A016297 -id:A016297 - cp's OEIS frontend

A225466 Triangle read by rows, 3^k*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

1, 2, 3, 4, 21, 9, 8, 117, 135, 27, 16, 609, 1431, 702, 81, 32, 3093, 13275, 12015, 3240, 243, 64, 15561, 115479, 171990, 81405, 13851, 729, 128, 77997, 970515, 2238327, 1655640, 479682, 56133, 2187, 256, 390369, 7998111, 27533142, 29893941, 13121514, 2561706

Offset: 0

Peter Luschny, May 08 2013

The definition of the Stirling-Frobenius subset numbers of order m is in A225468.
From Wolfdieter Lang, Apr 09 2017: (Start)
This is the Sheffer triangle (exp(2*x), exp(3*x) - 1), denoted by S2[3,2]. See also A282629 for S2[3,1]. The stirling2 triangle A048993 is in this notation denoted by S2[1,0].
The a-sequence for this Sheffer triangle has e.g.f. 3*x/log(1+x) and is 3*A006232(n)/A006233(n) (Cauchy numbers of the first kind). For a- and z-sequences for Sheffer triangles see the W. Lang link under A006232, also with references).
The z-sequence has e.g.f. (3/(log(1+x)))*(1 - 1/(1+x)^(2/3)) and gives 2*A284862/A284863.
The first column k sequences divided by 3^k are A000079, A016127, A016297, A025999. For the e.g.f.s and o.g.f.s see below.
The row sums give A284864. The alternating row sums give A284865.
This triangle appears in the o.g.f. G(n, x) of the sequence {(2 + 3*m)^n}{m>=0}, as G(n, x) = Sum{k=0..n} T(n, k)*k!*x^k/(1-x)^(k+1), n >= 0. Hence the corresponding e.g.f. is, by the linear inverse Laplace transform, E(n, t) = Sum_{m >=0} (2 + 3*m)^n t^m/m! = exp(t)*Sum_{k=0..n} T(n, k)*t^k.
The corresponding Eulerian number triangle is A225117(n, k) = Sum_{m=0..k} (-1)^(k-m)*binomial(n-m, k-m)*T(n, m)*m!, 0 <= k <= n. (End)

			[n\k][ 0,     1,      2,       3,       4,      5,     6,    7]
[0]    1,
[1]    2,     3,
[2]    4,    21,      9,
[3]    8,   117,    135,      27,
[4]   16,   609,   1431,     702,      81,
[5]   32,  3093,  13275,   12015,    3240,    243,
[6]   64, 15561, 115479,  171990,   81405,  13851,   729,
[7]  128, 77997, 970515, 2238327, 1655640, 479682, 56133, 2187.
...
From _Wolfdieter Lang_, Aug 11 2017: (Start)
Recurrence (see the Maple program): T(4, 2) = 3*T(3, 1) + (3*2+2)*T(3, 2) = 3*117 + 8*135 = 1431.
Boas-Buck recurrence for column k = 2, and n = 4: T(4,2) = (1/2)*(2*(4 + 3*2)*T(3, 2) + 2*6*(-3)^2*Bernoulli(2)*T(2, 2)) = (1/2)*(20*135 + 12*9*(1/6)*9) = 1431. (End)

Vincenzo Librandi, Rows n = 0..50, flattened
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 9.
Peter Luschny, Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.
Shi-Mei Ma, Toufik Mansour, and Matthias Schork, Normal ordering problem and the extensions of the Stirling grammar, Russian Journal of Mathematical Physics, 2014, 21(2), arXiv:1308.0169 [math.CO], 2013, p. 12.

Cf. A048993 (m=1), A154537 (m=2), A225467 (m=4), A225468.

Cf. A000079, A000244, A005057, A016127, A016297, A025999, A006232/A006233, A225117, A225472, A225468, A282629, A284862/A284863, A284864, A284865.

