A173187 -id:A173187 - cp's OEIS frontend

A173188 a(n) = binomial(n + 5, 5)*9^n.

1, 54, 1701, 40824, 826686, 14880348, 245525742, 3788111448, 55401129927, 775615818978, 10470813556203, 137072468372112, 1747673971744428, 21778706417122872, 266011342666286508, 3192136111995438096, 37707107822946112509, 439176902879019428046

Offset: 0

Zerinvary Lajos, Feb 12 2010

Number of n-permutations (n>=5) of 10 objects p, r, q, u, v, w, z, x, y, z with repetition allowed, containing exactly five (5) u's.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (54,-1215,14580,-98415,354294,-531441).

Cf. A081139, A173000, A173187.

[Binomial(n+5, 5)*9^n: n in [0..20]]; // Vincenzo Librandi, Oct 13 2011

Table[Binomial[n + 5, 5]*9^n, {n, 0, 20}]

a(n) = C(n + 5, 5)*9^n.
From Amiram Eldar, Aug 28 2022: (Start)
Sum_{n>=0} 1/a(n) = 184320*log(9/8) - 86835/4.
Sum_{n>=0} (-1)^n/a(n) = 450000*log(10/9) - 189645/4. (End)

A173191 a(n) = binomial(n + 6, 6)*9^n.

Original entry on oeis.org

1, 63, 2268, 61236, 1377810, 27280638, 491051484, 8207574804, 129269303163, 1939039547445, 27922169483208, 388371993720984, 5243021915233284, 68965903654222428, 886704475554288360, 11172476391984033336, 138259395350802412533, 1683511461036241140843

Offset: 0

Zerinvary Lajos, Feb 12 2010

Number of n-permutations (n>=6) of 10 objects p, r, q, u, v, w, z, x, y, z with repetition allowed, containing exactly six (6) u's.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (63,-1701,25515,-229635,1240029,-3720087,4782969).

Cf. A081139, A173000, A173187, A173188.

[Binomial(n+6, 6)*9^n: n in [0..20]]; // Vincenzo Librandi, Oct 13 2011

A173191:=n->binomial(n+6,6)*9^n: seq(A173191(n), n=0..25); # Wesley Ivan Hurt, Jul 24 2017

Table[Binomial[n + 6, 6]*9^n, {n, 0, 20}]

a(n) = C(n + 6, 6)*9^n.
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 1042074/5 - 1769472*log(9/8).
Sum_{n>=0} (-1)^n/a(n) = 5400000*log(10/9) - 2844729/5. (End)

A173192 a(n) = binomial(n + 7, 7)*9^n.

Original entry on oeis.org

1, 72, 2916, 87480, 2165130, 46766808, 911952756, 16415149608, 277005649635, 4432090394160, 67810983030648, 998670840996816, 14231059484204628, 197045439012064080, 2660113426662865080, 35113497231949819056, 454280870438350784037, 5772039294981398197176

Offset: 0

Zerinvary Lajos, Feb 12 2010

Number of n-permutations (n>=7) of 10 objects p, r, q, u, v, w, z, x, y, z with repetition allowed, containing exactly 7 u's.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (72,-2268,40824,-459270,3306744,-14880348,38263752,-43046721).

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

[Binomial(n+7, 7)*9^n: n in [0..20]]; // Vincenzo Librandi, Oct 13 2011

A173192:=n->binomial(n+7,7)*9^n: seq(A173192(n), n=0..25); # Wesley Ivan Hurt, Jul 24 2017

Table[Binomial[n + 7, 7]*9^n, {n, 0, 20}]

a(n) = C(n + 7, 7)*9^n.
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 16515072*log(9/8) - 19451943/10.
Sum_{n>=0} (-1)^n/a(n) = 63000000*log(10/9) - 13275423/2. (End)

A317052 Triangle read by rows: T(0,0) = 1; T(n,k) = 9*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 9, 81, 1, 729, 18, 6561, 243, 1, 59049, 2916, 27, 531441, 32805, 486, 1, 4782969, 354294, 7290, 36, 43046721, 3720087, 98415, 810, 1, 387420489, 38263752, 1240029, 14580, 45, 3486784401, 387420489, 14880348, 229635, 1215, 1, 31381059609, 3874204890, 172186884, 3306744, 25515, 54

Offset: 0

Zagros Lalo, Jul 20 2018

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013616 ((1+9*x)^n) and along skew diagonals pointing top-right in center-justified triangle given in A038291 ((9+x)^n).
The coefficients in the expansion of 1/(1-9*x-x^2) are given by the sequence generated by the row sums (see A099371).
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 9.109772228646443655... (a metallic mean), when n approaches infinity; (see A176522: ((9+sqrt(85))/2)).

			Triangle begins:
  1;
  9;
  81, 1;
  729, 18;
  6561, 243, 1;
  59049, 2916, 27;
  531441, 32805, 486, 1;
  4782969, 354294, 7290, 36;
  43046721, 3720087, 98415, 810, 1;
  387420489, 38263752, 1240029, 14580, 45;
  3486784401, 387420489, 14880348, 229635, 1215, 1;
  31381059609, 3874204890, 172186884, 3306744, 25515, 54;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 100.

Zagros Lalo, Left-justified triangle
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1 + 9x)^n
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (9 + x)^n

Row sums give A099371.
Cf. A013616, A038291, A176522.
Cf. A001019 (column 0), A053540 (column 1), A081139 (column 2), A173187 (column 3), A173000 (column 4).

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 9 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten

T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, 9*T(n-1, k)+T(n-2, k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 20 2018

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)

cp's OEIS Frontend

A173188 a(n) = binomial(n + 5, 5)*9^n.

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A173191 a(n) = binomial(n + 6, 6)*9^n.

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A173192 a(n) = binomial(n + 7, 7)*9^n.

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A317052 Triangle read by rows: T(0,0) = 1; T(n,k) = 9*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

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

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Magma

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

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Magma

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