A193685 -id:A193685 - cp's OEIS frontend

A196835 Alternating row sums of Sheffer triangle A193685 (5-restricted Stirling2 numbers).

1, 4, 15, 51, 146, 273, -319, -6374, -36235, -113833, 69388, 3772035, 28631669, 112704452, -96418909, -5652669753, -50538496446, -230554460867, 281597003109, 16303457144146, 166512491229617, 872578914956059, -1111135578108284, -78512971676777833, -919653124088665479

Offset: 0

Wolfdieter Lang, Oct 07 2011

			a(2) = 25 - 11 + 1 = 15.

Seiichi Manyama, Table of n, a(n) for n = 0..589

Cf. A193685, A196834 (row sums), A193683, A193684, A293037, A346740.

my(x='x+O('x^30)); Vec(serlaplace(exp(-exp(x)+5*x+1))) \\ Michel Marcus, Aug 02 2021

a(n) = Sum_{m=0..n} (-1)^m * A193685(n,m), n>=0.
E.g.f.: exp(-exp(x)+5*x+1).
a(n) = exp(1) * Sum_{k>=0} (-1)^k * (k + 5)^n / k!. - Ilya Gutkovskiy, Dec 20 2019
a(0) = 1; a(n) = 5 * a(n-1) - Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Seiichi Manyama, Aug 02 2021

A196834 Row sums of Sheffer triangle A193685 (5-restricted Stirling2 numbers).

Original entry on oeis.org

1, 6, 37, 235, 1540, 10427, 73013, 529032, 3967195, 30785747, 247126450, 2050937445, 17585497797, 155666739742, 1421428484337, 13377704321695, 129659127547372, 1293095848212799, 13259069937250169, 139671750579429512, 1510382932875294447, 16754464511605466311

Offset: 0

Wolfdieter Lang, Oct 07 2011

			a(2) = 25 + 11 + 1 = 37.

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A000110, A005493, A005494, A045379, A196835 (alternating row sums).

b:= proc(n, m) option remember;
     `if`(n=0, 1, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 5):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 22 2021

nmax = 20; CoefficientList[Series[E^(E^x + 5*x - 1), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 10 2020 *)

a(n) = Sum_{m=0..n} A193685(n,m).
E.g.f.: exp(exp(x)+5*x-1).
a(n) ~ exp(n/LambertW(n) - n - 1) * n^(n + 5) / LambertW(n)^(n + 11/2). - Vaclav Kotesovec, Jun 10 2020
a(0) = 1; a(n) = 5 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Jul 03 2020

A016269 Number of monotone Boolean functions of n variables with 2 mincuts. Also number of Sperner systems with 2 blocks.

Original entry on oeis.org

1, 9, 55, 285, 1351, 6069, 26335, 111645, 465751, 1921029, 7859215, 31964205, 129442951, 522538389, 2104469695, 8460859965, 33972448951, 136276954149, 546269553775, 2188563950925, 8764714059751, 35090233104309, 140455067207455, 562102681589085, 2249257981411351

Offset: 0

N. J. A. Sloane

Half the number of 2 X (n+2) binary arrays with both a path of adjacent 1's and a path of adjacent 0's from top row to bottom row. - R. H. Hardin, Mar 21 2002
As (0,0,1,9,55,...) this is the third binomial transform of cosh(x)-1. It is the binomial transform of A000392, when this has two leading zeros. Its e.g.f. is then exp(3x)cosh(x) - exp(3x) and a(n) = (4^n - 2*3^n + 2^n)/2. - Paul Barry, May 13 2003
Let P(A) be the power set of an n-element set A. Then a(n-2) is the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which x is not a subset of y and y is not 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. - Ross La Haye, Jan 10 2008
a(n) also gives the third column sequence of the Sheffer triangle A143494 (2-restricted Stirling2 numbers). See the e.g.f. given below, and comments on the general case under A193685. - Wolfdieter Lang, Oct 08 2011
a(n) is also the number of even binomial coefficients in rows 0 through 2^(n+1)-1 of Pascal's triangle. - Aaron Meyerowitz, Oct 29 2013

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 292, #8, s(n,2).