SF_SS := proc(n, k, m) option remember;
if n = 0 and k = 0 then return(1) fi;
if k > n or  k < 0 then return(0) fi;
m*SF_SS(n-1, k-1, m) + (m*(k+1)-1)*SF_SS(n-1, k, m) end:
seq(print(seq(SF_SS(n, k, 3), k=0..n)), n=0..5);

EulerianNumber[n_, k_, m_] := EulerianNumber[n, k, m] = (If[ n == 0, Return[If[k == 0, 1, 0]]]; Return[(m*(n-k)+m-1)*EulerianNumber[n-1, k-1, m] + (m*k+1)*EulerianNumber[n-1, k, m]]); SFSS[n_, k_, m_] := Sum[ EulerianNumber[n, j, m]*Binomial[j, n-k], {j, 0, n}]/k!; Table[ SFSS[n, k, 3], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 29 2013, translated from Sage *)

T(n, k) = sum(j=0, k, binomial(k, j)*(-1)^(j - k)*(2 + 3*j)^n/k!);
for(n=0, 10, for(k=0, n, print1(T(n, k),", ");); print();) \\ Indranil Ghosh, Apr 10 2017

from sympy import binomial, factorial
def T(n, k): return sum(binomial(k, j)*(-1)**(j - k)*(2 + 3*j)**n//factorial(k) for j in range(k + 1))
for n in range(11): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Apr 10 2017

@CachedFunction
def EulerianNumber(n, k, m) :
    if n == 0: return 1 if k == 0 else 0
    return (m*(n-k)+m-1)*EulerianNumber(n-1,k-1,m) + (m*k+1)*EulerianNumber(n-1,k,m)
def SF_SS(n, k, m):
    return add(EulerianNumber(n,j,m)*binomial(j,n-k) for j in (0..n))/ factorial(k)
def A225466(n): return SF_SS(n, k, 3)

T(n, k) = (1/k!)*Sum_{j=0..n} binomial(j, n-k)*A_3(n, j) where A_m(n, j) are the generalized Eulerian numbers A225117.
For a recurrence see the Maple program.
T(n, 0) ~ A000079; T(n, 1) ~ A005057; T(n, n) ~ A000244.
From Wolfdieter Lang, Apr 09 2017: (Start)
T(n, k) = Sum_{j=0..k} binomial(k,j)*(-1)^(j-k)*(2 + 3*j)^n/k!, 0 <= k <= n.
E.g.f. of triangle: exp(2*z)*exp(x*(exp(3*z)-1)) (Sheffer type).
E.g.f. for sequence of column k is exp(2*x)*((exp(3*x) - 1)^k)/k! (Sheffer property).
O.g.f. for sequence of column k is 3^k*x^k/Product_{j=0..k} (1 - (2+3*j)*x).
A nontrivial recurrence for the column m=0 entries T(n, 0) = 2^n from the z-sequence given above: T(n,0) = n*Sum_{k=0..n-1} z(k)*T(n-1,k), n >= 1, T(0, 0) = 1.
The corresponding recurrence for columns k >= 1 from the a-sequence is T(n, k) = (n/k)* Sum_{j=0..n-k} binomial(k-1+j, k-1)*a(j)*T(n-1, k-1+j).
Recurrence for row polynomials R(n, x) (Meixner type): R(n, x) = ((3*x+2) + 3*x*d_x)*R(n-1, x), with differentiation d_x, for n >= 1, with input R(0, x) = 1.
(End)
Boas-Buck recurrence for column sequence k: T(n, k) = (1/(n - k))*((n/2)*(4 + 3*k)*T(n-1, k) + k*Sum_{p=k..n-2} binomial(n, p)*(-3)^(n-p)*Bernoulli(n-p)*T(p, k)), for n > k >= 0, with input T(k, k) = 3^k. See a comment and references in A282629, An example is given below. - Wolfdieter Lang, Aug 11 2017

A225468 Triangle read by rows, S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

Original entry on oeis.org

1, 2, 1, 4, 7, 1, 8, 39, 15, 1, 16, 203, 159, 26, 1, 32, 1031, 1475, 445, 40, 1, 64, 5187, 12831, 6370, 1005, 57, 1, 128, 25999, 107835, 82901, 20440, 1974, 77, 1, 256, 130123, 888679, 1019746, 369061, 53998, 3514, 100, 1

