A016129 -id:A016129 - cp's OEIS frontend

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

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

A016173 Expansion of 1/((1-6x)(1-10*x)).

Original entry on oeis.org

1, 16, 196, 2176, 23056, 238336, 2430016, 24580096, 247480576, 2484883456, 24909300736, 249455804416, 2496734826496, 24980408958976, 249882453753856, 2499294722523136, 24995768335138816, 249974610010832896, 2499847660064997376, 24999085960389984256, 249994515762339905536, 2499967094574039433216

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..990
Index entries for linear recurrences with constant coefficients, signature (16,-60).

Cf. A016129.

[(10^(n+1) - 6^(n+1))/4: n in [0..40]]; // G. C. Greubel, Nov 13 2024

Table[(10^(n+1) - 6^(n+1))/4, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2011 *)
CoefficientList[Series[1/((1-6x)(1-10x)), {x,0,40}], x]  (* Harvey P. Dale, Mar 12 2011 *)

def A016173(n): return (pow(10,n+1) - pow(6,n+1))//4
print([A016173(n) for n in range(41)]) # G. C. Greubel, Nov 13 2024

a(n) = (10^(n+1) - 6^(n+1))/4. - Al Hakanson (hawkuu(AT)gmail.com), Dec 31 2008
a(n) = 16*a(n-1) - 60*a(n-2). - Philippe Deléham, Jan 01 2009
a(n) = 10*a(n-1) + 6^n, a(0)=1. - Vincenzo Librandi, Feb 09 2011
E.g.f.: (1/2)*(5*exp(10*x) - 3*exp(6*x)). - G. C. Greubel, Nov 13 2024

More terms added by G. C. Greubel, Nov 13 2024

A089274 Fifth column of the Legendre-Stirling triangle A071951.

Original entry on oeis.org

1, 70, 3192, 121424, 4203824, 137922336, 4380918784, 136378114048, 4191383868672, 127754693361152, 3873052857829376, 117001609550671872, 3526270158211870720, 106112798944292282368, 3189880933574260359168

Offset: 0

Wolfdieter Lang, Nov 07 2003

This is the fifth member of the family A000079 (powers of 2), A016129, A016309, A071952, etc.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (70,-1708,17544,-72000,86400).

Cf. A016129, A016309, A071951, A071952, A089278, A089500.

Cf. A000079 (powers of 2).

[(16875*(6*5)^n - 20000*(5*4)^n + 6048*(4*3)^n - 405*(3*2)^n + 2*(2*1)^n)/2520: n in [0..20]]; // Vincenzo Librandi, Sep 02 2011

Table[2^(n-3)*(5*(15)^(n+3) -2*(10)^(n+4) +28*6^(n+3) -5*3^(n+4) +2)/315, {n,0,30}] (* G. C. Greubel, Nov 10 2024 *)

def A089274(n): return 2^n*(5*(15)^(n+3) -2*(10)^(n+4) +28*6^(n+3) -5*3^(n+4) +2)//2520
[A089274(n) for n in range(31)] # G. C. Greubel, Nov 10 2024

G.f.: 1/((1-2*1*x)*(1-3*2*x)*(1-4*3*x)*(1-5*4*x)*(1-6*5*x)).
a(n) = (16875*(6*5)^n -20000*(5*4)^n +6048*(4*3)^n -405*(3*2)^n +2*(2*1)^n)/2520.
a(n) = A071951(n+5, 5), n>=0.
a(n) = det(|ps(i+5,j+4)|, 1 <= i,j <= n), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467). [Mircea Merca, Apr 06 2013]
E.g.f.: (1/2520)*(2*exp(2*x) - 405*exp(6*x) + 6048*exp(12*x) - 20000*exp(20*x) + 16875*exp(30*x)). - G. C. Greubel, Nov 10 2024

A100851 Triangle read by rows: T(n,k) = 2^n * 3^k, 0 <= k <= n, n >= 0.

