A000453 -id:A000453 - cp's OEIS frontend

A305862 a(n) = 3844^n - 5763^n + 220*2^n - 14.

14, 234, 1826, 10770, 55154, 260274, 1167026, 5059890, 21442994, 89438514, 368866226, 1509026610, 6137242034, 24853275954, 100327829426, 404059098930, 1624486948274, 6522713868594, 26165182536626, 104883769004850, 420204307937714, 1682825158192434, 6737324873467826

Offset: 0

Vincenzo Librandi, Jun 15 2018

From Bruno Berselli, Jun 15 2018: (Start)
a(0) = 2*7 and a(40) = 2*232110255958477539427146457 are semiprimes. For which values of n > 40 is a(n) semiprime?
For odd n, a(n) is divisible by 2*3.
For n == 3 (mod 4), a(n) is divisible by 2*3*5.
For n == 0 or 5 (mod 6), a(n) is divisible by 2*7.
For n == 2 or 4 (mod 5), a(n) is divisible by 2*11.
For n == 1 or 11 (mod 12), a(n) is divisible by 2*3*13.
For n == 15 (mod 16), a(n) is divisible by 2*3*5*17^2, etc.
If a(n) is divisible by 37 then it is also divisible by 3*5*7*13*19*73. (End)

Takao Komatsu, On poly-Euler numbers of the second kind, arXiv:1806.05515 [math.NT], 2018, page 11 (Lemma 3.4).
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A000453, A000918, A305861.

[384*4^n-576*3^n+220*2^n-14: n in [0..30]];

Table[384 4^n - 576 3^n + 220 2^n - 14, {n, 0, 30}]

a(n) = 384*4^n - 576*3^n + 220*2^n - 14; \\ Michel Marcus, Jul 03 2018

G.f.: 2*(7 + 47*x - 12*x^2)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)).
a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4).
a(n) = 14*A000453(n+4) + 94*A000453(n+3) - 24*A000453(n+2) for n>1. - Bruno Berselli, Jun 15 2018

A327611 Number of length n reversible string structures that are not palindromic using exactly four different colors.

Original entry on oeis.org

0, 0, 0, 1, 6, 37, 182, 876, 3920, 17175, 73030, 306296, 1266916, 5198207, 21180642, 85909216, 347179440, 1399443775, 5629876910, 22616222616, 90754709276, 363889980927, 1458171985402, 5840531023856, 23385647663560, 93613189390175, 374664530448390

Offset: 1

Andrew Howroyd, Sep 18 2019

Andrew Howroyd, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (8,-10,-60,145,100,-470,120,456,-288).

Column k=4 of A309748.

Cf. A000453, A056328, A284949, A327610.

concat([0,0,0], Vec((1 - 2*x - x^2 + 6*x^3 + 5*x^4 - 18*x^5)/((1 - x)*(1 - 2*x)*(1 + 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 2*x^2)*(1 - 3*x^2)) + O(x^30))) \\ Andrew Howroyd, Sep 18 2019

a(n) = A056328(n) - A000453(ceiling(n/2), 4).
a(n) = 8*a(n-1) - 10*a(n-2) - 60*a(n-3) + 145*a(n-4) + 100*a(n-5) - 470*a(n-6) + 120*a(n-7) + 456*a(n-8) - 288*a(n-9) for n > 9.
G.f.: x^4*(1 - 2*x - x^2 + 6*x^3 + 5*x^4 - 18*x^5)/((1 - x)*(1 - 2*x)*(1 + 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 2*x^2)*(1 - 3*x^2)).

A346954 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^4 / 4! ).