Offset: 0

Peter Luschny, May 16 2013

The definition of the Stirling-Frobenius subset numbers: S_m(n, k) = (Sum_{j=0..n} binomial(j, n-k)*A_m(n, j)) / (m^k*k!) where A_m(n, j) are the generalized Eulerian numbers. For m = 1 this gives the classical Stirling set numbers A048993. (See the links for details.)
From Peter Bala, Jan 27 2015: (Start)
Exponential Riordan array [ exp(2*z), 1/3*(exp(3*z) - 1)].
Triangle equals P * A111577 = P^(-1) * A075498, where P is Pascal's triangle A007318.
Triangle of connection constants between the polynomial basis sequences {x^n}n>=0 and { n!*3^n*binomial((x - 2)/3,n) }n>=0. An example is given below.
This triangle is the particular case a = 3, b = 0, c = 2 of the triangle of generalized Stirling numbers of the second kind S(a,b,c) defined in the Bala link. (End)

			[n\k][ 0,    1,     2,    3,    4,  5,  6]
[0]    1,
[1]    2,    1,
[2]    4,    7,     1,
[3]    8,   39,    15,    1,
[4]   16,  203,   159,   26,    1,
[5]   32, 1031,  1475,  445,   40,  1,
[6]   64, 5187, 12831, 6370, 1005, 57,  1.
Connection constants: Row 3: [8, 39, 15, 1] so
x^3 = 8 + 39*(x - 2) + 15*(x - 2)*(x - 5) + (x - 2)*(x - 5)*(x - 8). - _Peter Bala_, Jan 27 2015

Vincenzo Librandi, Rows n = 0..50, flattened
Peter Bala, A 3 parameter family of generalized Stirling numbers
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 8.
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.
Peter Luschny, Generalized Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.
Shi-Mei Ma, Toufik Mansour, and Matthias Schork, Normal ordering problem and the extensions of the Stirling grammar, Russian Journal of Mathematical Physics, 2014, 21(2), arXiv 1308.0169 p. 12.

Cf. A048993 (m=1), A039755 (m=2), A225469 (m=4).

Cf. A075498, A111577. Columns: A000079, A016127, A016297, A025999. A225466, A225472.

SF_S := proc(n, k, m) option remember;
if n = 0 and k = 0 then return(1) fi;
if k > n or k < 0 then return(0) fi;
SF_S(n-1, k-1, m) + (m*(k+1)-1)*SF_S(n-1, k, m) end:
seq(print(seq(SF_S(n, k, 3), k=0..n)), n = 0..5);

EulerianNumber[n_, k_, m_] := EulerianNumber[n, k, m] = (If[ n == 0, Return[If[k == 0, 1, 0]]]; Return[(m*(n-k)+m-1)*EulerianNumber[n-1, k-1, m] + (m*k+1)*EulerianNumber[n-1, k, m]]); SFS[n_, k_, m_] := Sum[ EulerianNumber[n, j, m]*Binomial[j, n-k], {j, 0, n}]/(k!*m^k); Table[ SFS[n, k, 3], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 29 2013, translated from Sage *)

@CachedFunction
def EulerianNumber(n, k, m) :
    if n == 0: return 1 if k == 0 else 0
    return (m*(n-k)+m-1)*EulerianNumber(n-1,k-1,m) + (m*k+1)*EulerianNumber(n-1,k,m)
def SF_S(n, k, m):
    return add(EulerianNumber(n, j, m)*binomial(j, n - k) for j in (0..n))/ (factorial(k)*m^k)
for n in (0..6): [SF_S(n, k, 3) for k in (0..n)]