Original entry on oeis.org

1, 2, 6, 4, 12, 36, 8, 24, 72, 216, 16, 48, 144, 432, 1296, 32, 96, 288, 864, 2592, 7776, 64, 192, 576, 1728, 5184, 15552, 46656, 128, 384, 1152, 3456, 10368, 31104, 93312, 279936, 256, 768, 2304, 6912, 20736, 62208, 186624, 559872, 1679616, 512, 1536, 4608, 13824, 41472, 124416, 373248, 1119744, 3359232, 10077696

Offset: 0

Reinhard Zumkeller, Nov 20 2004

			From _Stefano Spezia_, Apr 28 2024: (Start)
Triangle begins:
   1;
   2,  6;
   4, 12,  36;
   8, 24,  72, 216;
  16, 48, 144, 432, 1296;
  32, 96, 288, 864, 2592, 7776;
  ...
(End)

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A000079, A000400, A001021, A003586, A005010, A007283, A016129.
Cf. A016149, A051958, A053524, A100852, A248337.
Cf. A065333(T(n, k))=1.

A100851:= func< n,k | 2^n*3^k >;
[A100851(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 11 2024

A100851[n_, k_]= 2^n*3^k;
Table[A100851[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 11 2024 *)

def A100851(n,k): return 2^n*3^k
flatten([[A100851(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 11 2024

T(n,0) = A000079(n).
T(n,1) = A007283(n) for n>0.
T(n,2) = A005010(n) for n>1.
T(n,n) = A000400(n) = A100852(n,n).
Sum_{k=0..n} T(n, k) = A016129(n).
T(2*n, n) = A001021(n). - Reinhard Zumkeller, Mar 04 2006
G.f.: 1/((1 - 2*x)*(1 - 6*x*y)). - Stefano Spezia, Apr 28 2024
From G. C. Greubel, Nov 11 2024: (Start)
Sum_{k=0..n} (-1)^k*T(n, k) = A053524(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = (1/2)*((1-(-1)^n)*A248337((n+1)/2) + (1 + (-1)^n)*A016149(n/2)).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1/2)*(-1)^floor(n/2)*( (1+(-1)^n) *A051958((n+2)/2) + 2*(1-(-1)^n)*A051958((n+1)/2)). (End)
Sum_{n>=0, k=0..n} 1/T(n,k) = 12/5. - Amiram Eldar, May 12 2025

A016200 Expansion of g.f. 1/((1-x)(1-2x)(1-6x)).

Original entry on oeis.org

1, 9, 61, 381, 2317, 13965, 83917, 503757, 3023053, 18139341, 108838093, 653032653, 3918204109, 23509241037, 141055478989, 846332939469, 5077997767885, 30467986869453, 182807921741005, 1096847531494605, 6581085191064781, 39486511150582989, 236919066911886541, 1421514401488096461

Offset: 0

N. J. A. Sloane

Muniru A Asiru, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (9,-20,12).

Cf. A003464, A016129, A241275.

List([0..100],n->(9*6^n-5*2^n+1)/5); # Muniru A Asiru, Feb 06 2018

seq((9*6^n-5*2^n+1)/5, n=0..100); # Muniru A Asiru, Feb 06 2018

CoefficientList[Series[1/((1-x)(1-2x)(1-6x)),{x,0,30}],x] (* or *) LinearRecurrence[{9,-20,12},{1,9,61},30] (* Harvey P. Dale, Aug 23 2025 *)

a(n) = (9*6^n - 5*2^n + 1)/5. - Bruno Berselli, Feb 09 2011
a(0)=1, a(n) = 6*a(n-1) + 2^(n+1) - 1. - Vincenzo Librandi, Feb 09 2011
a(n) = Sum_{k=0..n} 2^(n-1-k)*(3^(n+1-k) - 1). - J. M. Bergot, Feb 06 2018
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(x)*(9*exp(5*x) - 5*exp(x) + 1)/5.
a(n) = A016129(n+1) - A003464(n+2) = A241275(n+2)/5.
a(n) = 9*a(n-1) - 20*a(n-2) + 12*a(n-3). (End)

A016304 Expansion of 1/((1-2x)(1-6x)(1-7*x)).

Original entry on oeis.org

1, 15, 157, 1419, 11869, 94731, 733069, 5551323, 41378557, 304766187, 2224062061, 16112628987, 116053574365, 831966057483, 5941308640333, 42294437942811, 300292730428093, 2127439102098219, 15044413649559085

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 (15,-68,84).