Original entry on oeis.org

1, 10, 65, 350, 1736, 9030, 60355, 561550, 6188996, 69919850, 781211795, 8854058850, 106994019406, 1433756147470, 21287253921635, 339206526695750, 5630710652048216, 96341917117951890, 1708973354556320875, 31787279786739738250, 623964823224788294426

Offset: 4

Ilya Gutkovskiy, Aug 08 2021

nonn

Cf. A000453, A000629, A003704, A327505, A346390, A346895, A346955.

nmax = 24; CoefficientList[Series[-Log[1 - (Exp[x] - 1)^4/4!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 4] &
a[n_] := a[n] = StirlingS2[n, 4] + (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 4] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 4, 24}]

a(n) = Stirling2(n,4) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,4) * k * a(k).
a(n) ~ (n-1)! / (log(2^(3/4)*3^(1/4) + 1))^n. - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/4)} (4*k)! * Stirling2(n,4*k)/(k * 24^k). - Seiichi Manyama, Jan 23 2025

A346843 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^4 / 4!.

Original entry on oeis.org

1, 15, 155, 1400, 11991, 101031, 853315, 7300260, 63641006, 567304452, 5181338526, 48538121450, 466611951261, 4603782469653, 46613101232933, 484188586821376, 5157850655391981, 56321812548867229, 630125374420189131, 7219368394888423554, 84658119388335562972

Offset: 4

Ilya Gutkovskiy, Aug 05 2021

nonn

Cf. A000110, A000332, A000453, A003128, A005493, A049020, A346842, A346844.

b:= proc(n, m) option remember;
     `if`(n=0, binomial(m, 4), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=4..24);  # Alois P. Heinz, Aug 05 2021

nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] (Exp[x] - 1)^4/4!, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 4] &
Table[Sum[StirlingS2[n, k] Binomial[k, 4], {k, 0, n}], {n, 4, 24}]
Table[Sum[Binomial[n, k] StirlingS2[k, 4] BellB[n - k], {k, 0, n}], {n, 4, 24}]
Table[(BellB[n] - 24*BellB[n+1] + 29*BellB[n+2] - 10*BellB[n+3] + BellB[n+4])/24, {n, 4, 24}] (* Vaclav Kotesovec, Aug 06 2021 *)
With[{nn=30},Drop[CoefficientList[Series[(Exp[Exp[x]-1](Exp[x]-1)^4)/4!,{x,0,nn}],x] Range[0,nn]!,4]] (* Harvey P. Dale, Oct 03 2024 *)

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(x)-1)*(exp(x)-1)^4/4!)) \\ Michel Marcus, Aug 06 2021

a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(k,4).
a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(k,4) * Bell(n-k).
a(n) = (Bell(n) - 24*Bell(n+1) + 29*Bell(n+2) - 10*Bell(n+3) + Bell(n+4))/24. - Vaclav Kotesovec, Aug 06 2021

A383842 Expansion of 1/((1-x) * (1-2x) (1-3x) (1-4*x))^2.

Original entry on oeis.org

1, 20, 230, 2000, 14627, 95060, 567240, 3174400, 16904053, 86549620, 429352330, 2075659600, 9822847079, 45665147700, 209129160300, 945597624000, 4229196800505, 18738054705300, 82347219011950, 359322115058000, 1558151553849131, 6719660438870420, 28838298857544080

Offset: 0

Seiichi Manyama, May 12 2025

Index entries for linear recurrences with constant coefficients, signature (20,-170,800,-2273,3980,-4180,2400,-576).

Column k=4 of A383843.

Cf. A000453.

a(n) = sum(k=0, n, stirling(k+4, 4, 2)*stirling(n-k+4, 4, 2));

a(n) = 20*a(n-1) - 170*a(n-2) + 800*a(n-3) - 2273*a(n-4) + 3980*a(n-5) - 4180*a(n-6) + 2400*a(n-7) - 576*a(n-8).

a(n) = Sum_{k=0..n} Stirling2(k+4,4) * Stirling2(n-k+4,4).

A384988 a(n) = Stirling2(n,2)^2 + Stirling2(n,3).

Original entry on oeis.org

0, 1, 10, 55, 250, 1051, 4270, 17095, 68050, 270451, 1075030, 4276735, 17030650, 67881451, 270777790, 1080817975, 4316294050, 17244046051, 68912400550, 275457464815, 1101251874250, 4403270396251, 17607863991310, 70415790601255, 281616141147250, 1126323450484051

Offset: 1

Julian Allagan, Jun 14 2025

Also, one third of the number of proper vertex colorings of the n-complete tripartite graph using exactly 5 interchangeable colors.