T(n, k) = (Sum_{j=0..n} binomial(j, n-k)*A_3(n, j)) / (3^k*k!) with A_3(n,j) = A225117.
For a recurrence see the Maple program.
T(n, 0) ~ A000079; T(n, 1) ~ A016127; T(n, 2) ~ A016297; T(n, 3) ~ A025999;
T(n, n) ~ A000012; T(n, n-1) ~ A005449; T(n, n-2) ~ A024212.
From Peter Bala, Jan 27 2015: (Start)
T(n,k) = Sum_{i = 0..n} (-1)^(n+i)*3^(i-k)*binomial(n,i)*Stirling2(i+1,k+1).
E.g.f.: exp(2*z)*exp(x/3*(exp(3*z) - 1)) = 1 + (2 + x)*z + (4 + 7*x + x^2)*z^2/2! + ....
T(n,k) = 1/(3^k*k!)*Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*(3*j + 2)^n.
O.g.f. for n-th diagonal: exp(-2*x/3)*Sum_{k >= 0} (3*k + 2)^(k + n - 1)*((x/3*exp(-x))^k)/k!.
O.g.f. column k: 1/( (1 - 2*x)*(1 - 5*x)...(1 - (3*k + 2)*x) ). (End)
E.g.f. column k: exp(2*x)*((exp(3*x) - 1)/3)^k, k >= 0. See the Bala link for the S(3,0,2) exponential Riordan aka Sheffer triangle. - Wolfdieter Lang, Apr 10 2017

A225472 Triangle read by rows, k!*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

Original entry on oeis.org

1, 2, 3, 4, 21, 18, 8, 117, 270, 162, 16, 609, 2862, 4212, 1944, 32, 3093, 26550, 72090, 77760, 29160, 64, 15561, 230958, 1031940, 1953720, 1662120, 524880, 128, 77997, 1941030, 13429962, 39735360, 57561840, 40415760, 11022480, 256, 390369, 15996222, 165198852

Offset: 0

Peter Luschny, May 17 2013

The Stirling-Frobenius subset numbers are defined in A225468 (see also the Sage program).

			[n\k][0,     1,      2,       3,       4,       5,      6 ]
[0]   1,
[1]   2,     3,
[2]   4,    21,     18,
[3]   8,   117,    270,     162,
[4]  16,   609,   2862,    4212,    1944,
[5]  32,  3093,  26550,   72090,   77760,   29160,
[6]  64, 15561, 230958, 1031940, 1953720, 1662120, 524880.

Vincenzo Librandi, Rows n = 0..50, flattened
P. Bala, A 3 parameter family of generalized Stirling numbers.
Peter Luschny, Generalized Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.

Cf. A131689 (m=1), A145901 (m=2), A225473 (m=4).

Cf. A225466, A225468, columns: A000079, 3*A016127, 3^2*2!*A016297, 3^3*3!*A025999.

SF_SO := proc(n, k, m) option remember;
if n = 0 and k = 0 then return(1) fi;
if k > n or k < 0 then return(0) fi;
m*k*SF_SO(n-1, k-1, m) + (m*(k+1)-1)*SF_SO(n-1, k, m) end:
seq(print(seq(SF_SO(n, k, 3), k=0..n)), n = 0..5);

EulerianNumber[n_, k_, m_] := EulerianNumber[n, k, m] = (If[ n == 0, Return[If[k == 0, 1, 0]]]; Return[(m*(n-k)+m-1)*EulerianNumber[n-1, k-1, m] + (m*k+1)*EulerianNumber[n-1, k, m]]); SFSO[n_, k_, m_] := Sum[ EulerianNumber[n, j, m]*Binomial[j, n-k], {j, 0, n}]; Table[ SFSO[n, k, 3], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 29 2013, translated from Sage *)

@CachedFunction
def EulerianNumber(n, k, m) :
    if n == 0: return 1 if k == 0 else 0
    return (m*(n-k)+m-1)*EulerianNumber(n-1, k-1, m)+ (m*k+1)*EulerianNumber(n-1, k, m)
def SF_SO(n, k, m):
    return add(EulerianNumber(n, j, m)*binomial(j, n - k) for j in (0..n))
for n in (0..6): [SF_SO(n, k, 3) for k in (0..n)]

