06 2017

LinearRecurrence[{6,-11,6}, {1, 3, 10}, 50] (* or *) Table[(3^n - 2^n + 1)/2, {n,1,50}] (* G. C. Greubel, Oct 06 2017 *)

a(n) = (3^n-2^n+1)/2; \\ Michel Marcus, Nov 30 2015

a(n) = (3^n - 2^n + 1)/2.
a(n) = StirlingS2(n+2,3) + StirlingS2(n+1,2) + 1. - Ross La Haye, Jan 12 2008
From Colin Barker, Jul 29 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n > 3.
G.f.: x*(1-3*x+3*x^2)/((1-x)*(1-2*x)*(1-3*x)). (End)

A053440 Number of k-simplices in the first derived complex of the standard triangulation of an n-simplex. Equivalently, T(n,k) is the number of ascending chains of length k+1 of nonempty subsets of the set {1, 2, ..., n+1}.

Original entry on oeis.org

1, 3, 2, 7, 12, 6, 15, 50, 60, 24, 31, 180, 390, 360, 120, 63, 602, 2100, 3360, 2520, 720, 127, 1932, 10206, 25200, 31920, 20160, 5040, 255, 6050, 46620, 166824, 317520, 332640, 181440, 40320, 511, 18660, 204630, 1020600, 2739240, 4233600, 3780000, 1814400, 362880

Offset: 0

Rob Arthan, Jan 12 2000

T(n,k) is the number of length k+1 sequences of nonempty mutually disjoint subsets of {1,2,...,n+1}. The e.g.f. for the column corresponding to k is exp(x)*(exp(x)-1)^(k+1). - Geoffrey Critzer, Dec 20 2011

			T(2,1) = 12 because there are 12 such length 2 sequences of subsets of {1,2,3}: ({1},{2}), ({1},{3}), ({2},{3}), ({1},{2,3}), ({2},{1,3}), ({3},{1,2}) with two orderings for each. - _Geoffrey Critzer_, Dec 20 2011
Triangle begins:
   1
   3      2
   7     12      6
  15     50     60     24
  31    180    390    360    120

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
F. Brenti and V. Welker, f-vectors of barycentric subdivisions Math. Z., 259(4), 849-865, 2008.
Wikipedia, Barycentric subdivision

Other versions are A028246, A142071.
Columns k=0..1 are A000225(n+1), A028243(n+2).
Cf. A000142 (main diagonal), A002050 (row sums), A019538.

a := (n, k) -> (k+1)!*Stirling2(n+2, k+2):
seq(print(seq(a(n, k), k = 0..n)), n = 0..10);

nn = 5; a = Exp[ x] - 1 ; f[list_] := Select[list, # > 0 &];Map[f, Transpose[Table[Drop[Range[0, nn]!CoefficientList[Series[a^k  Exp[x], {x, 0, nn}],x], 1], {k, 1, 5}]]] // Grid (* Geoffrey Critzer, Dec 20 2011 *)
Table[(k+1)!*StirlingS2[n+2,k+2], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 19 2017 *)

for(n=0,10, for(k=0,n, print1((k+1)!*stirling(n+2,k+2,2), ", "))) \\ G. C. Greubel, Nov 19 2017

T(0,k) = delta(0,k), T(n,k) = delta(0,k) + (k+1)(T(n-1,k-1) + (k+2)T(n-1,k)).
E.g.f.: exp(x)*(exp(x)-1)/(1-y*(exp(x)-1)). - Vladeta Jovovic, Apr 13 2003
T(n,k) = Sum_{i = 0..n} binomial(n+1,i+1)*(k+1)!*Stirling2(i+1,k+1) = (k+1)!*Stirling2(n+2,k+2) (Brenti and Welker). - Peter Bala, Jul 12 2014
T(n,k) = (k+1)!*Stirling2(n+2, k+2). - G. C. Greubel, Nov 19 2017

More terms from James Sellers, Jan 14 2000

A245019 Number of ordered n-tuples of positive integers, whose minimum is 0 and maximum is 4.

Original entry on oeis.org

0, 2, 24, 194, 1320, 8162, 47544, 266114, 1448520, 7727522, 40616664, 211117634, 1088079720, 5571427682, 28384443384, 144041002754, 728708854920, 3677645732642, 18524892775704, 93171895169474, 468051525534120, 2349032799986402

Offset: 1

Ovidiu Bagdasar, Sep 17 2014

For given k and n positive integers, let T(k,n) represent the number of n-tuples of positive integers, whose minimum is zero and maximum is k. In this notation, the sequence corresponds to a(n) = T(4,n).

			For n=2 the a(2)=2 solutions are (0,4) and (4,0).

O. Bagdasar, On Some Functions Involving the lcm and gcd of Integer Tuples, Scientific publications of the state university of Novi Pazar, Ser. A: Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91-100.
Index entries for linear recurrences with constant coefficients, signature (12,-47,60).

T(1,n) gives A000918, T(2,n-1) gives A028243, T(n,3) gives A008588, T(n,4) gives A005914.

concat(0, Vec(-2*x^2/((3*x-1)*(4*x-1)*(5*x-1)) + O(x^100))) \\ Colin Barker, Sep 17 2014

a(n) = 5^n-2*4^n+3^n.
a(n) = 12*a(n-1)-47*a(n-2)+60*a(n-3). G.f.: -2*x^2 / ((3*x-1)*(4*x-1)*(5*x-1)). - Colin Barker, Sep 17 2014
a(n) = 2*A016753(n-2) for n>1. - Colin Barker, Sep 17 2014

A249163 Triangle read by rows: the positive terms of A163626.