The complete 3-partite graph K(n,n,n) has 3n vertices partitioned into three sets of size n each, with edges between every pair of vertices from different sets. 3*a(n) = 0 for n < 2 because we need at least 2 vertices per partition to create 5 nonempty independent sets.

			3*a(2) = 3 because K(2,2,2) can be partitioned into 5 nonempty independent sets in exactly 3 ways.

Vincenzo Librandi, Table of n, a(n) for n = 1..500
Richard P. Stanley, Enumerative Combinatorics, Cambridge University Press.
Eric Weisstein's World of Mathematics, Complete Multipartite Graph.
Eric Weisstein's World of Mathematics, Stirling Number of the Second Kind.
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A000392, A000453, A060867, A385432.

[(6 - 3*2^(n+1) + 2*3^(n-1) + 4^n)/4: n in [1..30]]; // Vincenzo Librandi, Jul 24 2025

Table[(StirlingS2[n, 3] + StirlingS2[n, 2]^2), {n, 1, 20}]

3*a(n) = 2^(2*n - 2) + (1/2)*3^(n - 1) - 3*2^(n - 1) + 3/2 for n >= 1.
G.f.: 1/(4*(1 - 4*x)) + 1/(6*(1 - 3*x)) - 3/(2*(1 - 2*x)) + 3/(2*(1 - x)).
a(n) = A385432(n, 5) / 3 = A060867(n-1) + A000392(n).
From Stefano Spezia, Jun 14 2025: (Start)
a(n) = (6 - 3*2^(n+1) + 2*3^(n-1) + 4^n)/4.
E.g.f.: (exp(x) - 1)^2*(3*exp(2*x) + 8*exp(x) - 5)/12. (End)
a(n) = A000453(n+2) -10*A000453(n). - R. J. Mathar, Jul 20 2025

A385312 a(n) is the number of ternary strings of length n with at least one 0, at least two 1's and at least three 2's.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 60, 455, 2268, 9366, 34800, 121077, 403392, 1304732, 4133220, 12900771, 39837684, 122064930, 371891592, 1128317489, 3412864056, 10299925992, 31033986588, 93394501983, 280818931020, 843832511150, 2534467085280, 7609793357805, 22843103816688, 68558705110836

Offset: 0

Enrique Navarrete, Jun 25 2025

			a(6) = 60 since the strings are the 60 permutations of 011222.
a(7) = 455 since the strings are the 210 permutations of 0011222, the 140 permutations of 0111222 and the 105 permutations of 0112222.

Index entries for linear recurrences with constant coefficients, signature (13,-72,222,-417,489,-350,140,-24).

Cf. A000453, A000478, A358341.

LinearRecurrence[{13, -72, 222, -417, 489, -350, 140, -24}, {0, 0, 0, 0, 0, 0, 60, 455, 2268, 9366, 34800, 121077}, 30] (* Amiram Eldar, Jun 28 2025 *)

a(n) = 3^n - 2^(n-2)*(binomial(n,2) + 4*n + 12) + 3*binomial(n,3) + 4*binomial(n,2) + 4*n + 3 for n>=4.
E.g.f.: (exp(x) - x^2/2 - x - 1)*(exp(x) - x - 1)*(exp(x) - 1).
G.f.: x^6*(60 - 325*x +673*x^2 - 678*x^3 + 348*x^4 - 72*x^5)/((1 - x)^4*(1 - 2*x)^3*(1 - 3*x)). - Stefano Spezia, Jun 25 2025

A056473 Number of palindromic structures using exactly four different symbols.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 10, 10, 65, 65, 350, 350, 1701, 1701, 7770, 7770, 34105, 34105, 145750, 145750, 611501, 611501, 2532530, 2532530, 10391745, 10391745, 42355950, 42355950, 171798901

Offset: 1

Marks R. Nester

Permuting the symbols will not change the structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Index entries for linear recurrences with constant coefficients, signature (1,9,-9,-26,26,24,-24).

Cf. A000453, A007581.