For a recurrence see the Maple program.
T(n, 0) ~ A000079; T(n, 1) ~ A005057; T(n, n) ~ A032031.
From Wolfdieter Lang, Apr 10 2017: (Start)
E.g.f. for sequence of column k: exp(2*x)*(exp(3*x) - 1)^k, k >= 0. From the Sheffer triangle S2[3,2] = A225466 with column k multiplied with k!.
O.g.f. for sequence of column k is 3^k*k!*x^k/Product_{j=0..k} (1 - (2+3*j)*x), k >= 0.
T(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k, j)*(2+3*j)^n, 0 <= k <= n.
Three term recurrence (see the Maple program): T(n, k) = 0 if n < k , T(n, -1) = 0, T(0,0) = 1, T(n, k) = 3*k*T(n-1, k-1) + (2 + 3*k)*T(n-1, k) for n >= 1, k=0..n.
For the column scaled triangle (with diagonal 1s) see A225468, and the Bala link with (a,b,c) = (3,0,2), where Sheffer triangles are called exponential Riordan triangles.
(End)
The e.g.f. of the row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k is exp(2*z)/(1 - x*(exp(3*z) - 1)). - Wolfdieter Lang, Jul 12 2017

A025999 Expansion of g.f. 1/((1-2x) (1-5x) (1-8x) (1-11*x)).

Original entry on oeis.org

1, 26, 445, 6370, 82901, 1019746, 12105885, 140404290, 1603014501, 18104952866, 202945103725, 2262802497410, 25134485221301, 278430633932386, 3078357517755965, 33986947913921730, 374856803115095301, 4131429114327366306, 45509760855920174605, 501119725990818613250

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (26,-231,806,-880).

Cf. A016127, A016297.

CoefficientList[Series[1/((1-2x)(1-5x)(1-8x)(1-11x)),{x,0,20}],x] (* or *) LinearRecurrence[{26,-231,806,-880},{1,26,445,6370},20] (* Harvey P. Dale, May 24 2014 *)

a(n) = -4*2^n/81 +125*5^n/54 -256*8^n/27 +1331*11^n/162. - R. J. Mathar, Jun 20 2013
a(0)=1, a(1)=26, a(2)=445, a(3)=6370, a(n)=26*a(n-1)-231*a(n-2)+ 806*a(n-3)- 880*a(n-4). - Harvey P. Dale, May 24 2014
From Seiichi Manyama, May 04 2025: (Start)
a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(n+3,k+3) * Stirling2(k+3,3).
a(n) = Sum_{k=0..n} (-3)^k * 11^(n-k) * binomial(n+3,k+3) * Stirling2(k+3,3). (End)
E.g.f.: exp(2*x)*(1331*exp(9*x) - 1536*exp(6*x) + 375*exp(3*x) - 8)/162. - Stefano Spezia, May 04 2025

A016307 Expansion of g.f. 1/((1-2x)(1-6x)(1-10*x)).

Original entry on oeis.org

1, 18, 232, 2640, 28336, 295008, 3020032, 30620160, 308720896, 3102325248, 31113951232, 311683706880, 3120102240256, 31220613439488, 312323680632832, 3123942083788800, 31243652502716416, 312461915016265728, 3124771490097528832, 31248628940585041920

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (18,-92,120).

Cf. A016297, A016315.

[(2^n-18*6^n+25*10^n)/8: n in [0..20]]; // Vincenzo Librandi, Sep 01 2011

CoefficientList[Series[1/((1-2x)(1-6x)(1-10x)),{x,0,30}],x] (* or *) LinearRecurrence[{18,-92,120},{1,18,232},30] (* Harvey P. Dale, Nov 06 2019 *)

From Vincenzo Librandi, Sep 01 2011: (Start)
  a(n) = (2^n - 18*6^n + 25*10^n)/8.
  a(n) = 18*a(n-1) - 92*a(n-2) + 120*a(n-3) for n > 2.
  a(n) = 16*a(n-1) - 60*a(n-2) + 2^n for n > 1. (End)
From Seiichi Manyama, May 04 2025: (Start)
a(n) = Sum_{k=0..n} 4^k * 2^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).
a(n) = Sum_{k=0..n} (-4)^k * 10^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2). (End)
E.g.f.: exp(2*x)*(1 - 18*exp(4*x) + 25*exp(8*x))/8. - Stefano Spezia, May 04 2025

A019618 Expansion of 1/((1-4x)(1-7x)(1-10*x)).

