A005057 -id:A005057 - cp's OEIS frontend

A155595 11^n+2^n-1.

1, 12, 124, 1338, 14656, 161082, 1771624, 19487298, 214359136, 2357948202, 25937425624, 285311672658, 3138428380816, 34522712152122, 379749833599624, 4177248169448418, 45949729863637696, 505447028499424842

Offset: 0

Mohammad K. Azarian, Jan 24 2009

Index entries for linear recurrences with constant coefficients, signature (14,-35,22).

Cf. A074501, A074600, A005057, A005056, A099393, A155588, A155590, A155592, A155593, A155594.

a(n)=11^n+2^n-1 \\ Charles R Greathouse IV, Jun 11 2015

G.f.: 1/(1-11*x)+1/(1-2*x)-1/(1-x). E.g.f.: e^(11*x)+e^(2*x)-e^x.

a(n)=13*a(n-1)-22*a(n-2)-10 with a(0)=1, a(1)=12 - Vincenzo Librandi, Jul 21 2010

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

A248216 a(n) = 6^n - 2^n.

Original entry on oeis.org

0, 4, 32, 208, 1280, 7744, 46592, 279808, 1679360, 10077184, 60465152, 362795008, 2176778240, 13060685824, 78364147712, 470184951808, 2821109841920, 16926659313664, 101559956406272, 609359739486208, 3656158439014400, 21936950638280704

Offset: 0

Vincenzo Librandi, Oct 04 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-12).

Sequences of the form k^n - 2^n: A001047 (k=3), A020522 (k=4), A005057 (k=5), this sequence (k=6), A190540 (k=7), A248217 (k=8), A191465 (k=9), A060458 (k=10), A139740 (k=11).

Cf. A000079, A000400, A024023.

Magma
```
[6^n-2^n: n in [0..25]];
```

Table[6^n - 2^n, {n, 0, 25}] (* or *) CoefficientList[Series[4x/((1-2x)(1-6x)), {x, 0, 30}], x]
LinearRecurrence[{8,-12},{0,4},30] (* Harvey P. Dale, Dec 21 2019 *)

[2^n*(3^n -1) for n in (0..25)] # G. C. Greubel, Feb 09 2021

G.f.: 4*x/((1-2*x)*(1-6*x)).
a(n) = 8*a(n-1) - 12*a(n-2).
a(n) = 2^n*(3^n - 1) = A000079(n) * A024023(n).
E.g.f.: exp(6*x) - exp(2*x) = 2*exp(4*x)*sinh(2*x). - G. C. Greubel, Feb 09 2021
a(n) = 4*A016129(n-1). - R. J. Mathar, Mar 10 2022
a(n) = A000400(n) - A000079(n). - Bernard Schott, Mar 27 2022

A248338 a(n) = 10^n - 4^n.

Original entry on oeis.org

0, 6, 84, 936, 9744, 98976, 995904, 9983616, 99934464, 999737856, 9998951424, 99995805696, 999983222784, 9999932891136, 99999731564544, 999998926258176, 9999995705032704, 99999982820130816, 999999931280523264, 9999999725122093056, 99999998900488372224

Offset: 0

Vincenzo Librandi, Oct 05 2014

G. C. Greubel, Table of n, a(n) for n = 0..995
Index entries for linear recurrences with constant coefficients, signature (14,-40).

Cf. A000079, A000302, A005057, A011557, A016157.

Cf. similar sequences listed in A248337.

Magma
```
[10^n-4^n: n in [0..30]];
```

Table[10^n - 4^n, {n, 0, 30}] (* or *)
CoefficientList[Series[(6 x)/((1-4 x)(1-10 x)), {x, 0, 30}], x]

vector(20,n,10^(n-1)-4^(n-1)) \\ Derek Orr, Oct 05 2014

def A248338(n): return pow(10,n) - pow(4,n)
print([A248338(n) for n in range(41)]) # G. C. Greubel, Nov 13 2024

G.f.: 6*x/((1-4*x)*(1-10*x)).
a(n) = 14*a(n-1) - 40*a(n-2).
a(n) = 2^n*(5^n - 2^n) = A000079(n) * A005057(n) = A011557(n) - A000302(n).
a(n+1) = 6*A016157(n). [Bruno Berselli, Oct 05 2014]
E.g.f.: 2*exp(7*x)*sinh(3*x). - G. C. Greubel, Nov 13 2024

A356198 Number of edge covers in the n-book graph.

Original entry on oeis.org

5, 41, 233, 1217, 6185, 31121, 155993, 780737, 3905225, 19529201, 97652153, 488273057, 2441389865, 12206998481, 61035090713, 305175650177, 1525878644105, 7629394006961, 38146971607673, 190734861184097, 953674312211945, 4768371573642641

Offset: 1

Eric W. Weisstein, Jul 29 2022

Eric Weisstein's World of Mathematics, Book Graph
Eric Weisstein's World of Mathematics, Edge Cover
Index entries for linear recurrences with constant coefficients, signature (8,-17,10).

Cf. A005057, A016127.

Table[2 5^n - 2^(n + 1) - 1, {n, 20}]
LinearRecurrence[{8, -17, 10}, {5, 41, 233}, 20]
CoefficientList[Series[(10 x^2 - x - 5)/((x - 1) (2 x - 1) (5 x - 1)), {x, 0, 20}], x]

a(n) = 2*5^n - 2^(n + 1) - 1.
G.f.: x*(10*x^2-x-5)/((x-1)*(2*x-1)*(5*x-1)).
a(n) = 8*a(n-1) - 17*a(n-2) + 10*a(n-3).
a(n) = 2*A005057(n) - 1 = 6*A016127(n-1) - 1. - Hugo Pfoertner, Jul 29 2022

cp's OEIS Frontend

A155595 11^n+2^n-1.

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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.

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A248216 a(n) = 6^n - 2^n.

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A248338 a(n) = 10^n - 4^n.

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A356198 Number of edge covers in the n-book graph.

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