Original entry on oeis.org

1, 1, 1, 2, 1, 12, 1, 50, 24, 1, 180, 360, 1, 602, 3360, 720, 1, 1932, 25200, 20160, 1, 6050, 166824, 332640, 40320, 1, 18660, 1020600, 4233600, 1814400, 1, 57002, 5921520, 46070640, 46569600, 3628800, 1, 173052, 33105600, 451725120, 898128000, 239500800

Offset: 0

Paul Curtz, Dec 15 2014

We have two possibilities: with or without 0's.
Without 0's:
1,
1,
1,   2,
1,  12,
1,  50,  24,
1, 180, 360,
etc.
Sum of every row: A000670(n).
First two terms of successive columns: 1, 1, 2, 12, 24, 360, ... = A211374.
With 0's:
1,   0,    0,   0,
1,   0,    0,   0,
1,   2,    0,   0,
1,  12,    0,   0,
1,  50,   24,   0,
1, 180,  360,   0,
1, 602, 3360, 720,
etc.
The columns are essentially A000012, A028243, A028246, A228909, A228911, A228913, from Stirling numbers of the second kind S(n,3), S(n,5), S(n,7), S(n,9), S(n,11), ... .

Cf. A163626, A000670, A211374; also A000012, A000392, A000481, A000771, A049447, A028243, A028246, A091137, A228909, A163626, A228911, A228913 and Worpitzky numbers for the second Bernoulli numbers A164555(n)/A027642(n).

Derivative[0][y][x] = y[x]; Derivative[1][y][x] = y[x]*(1 - y[x]); Derivative[n_][y][x] := Derivative[n][y][x] = D[Derivative[n - 1][y][x], x]; row[n_] := CoefficientList[Derivative[n][y][x], y[x]] // Rest; Table[ Select[row[n], Positive] , {n, 0, 12}] // Flatten
(* or, simply: *) Table[(-1)^k*k!*StirlingS2[n+1, k+1], {n, 0, 12}, {k, 0, n}] // Flatten // Select[#, Positive]& (* Jean-François Alcover, Dec 16 2014 *)

A260217 Number of base-3 n-digit pandigital numbers.

Original entry on oeis.org

0, 0, 4, 24, 100, 360, 1204, 3864, 12100, 37320, 114004, 346104, 1046500, 3155880, 9500404, 28566744, 85831300, 257756040, 773792404, 2322425784, 6969374500, 20912317800, 62745342004, 188252803224, 564791964100, 1694443001160, 5083463221204, 15250658099064

Offset: 1

Jon E. Schoenfield, Jul 19 2015

From Manfred Boergens, Aug 02 2023: (Start)
a(n+1) is the number of pairs (A,B) where A and B are nonempty subsets of {1,2,...,n} and one of these subsets is a proper subset of the other.
If "proper" is omitted, see A091344.
If empty subsets are included, see A027649 (all subsets) and A056182 (proper subsets). (End)

			a(3)=4 because, in base 3, there are four 3-digit pandigital numbers (11=102_3, 15=120_3, 19=201_3, and 21=210_3).
a(4)=24 because, in base 3, there are 24 4-digit pandigital numbers (1002_3, 1012_3, 1020_3, 1021_3, 1022_3, 1102_3, 1120_3, 1200_3, 1201_3, 1202_3, 1210_3, 1220_3, 2001_3, 2010_3, 2011_3, 2012_3, 2021_3, 2100_3, 2101_3, 2102_3, 2110_3, 2120_3, 2201_3, and 2210_3).

Svenja Huntemann, Values, Temperatures, and Enumeration of Placement Games, Slides, Alberta-Montana Combinatorics and Algorithms Day, Banff, Canada, 23-25 June 2023. See p. 105/109.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A000392, A027649, A028243, A056182, A091344, A171102.

[2*3^(n-1) - 2^(n+1) + 2: n in [1..30]]; // Vincenzo Librandi, Jul 20 2015

Table[2 3^(n - 1) - 2^(n + 1) + 2, {n, 30}] (* Vincenzo Librandi, Jul 20 2015 *)

a(n) = 2*A028243(n) = 2*3^(n-1) - 2^(n+1) + 2.
a(n) = 4*A000392(n).
G.f.: 4*x^3/((1-x)*(1-2*x)*(1-3*x)).
E.g.f.: 2/3*((exp(x)-1)^3).

cp's OEIS Frontend

A228910 a(n) = 8^n - 7*7^n + 21*6^n - 35*5^n + 35*4^n - 21*3^n + 7*2^n - 1.

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A028245 a(n) = 5^(n-1) - 4*4^(n-1) + 6*3^(n-1) - 4*2^(n-1) + 1 (essentially Stirling numbers of second kind).

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

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A053154 Number of 2-element intersecting families (with not necessarily distinct sets) of an n-element set.

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A134169 a(n) = 2^(n-1)*(2^n - 1) + 1.

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A053156 Number of 2-element intersecting families (with not necessarily distinct sets) whose union is an n-element set.

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A053440 Number of k-simplices in the first derived complex of the standard triangulation of an n-simplex. Equivalently, T(n,k) is the number of ascending chains of length k+1 of nonempty subsets of the set {1, 2, ..., n+1}.

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A228910 a(n) = 8^n - 77^n + 216^n - 355^n + 354^n - 213^n + 72^n - 1.

A028245 a(n) = 5^(n-1) - 44^(n-1) + 63^(n-1) - 4*2^(n-1) + 1 (essentially Stirling numbers of second kind).