StirlingS2[Floor[(Range[40]+1)/2],4] (* or *) LinearRecurrence[ {1,9,-9,-26,26,24,-24},{0,0,0,0,0,0,1},40] (* Harvey P. Dale, Mar 08 2013 *)

stirling2( [(n+1)/2], 4).
G.f.: x^7/((x-1)*(2*x-1)*(2*x+1)*(2*x^2-1)*(3*x^2-1)). [Colin Barker, Jul 24 2012]
a(1)=a(2)=a(3)=a(4)=a(5)=a(6)=0, a(7)=1, a(n)=a(n-1)+9*a(n-2)-9*a(n-3)- 26*a(n-4)+ 26*a(n-5)+24*a(n-6)-24*a (n-7). - Harvey P. Dale, Mar 08 2013

A126680 Product_{i=4..n} Stirling_2(i,4).

Original entry on oeis.org

1, 10, 650, 227500, 386977500, 3006815175000, 102547431543375000, 14946288147446906250000, 9139670148451930618781250000, 23146488841058967849982079062500000, 240532409681630323860212020187339062500000, 10187978717854649915906947316453923964296875000000, 1750283547138817933292555967231683354203766362734375000000

Offset: 4

N. J. A. Sloane, Feb 13 2007

nonn

Partial products of A000453.

A337314 a(n) is the number of n-digit positive integers with exactly four distinct base 10 digits.

Original entry on oeis.org

0, 0, 0, 4536, 45360, 294840, 1587600, 7715736, 35244720, 154700280, 661122000, 2773768536, 11487556080, 47136955320, 192126589200, 779279814936, 3149513947440, 12695388483960, 51073849285200, 205172877726936, 823325141746800, 3301203837670200, 13228529919066000

Offset: 1

Stefano Spezia, Sep 26 2020

a(n) is the number of n-digit numbers in A031969.

			a(1) = a(2) = a(3) = 0 since the positive integers must have at least four digits;
a(4) = #{wxyz in N | w,x,y,z are four different digits with w != 0} = A073531(4) = 4536;
a(5) = 45360 since #[99999] - #[9999] - #(11111*[9]) - A335843(5) - A337313(5) - #{vwxyz in N | v,w,x,y,z are five different digits with v != 0} = 99999 - 9999 - 9 - 1215 - 16200 - 9*9*8*7*6 = 45360;
...

Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).
Index entries for sequences related to digits.

Cf. A000079, A000244, A000302, A000453, A031969, A073531, A335843, A337313, A337127.

Cf. A055642, A180599, A235155, A235691, A235718.

LinearRecurrence[{10,-35,50,-24},{0,0,0,4536},23]

concat([0,0,0],Vec(4536*x^4/(1-10*x+35*x^2-50*x^3+24*x^4)+O(x^24)))

O.g.f.: 4536*x^4/(1 - 10*x + 35*x^2 - 50*x^3 + 24*x^4).
E.g.f.: 189*(exp(x) - 1)^4.
a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4) for n > 4.
a(n) = 4536*S2(n, 4) where S2(n, 4) = A000453(n).
a(n) = 189*(4^n - 4*3^n + 3*2^(n+1) - 4).
a(n) ~ 189 * 4^n.
a(n) = 189*(A000302(n) - 4*A000244(n) + 3*A000079(n+1) - 4).
a(n) = A337127(n, 4).

cp's OEIS Frontend

A305862 a(n) = 384*4^n - 576*3^n + 220*2^n - 14.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A327611 Number of length n reversible string structures that are not palindromic using exactly four different colors.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A346954 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^4 / 4! ).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A346843 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^4 / 4!.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A383842 Expansion of 1/((1-x) * (1-2*x) * (1-3*x) * (1-4*x))^2.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A384988 a(n) = Stirling2(n,2)^2 + Stirling2(n,3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A385312 a(n) is the number of ternary strings of length n with at least one 0, at least two 1's and at least three 2's.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A056473 Number of palindromic structures using exactly four different symbols.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

A305862 a(n) = 3844^n - 5763^n + 220*2^n - 14.

A383842 Expansion of 1/((1-x) * (1-2x) (1-3x) (1-4*x))^2.