Original entry on oeis.org

1, 21, 303, 3745, 42711, 464961, 4918663, 51086385, 524227671, 5336085601, 54018566823, 544793838225, 5480212349431, 55028108373441, 551863246323783, 5529708675105265, 55374624529091991, 554289026917064481, 5546689809273133543, 55493495148326663505, 555121131971945559351

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (21,-138,280).

Cf. A016223, A016297, A017933, A020447.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-4*x)*(1-7*x)*(1-10*x)))); // Vincenzo Librandi, Jul 03 2013

I:=[1, 21, 303]; [n le 3 select I[n] else 21*Self(n-1)-138*Self(n-2)+280*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jul 03 2013

CoefficientList[Series[1 / ((1 - 4 x) (1 - 7 x) (1 - 10 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jul 03 2013 *)
LinearRecurrence[{21,-138,280},{1,21,303},30] (* Harvey P. Dale, Mar 09 2017 *)

x='x+O('x^30); Vec(1/((1-4*x)*(1-7*x)*(1-10*x))) \\ G. C. Greubel, Aug 24 2018

a(n) = (2*4^(n+1) -7^(n+2) +5*10^(n+1))/9. - R. J. Mathar, Nov 11 2012
a(0)=1, a(1)=21, a(2)=303; for n>2, a(n) = 21*a(n-1) -138*a(n-2) +280*a(n-3). - Vincenzo Librandi, Jul 03 2013
a(n) = 17*a(n-1) -70*a(n-2) +4^n. - Vincenzo Librandi, Jul 03 2013
From Seiichi Manyama, May 05 2025: (Start)
a(n) = Sum_{k=0..n} 3^k * 4^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).
a(n) = Sum_{k=0..n} (-3)^k * 10^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2). (End)

A020447 Expansion of 1/((1-5x) (1-8x) (1-11*x)).

Original entry on oeis.org

1, 24, 393, 5480, 70161, 853944, 10066393, 116192520, 1322205921, 14898923864, 166735197993, 1856912289960, 20608880226481, 228161663489784, 2521496249891193, 27830232878409800, 306882907287251841, 3381715508097175704, 37246902627265441993, 410100204278978264040

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (24,-183,440).

Cf. A016223, A016297, A017933, A019618.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-5*x)*(1-8*x)*(1-11*x)))); // Vincenzo Librandi, Jul 03 2013

I:=[1, 24, 393]; [n le 3 select I[n] else 24*Self(n-1)-183*Self(n-2)+440*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jul 03 2013

CoefficientList[Series[1 / ((1 - 5 x) (1 - 8 x) (1 - 11 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jul 03 2013 *)
LinearRecurrence[{24,-183,440},{1,24,393},30] (* Harvey P. Dale, Jun 20 2015 *)

a(n) = 25*5^n/18 -64*8^n/9 +121*11^n/18. - R. J. Mathar, Jun 29 2013
a(0)=1, a(1)=24, a(2)=393; for n>2, a(n) = 24*a(n-1) -183*a(n-2) +440*a(n-3). - Vincenzo Librandi, Jul 03 2013
a(n) = 19*a(n-1) -88*a(n-2) +5^n. - Vincenzo Librandi, Jul 03 2013
From Seiichi Manyama, May 05 2025: (Start)
a(n) = Sum_{k=0..n} 3^k * 5^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).
a(n) = Sum_{k=0..n} (-3)^k * 11^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2). (End)

A025992 Expansion of 1/((1-2x)(1-5x)(1-7x)(1-8*x)).