Cf. A016129, A016130, A016311, A016316, A016321, A016325. - Zerinvary Lajos, Jun 05 2009

[ n eq 1 select 1 else n eq 2 select 15 else n eq 3 select 157 else 15*Self(n-1)-68*Self(n-2) +84*Self(n-3): n in [1..20] ]; // Vincenzo Librandi, Aug 25 2011

CoefficientList[Series[1/((1-2x)(1-6x)(1-7x)), {x, 0, 30}], x] (* or *) LinearRecurrence[{15, -68, 84}, {1, 15, 157}, 30]

Vec(1/((1-2*x)*(1-6*x)*(1-7*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

[(7^n - 2^n)/5-(6^n - 2^n)/4 for n in range(2,21)] # Zerinvary Lajos, Jun 05 2009

a(n) = (7^(n+2) - 2^(n+2))/5-(6^(n+2) - 2^(n+2))/4. - Zerinvary Lajos, Jun 05 2009 [corrected by Joerg Arndt, Aug 25 2011]
From Vincenzo Librandi, Aug 25 2011: (Start)
a(n) = 15*a(n-1) - 68*a(n-2) + 84*a(n-3) for n > 2;
a(n) = 13*a(n-1) - 42*a(n-2) + 2^n for n > 1. (End)
E.g.f.: exp(2*x)*(1 - 45*exp(4*x) + 49*exp(5*x))/5. - Stefano Spezia, Aug 25 2025

A089511 Triangle of integers used to compute column sequences of array A078739 ((2,2)-Stirling2).

Original entry on oeis.org

1, -1, 3, 1, -6, 6, -1, 27, -108, 100, 1, -36, 216, -400, 225, -1, 135, -2160, 10000, -16875, 9261, 1, -162, 3240, -20000, 50625, -55566, 21952, -1, 567, -27216, 350000, -1771875, 4084101, -4302592, 1679616, 1, -648, 36288, -560000, 3543750, -10890936, 17210368, -13436928

Offset: 2

Wolfdieter Lang, Dec 01 2003

The k-th column sequence (without leading zeros) of A078739 is for even k: sum(a(k,m)*((m+1)*m)^n,m=1..k-1)/D(k) and for odd k it is: ((k^2-1)/2)*sum(a(k,m)*((m+1)*m)^n,m=1..k-1)/D(k), where D(k) := A089512(k) and n>=0, k>=2.

			[1]; [ -1,3]; [1,-6,6]; [ -1,27,-108,100]; ...
a(2,1)=A089512(2)*A089275(1,0)*A089278(1,1)/A089500(1)=1*1*1/1=1;
a(3,2)=A089512(3)*A089276(1,0)*A089278(2,2)/A089500(2)=2*1*3/2=3.
a(4,3)=1*(1+18/(4*3))*24/10 =6; a(5,4)= 18*(1+8/(5*4))*2500/630=100.
k=2 column sequence of A078739 is (1*(2*1)^n)/1 = 2^n. k=3: 4*(-1*(2*1)^n + 3*(3*2)^n)/2 (see A016129).

W. Lang, First 10 rows.

a(n, m) triangle 2<=n, 1<= m <= (n-1), else 0, with a(2*k, m)= D(2*k)*sum(A089275(k, p)/((m+1)*m)^p, p=0..k-1)*A089278(2*k-1, m)/A089500(2*k-1) and a(2*k+1, m)= D(2*k+1)*sum(A089276(k, p)/((m+1)*m)^p, p=0..k-1)*A089278(2*k, m)/A089500(2*k), where D(n) := A089512(n).

A144843 a(n) = (6^n - 2^n)^2 / 16.

Original entry on oeis.org

1, 64, 2704, 102400, 3748096, 135675904, 4893282304, 176265625600, 6346852335616, 228502162898944, 8226263614357504, 296147719133593600, 10661344637077159936, 383808727914259677184, 13817118056668205154304, 497416296261117961830400, 17906987220053014721069056

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Sep 22 2008

G. C. Greubel, Table of n, a(n) for n = 1..630
Index entries for linear recurrences with constant coefficients, signature (52,-624,1728).

Cf. A016129.