G. C. Greubel, Table of n, a(n) for n = 0..1000
K. S. Brown, Dedekind's problem
John Elias, Illustration of Initial Terms: Inverse of the Sierpinski Triangle
Vladeta Jovovic, Illustration for A016269, A047707, A051112-A051118
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
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.
N. M. Rivière, Recursive formulas on free distributive lattices, J. Combinatorial Theory 5 1968 229--234. MR0231764 (38 #92). - _N. J. A. Sloane_, May 12 2012
Index entries for sequences related to Boolean functions
Index entries for linear recurrences with constant coefficients, signature (9,-26,24).

Equals (1/2) A038721(n+1). First differences of A000453. Partial sums of A027650. Pairwise sums of A099110. Odd part of A019333.

Cf. A000079, A000392, A001047, A006516, A032263, A143494, A193685.

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

a:= n-> Stirling2(n+4, 4)-Stirling2(n+3, 4): seq(a(n), n=0..24); # Zerinvary Lajos, Oct 05 2007

CoefficientList[1/((1-2x)(1-3x)(1-4x)) + O[x]^30, x] (* Jean-François Alcover, Nov 28 2015 *)
LinearRecurrence[{9, -26, 24}, {1, 9, 55}, 40] (* Vincenzo Librandi, Oct 06 2017 *)

a(n)=(2^n)*(2^n-1)/2-3^n+2^n \\ Charles R Greathouse IV, Mar 22 2016

G.f.: 1/((1-2*x)*(1-3*x)*(1-4*x)).
a(n-2) = (2^n)*(2^n - 1)/2 - 3^n + 2^n.
From Hieronymus Fischer, Jun 25 2007: (Start)
a(n) = Sum_{0<=i,j,k,<=n, i+j+k=n} 2^i*3^j*4^k.
a(n) = 2^(n+1)*(1+2^(n+2))-3^(n+2). (End)
a(n) = 3*StirlingS2(n+3,4) + StirlingS2(n+3,3). - Ross La Haye, Jan 10 2008
If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-2) = f(n,2,2), (n >= 2). - Milan Janjic, Apr 26 2009
E.g.f.: (d^2/dx^2) (exp(2*x)*((exp(x)-1)^2)/2!). See the Sheffer comment given above. - Wolfdieter Lang, Oct 08 2011
a(n) = A006516(n+2) - A001047(n+2). - Ross La Haye, Jan 26 2016
a(n) = A006516(n+1) + 3*a(n-1), n>=1, a(0)=1. - Carlos A. Rico A., Jun 22 2019

A003468 Number of minimal 3-covers of a labeled n-set.

Original entry on oeis.org

1, 22, 305, 3410, 33621, 305382, 2619625, 21554170, 171870941, 1337764142, 10216988145, 76862115330, 571247591461, 4203844925302, 30687029023865, 222518183370890, 1604626924403181, 11518132293452862

Offset: 3

N. J. A. Sloane

nonn

This is also the fourth column of the Sheffer triangle A143496 (4-restricted Stirling2 numbers). See the e.g.f. given below. See also the Sheffer comments in A193685. - Wolfdieter Lang, Oct 08 2011

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
T. Hearne and C. G. Wagner, Minimal covers of finite sets, Discr. Math. 5 (1973), 247-251.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Minimal cover.
Index entries for linear recurrences with constant coefficients, signature (22, -179, 638, -840).

Cf. A000392, A003468, A016111, A046166-A046169, A057668, A005783-A005786, A055066.

Cf. A143496, A000302, A005060, A016103.

