A196662 -id:A196662 - cp's OEIS frontend

A196663 Expansion of g.f. (1-4x)/(1-13x).

1, 9, 117, 1521, 19773, 257049, 3341637, 43441281, 564736653, 7341576489, 95440494357, 1240726426641, 16129443546333, 209682766102329, 2725875959330277, 35436387471293601, 460673037126816813, 5988749482648618569, 77853743274432041397, 1012098662567616538161

Offset: 0

Philippe Deléham, Oct 05 2011

Index entries for linear recurrences with constant coefficients, signature (13).

Cf. A002001, A193722, A196661, A196662.

a(0) = 1, a(n) = 9*13^(n-1) for n>0.
a(n) = Sum_{k=0..n} A193722(n,k)*4^k.
From Elmo R. Oliveira, Mar 18 2025: (Start)
E.g.f.: (9*exp(13*x) + 4)/13.
a(n) = 13*a(n-1). (End)

A196664 Expansion of g.f. (1-5x)/(1-16x).

Original entry on oeis.org

1, 11, 176, 2816, 45056, 720896, 11534336, 184549376, 2952790016, 47244640256, 755914244096, 12094627905536, 193514046488576, 3096224743817216, 49539595901075456, 792633534417207296, 12682136550675316736, 202914184810805067776, 3246626956972881084416

Offset: 0

Philippe Deléham, Oct 05 2011

Index entries for linear recurrences with constant coefficients, signature (16).

Cf. A001025, A193722, A196661, A196662, A196663.

Join[{1},Table[11 16^(n-1),{n,20}]] (* Harvey P. Dale, Oct 20 2011 *)

a(0) = 1, a(n) = 11*16^(n-1) for n>0.
a(n) = Sum_{k=0..n} A193722(n,k)*5^k.
From Elmo R. Oliveira, Mar 18 2025: (Start)
E.g.f.: (11*exp(16*x) + 5)/16.
a(n) = 16*a(n-1). (End)

A196665 Expansion of g.f. (1-6x)/(1-19x).

Original entry on oeis.org

1, 13, 247, 4693, 89167, 1694173, 32189287, 611596453, 11620332607, 220786319533, 4194940071127, 79703861351413, 1514373365676847, 28773093947860093, 546688785009341767, 10387086915177493573, 197354651388372377887, 3749738376379075179853

Offset: 0

Philippe Deléham, Oct 05 2011

Index entries for linear recurrences with constant coefficients, signature (19).

Cf. A002001, A193722, A196661, A196662, A196663, A196664.

a(0) = 1, a(n) = 13*19^(n-1) for n>0.
a(n) = Sum_{k=0..n} A193722(n,k)*6^k.
From Elmo R. Oliveira, Mar 18 2025: (Start)
E.g.f.: (13*exp(19*x) + 6)/19.
a(n) = 19*a(n-1) for n > 1. (End)

A196666 Expansion of g.f. (1 - 7x)/(1 - 22x).

Original entry on oeis.org

1, 15, 330, 7260, 159720, 3513840, 77304480, 1700698560, 37415368320, 823138103040, 18109038266880, 398398841871360, 8764774521169920, 192825039465738240, 4242150868246241280, 93327319101417308160, 2053201020231180779520, 45170422445085977149440

Offset: 0

Philippe Deléham, Oct 05 2011

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

Cf. A002001, A193722, A196661, A196662, A196663, A196664, A196665.

a(n)=if(n,15*22^n--,1) \\ Charles R Greathouse IV, Oct 06 2011

a(0) = 1, a(n) = 15*22^(n-1) for n>0.
a(n) = Sum_{k=0..n} A193722(n,k)*7^k.
From Elmo R. Oliveira, Mar 25 2025: (Start)
E.g.f.: (15*exp(22*x) + 7)/22.
a(n) = 22*a(n-1) for n > 1. (End)

A196512 Expansion of g.f. (1-9x)/(1-28x).

Original entry on oeis.org

1, 19, 532, 14896, 417088, 11678464, 326996992, 9155915776, 256365641728, 7178237968384, 200990663114752, 5627738567213056, 157576679881965568, 4412147036695035904, 123540117027461005312, 3459123276768908148736, 96855451749529428164608, 2711952648986823988609024

Offset: 0

Philippe Deléham, Oct 05 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (28).