[4^(n-2)*(3^n-1)^2: n in [1..30]]; // G. C. Greubel, Oct 03 2024

Table[(6^n-2^n)^2/16, {n,20}] (* Harvey P. Dale, Apr 15 2020 *)

[4^(n-2)*(3^n-1)^2 for n in range(1,31)] # G. C. Greubel, Oct 03 2024

From R. J. Mathar, Sep 24 2008: (Start)
a(n) = 81*36^(n-2) + 4^(n-2) - 18*12^(n-2).
G.f.: x*(1+12*x)/((1-4*x)*(1-12*x)*(1-36*x)). (End)
a(n) = A016129(n-1)^2. - Philippe Deléham, Nov 26 2008
a(n) = 4^(n-2) * (3^n - 1)^2. - Harvey P. Dale, Apr 15 2020
E.g.f.: (1/16)*exp(4*x)*(1 - 2*exp(8*x) + exp(32*x)). - G. C. Greubel, Oct 03 2024

More terms from R. J. Mathar, Sep 24 2008

A026542 Expansion of 1/((1-2x)(1-6x)(1-7x)(1-11*x)).

Original entry on oeis.org

1, 26, 443, 6292, 81081, 986622, 11585911, 132996344, 1504338341, 16852487938, 187601429379, 2079728352156, 22993065448081, 253755685986374, 2797253854490447, 30812086837337728, 339233247941143101, 3733693166454672330, 41085669244650954715

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..950
Index entries for linear recurrences with constant coefficients, signature (26,-233,832,-924).

Cf. A000079, A016129, A016304, A026543.

[(1/180)*(11^(n+3) -9*7^(n+3) +9*6^(n+3) -2^(n+3)): n in [0..30]]; // G. C. Greubel, Apr 09 2022

CoefficientList[Series[1/((1-2x)(1-6x)(1-7x)(1-11x)),{x,0,30}],x] (* Harvey P. Dale, May 27 2019 *)

[(1/180)*(11^(n+3) -9*7^(n+3) +9*6^(n+3) -2^(n+3)) for n in (0..30)] # G. C. Greubel, Apr 09 2022

a(n) = (1/180)*(11^(n+3) -9*7^(n+3) +9*6^(n+3) -2^(n+3)). - R. J. Mathar, Jun 23 2013

E.g.f.: (1/180)*(-8*exp(2*x) + 1944*exp(6*x) - 3087*exp(7*x) + 1331*exp(11*x)). - G. C. Greubel, Apr 09 2022

A026543 Expansion of 1/((1-2x)(1-6x)(1-7x)(1-12*x)).

Original entry on oeis.org

1, 27, 481, 7191, 98161, 1272663, 16005025, 197611623, 2412718033, 29257382583, 353312653057, 4255864465671, 51186427162417, 615069092006487, 7386770412718177, 88683539390560935, 1064502765417159313

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..915
Index entries for linear recurrences with constant coefficients, signature (27,-248,900,-1008)

Cf. A000079, A016129, A016304, A026542.

[(-2^n +225*6^n -7^(n+3) +12^(n+2))/25: n in [0..30]]; // G. C. Greubel, Apr 09 2022

CoefficientList[Series[1/((1-2x)(1-6x)(1-7x)(1-12x)),{x,0,30}],x] (* Harvey P. Dale, Apr 18 2019 *)

[(-2^n +225*6^n -7^(n+3) +12^(n+2))/25 for n in (0..30)] # G. C. Greubel, Apr 09 2022

a(n) = (-2^n + 225*6^n - 7^(n+3) + 12^(n+2))/25. - R. J. Mathar, Jun 23 2013

E.g.f.: 1/25 (-exp(2*x) + 225*exp(6*x) - 343*exp(7*x) + 144*exp(12*x)). - G. C. Greubel, Apr 09 2022

cp's OEIS Frontend

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

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A016173 Expansion of 1/((1-6*x)*(1-10*x)).

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A089274 Fifth column of the Legendre-Stirling triangle A071951.

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A100851 Triangle read by rows: T(n,k) = 2^n * 3^k, 0 <= k <= n, n >= 0.

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A016200 Expansion of g.f. 1/((1-x)*(1-2*x)*(1-6*x)).

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A016304 Expansion of 1/((1-2*x)*(1-6*x)*(1-7*x)).

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A089511 Triangle of integers used to compute column sequences of array A078739 ((2,2)-Stirling2).

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A144843 a(n) = (6^n - 2^n)^2 / 16.

A016173 Expansion of 1/((1-6x)(1-10*x)).

A016200 Expansion of g.f. 1/((1-x)(1-2x)(1-6x)).

A016304 Expansion of 1/((1-2x)(1-6x)(1-7*x)).

A026542 Expansion of 1/((1-2x)(1-6x)(1-7x)(1-11*x)).

A026543 Expansion of 1/((1-2x)(1-6x)(1-7x)(1-12*x)).