[7^n/6 - 6^n/2 + 5^n/2 - 4^n/6: n in [3..30]]; // Vincenzo Librandi, May 03 2013

A003468:=1/(6*z-1)/(4*z-1)/(7*z-1)/(5*z-1); # conjectured by Simon Plouffe in his 1992 dissertation

Table[7^n/6 - 6^n/2 + 5^n/2 - 4^n/6, {n, 3, 20}] (* Vaclav Kotesovec, Nov 19 2012 *)
LinearRecurrence[{22,-179,638,-840},{1,22,305,3410},20] (* Harvey P. Dale, Jan 09 2024 *)

G.f.: x^3/((1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)). - N. J. A. Sloane, May 12 1994, corrected by Vaclav Kotesovec, Nov 19 2012
E.g.f.: (exp(4*x)*(exp(x) - 1)^3)/6. More generally, e.g.f. for number of minimal m-covers of a labeled n-set is (exp((2^m - m - 1)*x)*(exp(x) - 1)^m)/m!. - Vladeta Jovovic, May 09 2004
If we define f(m, j, x) = sum(binomial(m, k)*stirling2(k, j)*x^(m - k),k = j .. m) then a(n) = f(n, 3, 4), (n >= 3). - Milan Janjic, Apr 26 2009
a(n) = 7^n/6 - 6^n/2 + 5^n/2 - 4^n/6. - Vaclav Kotesovec, Nov 19 2012

A001552 a(n) = 1^n + 2^n + ... + 5^n.

Original entry on oeis.org

5, 15, 55, 225, 979, 4425, 20515, 96825, 462979, 2235465, 10874275, 53201625, 261453379, 1289414505, 6376750435, 31605701625, 156925970179, 780248593545, 3883804424995, 19349527020825, 96470431101379, 481245667164585, 2401809362313955, 11991391850823225

Offset: 0

N. J. A. Sloane

a(n)*(-1)^n, n>=0, gives the z-sequence for the Sheffer triangle A049460 ((signed) 5-restricted Stirling1 numbers), which is the inverse triangle of A193685 (5-restricted Stirling2 numbers). See the W. Lang link under A006232 for a- and z-sequences for Sheffer matrices. The a-sequence for each (signed) r-restricted Stirling1 Sheffer triangle is A027641/A027642 (Bernoulli numbers). - Wolfdieter Lang, Oct 10 2011

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 365
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (15, -85, 225, -274, 120).

Column 5 of array A103438.

Table[Total[Range[5]^n], {n, 0, 40}] (* T. D. Noe, Oct 10 2011 *)

PARI
```
a(n)=if(n<0,0,sum(k=1,5,k^n))
```

[3**n + sigma(4, n) + 5**n for n in range(22)] # Zerinvary Lajos, Jun 04 2009

[1 + 2**n + 3**n + 4**n + 5**n for n in range(22)] # Zerinvary Lajos, Jun 04 2009

a(n) = Sum_{k=1..5} k^n, n >= 0.
O.g.f.: (5 - 60*x + 255*x^2 - 450*x^3 + 274*x^4)/Product_{j=1..5} (1 - j*x). - Simon Plouffe in his 1992 dissertation
E.g.f.: exp(x)*(1-exp(5*x))/(1-exp(x)) = Sum_{j=1..5} exp(j*x) (trivial). - Wolfdieter Lang, Oct 10 2011

A025211 Expansion of 1/((1-2x)(1-3x)(1-4x)(1-5x)).

Original entry on oeis.org

1, 14, 125, 910, 5901, 35574, 204205, 1132670, 6129101, 32566534, 170691885, 885423630, 4556561101, 23305343894, 118631189165, 601616805790, 3042056477901, 15346559343654, 77279066272045, 388583895311150, 1951684190615501, 9793511186181814, 49108010998116525

Offset: 0

N. J. A. Sloane

This gives the fourth column of the Sheffer triangle A143494 (2-restricted Stirling2 numbers). See the e.g.f. given below, and comments on the general case under A193685. - Wolfdieter Lang, Oct 08 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (14,-71,154,-120)

Cf. A000079, A001047, A016269.