Original entry on oeis.org

1, 22, 313, 3666, 38493, 377286, 3529681, 31947322, 282198565, 2447183310, 20920905369, 176852694018, 1481626607917, 12322682753494, 101879323774177, 838170485025354, 6867569457133749, 56077266261254238

Offset: 0

N. J. A. Sloane

From Bruno Berselli, May 09 2013: (Start)
a(n) - 2*a(n-1), for n>0, gives A019928 (after 1);
a(n) - 5*a(n-1), for n>0, gives A016311 (after 1);
a(n) - 7*a(n-1), for n>0, gives A016297 (after 1);
a(n) - 8*a(n-1), for n>0, gives A016296 (after 1);
a(n) - 7*a(n-1) + 10*a(n-2), for n>1, gives A016177 (after 15);
a(n) - 9*a(n-1) + 14*a(n-2), for n>1, gives A016162 (after 13);
a(n) - 10*a(n-1) + 16*a(n-2), for n>1, gives A016161 (after 12);
a(n) - 12*a(n-1) + 35*a(n-2), for n>1, gives A016131 (after 10);
a(n) - 13*a(n-1) + 40*a(n-2), for n>1, gives A016130 (after 9);
a(n) - 15*a(n-1) + 56*a(n-2), for n>1, gives A016127 (after 7);
a(n) - 20*a(n-1) +131*a(n-2) -280*a(n-3), for n>2, gives A000079 (after 4);
a(n) - 17*a(n-1) +86*a(n-2) -112*a(n-3), for n>2, gives A000351 (after 25);
a(n) - 15*a(n-1) +66*a(n-2) -80*a(n-3), for n>2, gives A000420 (after 49);
a(n) - 14*a(n-1) +59*a(n-2) -70*a(n-3), for n>2, gives A001018 (after 64),
and naturally: a(n) - 22*a(n-1) + 171*a(n-2) - 542*a(n-3) + 560*a(n-4), for n>3, gives 0 (see Harvey P. Dale in Formula lines). (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (22,-171,542,-560).

Cf. A000079, A000351, A000420, A001018, A016127, A016130, A016131, A016161, A016162, A016177, A016296, A016297, A016311, A019928.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!(1/((1-2*x)*(1-5*x)*(1-7*x)*(1-8*x)))); // Bruno Berselli, May 09 2013

CoefficientList[Series[1/((1-2x)(1-5x)(1-7x)(1-8x)),{x,0,30}],x] (* or *) LinearRecurrence[ {22,-171,542,-560},{1,22,313,3666},30] (* Harvey P. Dale, Jan 29 2013 *)

a(n) = n+=3; (5*8^n-9*7^n+5*5^n-2^n)/90 \\ Charles R Greathouse IV, Oct 03 2016

def A025992(n): return (5*pow(8,n+3)-9*pow(7,n+3)+pow(5,n+4)-pow(2,n+3))//90
print([A025992(n) for n in range(41)]) # G. C. Greubel, Dec 27 2024

a(0)=1, a(1)=22, a(2)=313, a(3)=3666, a(n) = 22*a(n-1) - 171*a(n-2) + 542*a(n-3) - 560*a(n-4). - Harvey P. Dale, Jan 29 2013
a(n) = (5*8^(n+3) - 9*7^(n+3) + 5^(n+4) - 2^(n+3))/90. - Yahia Kahloune, May 07 2013
E.g.f.: (1/90)*(-8*exp(2*x) + 625*exp(5*x) - 3087*exp(7*x) + 2560*exp(8*x)). - G. C. Greubel, Dec 27 2024

cp's OEIS Frontend

A225466 Triangle read by rows, 3^k*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Sage

Formula

A225468 Triangle read by rows, S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A225472 Triangle read by rows, k!*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A025999 Expansion of g.f. 1/((1-2*x) * (1-5*x) * (1-8*x) * (1-11*x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A016307 Expansion of g.f. 1/((1-2*x)*(1-6*x)*(1-10*x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A019618 Expansion of 1/((1-4*x)*(1-7*x)*(1-10*x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Magma

Mathematica

PARI

Formula

A020447 Expansion of 1/((1-5*x) * (1-8*x) * (1-11*x)).

Views

Author

Keywords

Links

Crossrefs

Programs

A025999 Expansion of g.f. 1/((1-2x) (1-5x) (1-8x) (1-11*x)).

A016307 Expansion of g.f. 1/((1-2x)(1-6x)(1-10*x)).

A019618 Expansion of 1/((1-4x)(1-7x)(1-10*x)).

A020447 Expansion of 1/((1-5x) (1-8x) (1-11*x)).

A025992 Expansion of 1/((1-2x)(1-5x)(1-7x)(1-8*x)).