A053108 -id:A053108 - cp's OEIS frontend

A053112 Expansion of (-1 + 1/(1-9x)^9)/(81x); related to A053108.

1, 45, 1485, 40095, 938223, 19702683, 379980315, 6839645670, 116273976390, 1883638417518, 29282015399598, 439230230993970, 6385731819835410, 90312492880529370, 1246312401751305306, 16825217423642621631

Offset: 0

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (81, -2916, 61236, -826686, 7440174, -44641044, 172186884, -387420489, 387420489).

Cf. A053108, A053110 - A053113.

Without signs: A078812. With zeros: A049310. Cf. A008310 (T(n, x)), A008312 (U(n, x)).

[9^(n-1)*Binomial(n+9, 8): n in [0..30]]; // G. C. Greubel, Aug 16 2018

CoefficientList[Series[(-1+1/(1-9*x)^9)/(81*x),{x,0,30}],x] (* or *) LinearRecurrence[{81,-2916,61236,-826686,7440174,-44641044,172186884,-387420489,387420489}, {1,45,1485,40095,938223,19702683, 379980315, 6839645670,116273976390},20] (* Harvey P. Dale, Apr 27 2013 *)
Table[9^(n - 1)*Binomial[n + 9, 8], {n, 0, 30}] (* G. C. Greubel, Aug 16 2018 *)

vector(30,n,n--; 9^(n-1)*binomial(n+9, 8)) \\ G. C. Greubel, Aug 16 2018

G.f.: (-1 + 1/(1-9*x)^9)/(81*x).
a(n) = 9^(n-1)*binomial(n+9, 8).
a(0)=1, a(1)=45, a(2)=1485, a(3)=40095, a(4)=938223, a(5)=19702683, a(6)=379980315, a(7)=6839645670, a(8)=116273976390, a(n)=81*a(n-1)- 2916*a(n-2)+ 61236*a(n-3)- 826686*a(n-4)+ 7440174*a(n-5)- 44641044*a(n-6)+ 172186884*a(n-7)- 387420489*a(n-8)+ 387420489*a(n-9). - Harvey P. Dale, Apr 27 2013

A196221 Binomial(n+10, 10)*9^n.

Original entry on oeis.org

1, 99, 5346, 208494, 6567561, 177324147, 4255779528, 93019181112, 1883638417518, 35789129932842, 644204338791156, 11068601821048044, 182631930047292726, 2908062270753045714, 44867246463046991016, 673008696945704865240, 9842752192830933654135, 140693457815171581056165, 1969708409412402134786310

Offset: 0

Vincenzo Librandi, Oct 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A053108, A081139, A173000, A173187, A173188, A173191, A173192.

Magma
```
[Binomial(n+10, 10)*9^n: n in [0..20]];
```

a(n) = C(n+10, 10)*9^n.

G.f. -1 / (9*x-1)^11 . - R. J. Mathar, Oct 13 2011

A197194 a(n) = binomial(n+9, 9)*9^n.

Original entry on oeis.org

1, 90, 4455, 160380, 4691115, 118216098, 2659862205, 54717165360, 1046465787510, 18836384175180, 322102169395578, 5270762771927640, 83014513657860330, 1264374900327411180, 18694686026269579590, 269203478778281946096, 3785673920319589866975, 52108688079693178168950, 703467289075857905280825

Offset: 0

Vincenzo Librandi, Oct 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A053108, A081139, A173000, A173187, A173188, A173191, A173192

Magma
```
[Binomial(n+9, 9)*9^n: n in [0..20]];
```

Table[Binomial[n+9,9]9^n,{n,0,20}] (* Harvey P. Dale, Feb 22 2020 *)

A197194_list, m, k = [], [1]*10, 1
for _ in range(10**2):
    A197194_list.append(k*m[-1])
    k *= 9
    for i in range(9):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

a(n) = C(n + 9, 9)*9^n.
G.f.: 1 / (9*x-1)^10 . - R. J. Mathar, Oct 13 2011
From Amiram Eldar, Apr 17 2022: (Start)
Sum_{n>=0} 1/a(n) = 1358954496*log(9/8) - 44817299757/280.
Sum_{n>=0} (-1)^n/a(n) = 8100000000*log(10/9) - 47791529847/56. (End)

A293270 a(n) = n^nbinomial(2n-1, n).

Original entry on oeis.org

1, 1, 12, 270, 8960, 393750, 21555072, 1413199788, 107961384960, 9418192087590, 923780000000000, 100633991211229476, 12055263261877075968, 1575041416811693275900, 222887966509090352332800, 33962507149515380859375000, 5543988061027763016035205120

Offset: 0

Ilya Gutkovskiy, Oct 04 2017

nonn

The n-th term of the n-fold convolution of the powers of n.

Cf. A000312, A001700, A088218.

Cf. A001787, A027472, A036071, A036084, A036226, A038846, A053107, A053108, A053109.

Join[{1}, Table[n^n Binomial[2 n - 1, n], {n, 1, 16}]]
Join[{1}, Table[(-1)^n n^n Binomial[-n, n], {n, 1, 16}]]
Table[SeriesCoefficient[1/(1 - n x)^n, {x, 0, n}], {n, 0, 16}]

a(n) = n^n*binomial(2*n-1, n); \\ Altug Alkan, Oct 04 2017

a(n) = [x^n] 1/(1 - n*x)^n.

a(n) ~ 2^(2*n-1)*n^n/sqrt(Pi*n).

cp's OEIS Frontend

A053112 Expansion of (-1 + 1/(1-9*x)^9)/(81*x); related to A053108.

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A196221 Binomial(n+10, 10)*9^n.

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A197194 a(n) = binomial(n+9, 9)*9^n.

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Formula

A293270 a(n) = n^n*binomial(2*n-1, n).

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A053112 Expansion of (-1 + 1/(1-9x)^9)/(81x); related to A053108.

A293270 a(n) = n^nbinomial(2n-1, n).