[3^(n+3)/2 -2*4^(n+2)-2^(n+2)/3+5^(n+3)/6: n in [0..30]]; // Vincenzo Librandi, Jun 21 2011

CoefficientList[Series[1/((1 - 2 x) (1 - 3 x) (1 - 4 x) (1 - 5 x)), {x, 0, 25}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)
LinearRecurrence[{14,-71,154,-120},{1,14,125,910},30] (* Harvey P. Dale, Feb 05 2020 *)

a(n)=n-=2;3^n*3/2-2*4^n-2^n/3+5^n*5/6 \\ Charles R Greathouse IV, Jun 21 2011

If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*stirling2(k,j)*x^(m-k) then a(n-3) = (-1)^(n-1)*f(n,3,-5), (n >= 3). - Milan Janjic, Apr 26 2009
If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*stirling2(k,j)*x^(m-k) then a(n-3) = f(n,3,2), (n >= 3). - Milan Janjic, Apr 26 2009
a(n) = 3^(n+3)/2 - 2*4^(n+2) - 2^(n+2)/3 + 5^(n+3)/6. - R. J. Mathar, Mar 22 2011
O.g.f.: 1/((1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)).
E.g.f.: (d^3/dx^3) (exp(2*x)*((exp(x)-1)^3)/3!). See the Sheffer comment given above. - Wolfdieter Lang, Oct 08 2011

A028025 Expansion of 1/((1-3x)(1-4x)(1-5x)*(1-6x)).

Original entry on oeis.org

1, 18, 205, 1890, 15421, 116298, 830845, 5709330, 38119741, 249026778, 1599719485, 10142356770, 63639854461, 396031348458, 2448208592125, 15053605980210, 92160458747581, 562225198873338, 3419937140824765

Offset: 0

N. J. A. Sloane

This gives the fourth column of the Sheffer triangle A143495 (3-restricted Stirling2 numbers). See the e.g.f. given below, and comments on the general case under A193685. - Wolfdieter Lang, Oct 08 2011

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (18,-119,342,-360).

Cf. A000453, A025211, A003468, A028165, A028200, A016109, A016075, A016094.

Cf. A143495, A193685.

CoefficientList[Series[1/((1-3x)(1-4x)(1-5x)(1-6x)),{x,0,30}],x] (* or *) LinearRecurrence[{18,-119,342,-360},{1,18,205,1890},30] (* Harvey P. Dale, Jan 29 2024 *)

Vec(1/((1-3*x)*(1-4*x)*(1-5*x)*(1-6*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-3) = f(n,3,3), (n >= 3). - Milan Janjic, Apr 26 2009
a(n) = -5^(n+3)/2 + 2*4^(n+2)+ 6^(n+2) - 3^(n+2)/2. - R. J. Mathar, Mar 22 2011
O.g.f.: 1/((1-3*x)*(1-4*x)*(1-5*x)*(1-6*x)).
E.g.f.: (d^3/dx^3)(exp(3*x)*((exp(x)-1)^3)/3!). - Wolfdieter Lang, Oct 08 2011

A028165 Expansion of 1/((1-5x)(1-6x)(1-7x)*(1-8x)).

Original entry on oeis.org

1, 26, 425, 5590, 64701, 688506, 6906145, 66324830, 616252901, 5580303586, 49508360265, 432061044870, 3720287489101, 31681154472266, 267320885100785, 2238337148081710, 18621251375573301, 154069635600426546

Offset: 0

N. J. A. Sloane

This is the column m=2 sequence (without leading zeros) of the Sheffer triangle (exp(5*x), exp(x)-1) of the 5-restricted Stirling2 numbers A193685. For a proof see the column o.g.f. formula there. - Wolfdieter Lang, Oct 07 2011

Index entries for linear recurrences with constant coefficients, signature (26, -251, 1066, -1680).