Cf. A002001, A193722, A196661, A196662, A196663, A196664, A196665, A196666, A196676.

[1] cat [19*28^(n-1) : n in [1..15]]; // Vincenzo Librandi, Oct 06 2011

Join[{1},19*28^Range[0,20]] (* Harvey P. Dale, Dec 06 2015 *)

a(n)=if(n,19*28^(n-1),1)  \\ M. F. Hasler, Oct 06 2011

a(n) = 19*28^(n-1) for n > 0, a(0)=1.
a(n) = Sum_{k=0..n} A193722(n,k)*9^k.
From Elmo R. Oliveira, Mar 25 2025: (Start)
E.g.f.: (19*exp(28*x) + 9)/28.
a(n) = 28*a(n-1) for n > 1. (End)

A196676 Expansion of g.f. (1-8x)/(1-25x).

Original entry on oeis.org

1, 17, 425, 10625, 265625, 6640625, 166015625, 4150390625, 103759765625, 2593994140625, 64849853515625, 1621246337890625, 40531158447265625, 1013278961181640625, 25331974029541015625, 633299350738525390625, 15832483768463134765625, 395812094211578369140625, 9895302355289459228515625

Offset: 0

Philippe Deléham, Oct 05 2011

Index entries for linear recurrences with constant coefficients, signature (25).

Cf. A002001, A193722, A196661, A196662, A196663, A196664, A196665, A196666.

CoefficientList[Series[(1-8x)/(1-25x),{x,0,30}],x] (* Harvey P. Dale, Apr 29 2014 *)

a(0) = 1, a(n) = 17*25^(n-1) for n>0.
a(n) = Sum_{k=0..n} A193722(n,k)*8^k.
From Elmo R. Oliveira, Mar 25 2025: (Start)
E.g.f.: (17*exp(25*x) + 8)/25.
a(n) = 25*a(n-1) for n > 1. (End)

More terms from Elmo R. Oliveira, Mar 25 2025

A346178 Expansion of (1-2x)/(1-10x).

Original entry on oeis.org

1, 8, 80, 800, 8000, 80000, 800000, 8000000, 80000000, 800000000, 8000000000, 80000000000, 800000000000, 8000000000000, 80000000000000, 800000000000000, 8000000000000000, 80000000000000000, 800000000000000000, 8000000000000000000, 80000000000000000000

Offset: 0

Felix Fröhlich, Jul 09 2021

Index entries for linear recurrences with constant coefficients, signature (10).

Cf. expansion of (1-k*x)/(1-10*x) A011557 (k=0), A196662 (k=3), A090019 (k=4), A093143 (k=5), A093141 (k=6), A093138 (k=7), A093136 (k=8).

PARI
```
Vec((1-2*x)/(1-10*x) + O(x^20))
```

a(n) = 8*10^(n-1), n>0.

E.g.f.: (1 + 4*exp(10*x))/5. - Stefano Spezia, Jul 09 2021

cp's OEIS Frontend

A196663 Expansion of g.f. (1-4*x)/(1-13*x).

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A196664 Expansion of g.f. (1-5*x)/(1-16*x).

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A196665 Expansion of g.f. (1-6*x)/(1-19*x).

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A196666 Expansion of g.f. (1 - 7*x)/(1 - 22*x).

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A196512 Expansion of g.f. (1-9*x)/(1-28*x).

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A196676 Expansion of g.f. (1-8*x)/(1-25*x).

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A346178 Expansion of (1-2*x)/(1-10*x).

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Formula

A196663 Expansion of g.f. (1-4x)/(1-13x).

A196664 Expansion of g.f. (1-5x)/(1-16x).

A196665 Expansion of g.f. (1-6x)/(1-19x).

A196666 Expansion of g.f. (1 - 7x)/(1 - 22x).

A196512 Expansion of g.f. (1-9x)/(1-28x).

A196676 Expansion of g.f. (1-8x)/(1-25x).

A346178 Expansion of (1-2x)/(1-10x).