Cf. A000351, A005062, A019757, A193685.

Cf. A000453, A025211, A028025, A003468, A028200, A016109, A016075, A016094.

Vec(1/((1-5*x)*(1-6*x)*(1-7*x)*(1-8*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-3) = f(n,3,5), (n >= 3). - Milan Janjic, Apr 26 2009
a(n) = 26*a(n-1) - 251*a(n-2) + 1066*a(n-3) - 1680*a(n-4), n >= 4. - Vincenzo Librandi, Mar 19 2011
a(n) = 15*a(n-1) - 56*a(n-2) + 6^(n+1) - 5^(n+1), a(0)=1, a(1)=26. - Vincenzo Librandi, Mar 19 2011
E.g.f.: (d^3/dx^3)(exp(5*x)*((exp(x)-1)^3)/3!). See the Sheffer triangle comment above. - Wolfdieter Lang, Oct 07 2011
a(n) = -125*5^n/6 + 108*6^n - 343*7^n/2 + 256*8^n/3. - R. J. Mathar, Jun 23 2013

A016103 Expansion of 1/((1-4x)(1-5x)(1-6x)).

Original entry on oeis.org

1, 15, 151, 1275, 9751, 70035, 481951, 3216795, 20991751, 134667555, 852639151, 5343198315, 33212784151, 205111785075, 1260114546751, 7708980203835, 46999640806951, 285743822630595, 1733261544204751

Offset: 0

Robert G. Wilson v

2*a(n-2) = 6^n - 2*5^n + 4^n is the number of 3 X n {0,1}-matrices such that: (a) first and second row have a common 1, (b) first and third row have a common 1, (c) second and third row have no common 1. - Andi Fugard and Vladeta Jovovic, Jul 26 2008

This is the third column of the Sheffer triangle A143496 (4-restricted Stirling2 numbers). See A193685 for general comments. - Wolfdieter Lang, Oct 08 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Andi Fugard, Counting first-order models (with n individuals) of syllogisms.
Index entries for linear recurrences with constant coefficients, signature (15,-74,120).

Cf. A051588, A000302, A005060, A003468.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-4*x)*(1-5*x)*(1-6*x)))); // Vincenzo Librandi, Jun 24 2013

I:=[1, 15, 151]; [n le 3 select I[n] else 15*Self(n-1)-74*Self(n-2)+120*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jun 24 2013

CoefficientList[Series[1 / ((1 - 4 x) (1 - 5 x) (1 - 6 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 24 2013 *)

Vec(1/((1-4*x)*(1-5*x)*(1-6*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 2^(3 + 2*n) + 2^(1 + n) * 3^(2 + n) - 5^(2 + n). - Andi Fugard, Jul 22 2008
If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-2) = f(n,2,4), n >= 2. - Milan Janjic, Apr 26 2009
O.g.f.: 1/((1-4*x)*(1-5*x)*(1-6*x)).
E.g.f.: (d^2/dx^2)(exp(4*x)*((exp(x)-1)^2)/2!). See the Sheffer triangle comment above. - Wolfdieter Lang, Oct 08 2011
a(n) = 15*a(n-1) - 74*a(n-2) + 120*a(n-3). - Vincenzo Librandi, Jun 24 2013

A016753 Expansion of 1/((1-3x)(1-4x)(1-5*x)).

Original entry on oeis.org

1, 12, 97, 660, 4081, 23772, 133057, 724260, 3863761, 20308332, 105558817, 544039860, 2785713841, 14192221692, 72020501377, 364354427460, 1838822866321, 9262446387852, 46585947584737

Offset: 0

N. J. A. Sloane

As (0,0,1,12,97,...) this is the fourth binomial transform of cosh(x)-1. It is the binomial transform of A016269, when this has two leading zeros. Its e.g.f. is then exp(4x)cosh(x) - exp(4x). - Paul Barry, May 13 2003
This gives the third column of the Sheffer triangle A143495 (3-restricted Stirling2 numbers). See the e.g.f. below, and A193685 for comments on the general case. - Wolfdieter Lang, Oct 08 2011
From Kevin Long, Mar 25 2017: (Start)
In the power set poset 2^(n+2), a(n) gives the number of size 3 subposets {A,B,C} such that A subset of C, B subset of C, and A||B. By symmetry, it also counts the size 3 subposets {A,B,C} such that C subset of A, C subset of B, and A||B.
By the power set poset, I mean the subsets of [n+2] ordered by inclusion. A||B means A and B are incomparable.
The result can be proved by showing that the formula holds. 5^n counts triples (A,B,C) of subsets of [n] where A subset of  C and B subset of C, since for each x in [n], it is either in C only, in A and C, in B and C, in all three, or in none. However, this also counts the cases where A subset of B and where B subset of A, and we want A||B.
Each case can be counted by 4^n, since if A subset of B⊆C, then each element x of [n] is either in all three, in B and C, in only C, or in none. Hence we subtract 2*4^n from 5^n. These two cases intersect, however, when A = B subset of C, which can be counted by 3^n, since each element x of [n] can be either in all three sets, in only C, or in none.
For the purposes of inclusion-exclusion, we add these sets back in to get 5^n-2*4^n+3^n to count all triples (A,B,C) where A subset of C, B subset of C, and A||B. We want sets, not triples, so this double-counts the sets since interchanging A and B give the same set, so we divide this by 2. Hence the formula for a(n) counts these subposets for 2^(n+2).  (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-47,60).

Cf. A000244, A005061, A016753, A132495, A193685, A245019.

[(5^(n+2) - 2*4^(n+2) + 3^(n+2))/2: n in [0..30]]; // G. C. Greubel, Sep 15 2018

CoefficientList[ Series[ 1/((1 - 3x)(1 - 4x)(1 - 5x)), {x, 0, 25} ], x ]
LinearRecurrence[{12,-47,60}, {1, 12, 97}, 30] (* G. C. Greubel, Sep 15 2018 *)

Vec(1/((1-3*x)*(1-4*x)*(1-5*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 5^(n+2)/2 - 4^(n+2) + 3^(n+2)/2. - Paul Barry, May 13 2003
If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*stirling2(k,j)*x^(m-k) then a(n-2) = f(n,2,3), (n >= 2). - Milan Janjic, Apr 26 2009
a(n) = 9*a(n-1) - 20*a(n-2) + 3^n, n >= 2. - Vincenzo Librandi, Mar 20 2011
O.g.f.: 1/((1-3*x)*(1-4*x)*(1-5*x)).
E.g.f.: (d^2/dx^2) (exp(3*x)*((exp(x)-1)^2)/2!). - Wolfdieter Lang, Oct 08 2011
a(n) = A245019(n+2)/2. - Kevin Long, Mar 24 2017

cp's OEIS Frontend

A196835 Alternating row sums of Sheffer triangle A193685 (5-restricted Stirling2 numbers).

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A196834 Row sums of Sheffer triangle A193685 (5-restricted Stirling2 numbers).

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Maple

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A016269 Number of monotone Boolean functions of n variables with 2 mincuts. Also number of Sperner systems with 2 blocks.

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A003468 Number of minimal 3-covers of a labeled n-set.

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A001552 a(n) = 1^n + 2^n + ... + 5^n.

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Sage

Sage

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A025211 Expansion of 1/((1-2x)(1-3x)(1-4x)(1-5x)).

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A028025 Expansion of 1/((1-3x)*(1-4x)*(1-5x)*(1-6x)).

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A028025 Expansion of 1/((1-3x)(1-4x)(1-5x)*(1-6x)).

A028165 Expansion of 1/((1-5x)(1-6x)(1-7x)*(1-8x)).

A016753 Expansion of 1/((1-3x)(1-4x)(1-5*x)).