A173191 -id:A173191 - cp's OEIS frontend

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

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)

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)

A362353 Triangle read by rows: T(n,k) = (-1)^(n-k)binomial(n, k)(k+3)^n, for n >= 0, and k = 0,1, ..., n. Coefficients of certain Sidi polynomials.

Original entry on oeis.org

1, -3, 4, 9, -32, 25, -27, 192, -375, 216, 81, -1024, 3750, -5184, 2401, -243, 5120, -31250, 77760, -84035, 32768, 729, -24576, 234375, -933120, 1764735, -1572864, 531441, -2187, 114688, -1640625, 9797760, -28824005, 44040192, -33480783, 10000000, 6561, -524288, 10937500, -94058496, 403536070, -939524096, 1205308188, -800000000, 214358881

Offset: 0

Harlan J. Brothers and Wolfdieter Lang, Apr 27 2023

This is the member N = 2 of a family of signed triangles with row sums n! = A000142(n): T(N; n, k) = (-1)^(n-k)*binomial(n, k)*(k + N + 1)^n, for integer N, n >= 0 and k = 0, 1, ..., n. The row polynomials PS(N; n, z) = Sum_{k=0..n} T(N; n, k)*z^k = ((-1)^n/z^N)*D_{n,N+1,n}(z) in [Sidi 1980].
For N = -1, 0 and 1 see A258773(n, k), A075513(n+1, k) and (-1)^(n-k) * A154715(n, k), respectively.
The column sequences, for k = 0, 1, ..., 6 and n >= k, are A141413(n+2), (-1)^(n+1)*A018215(n) = 4*(-1)^(n+1)*A002697(n), 5^2*(-1)^n*A081135(n), (-1)^(n+1)*A128964(n-1) = 6^3*(-1)^(n+1)*A081144(n), 7^4*(-1)^n*A139641(n-4), 2^15*(-1)^(n+1)*A173155(n-5), 3^12*(-1)^n*A173191(n-6), respectively.
The e.g.f. of the triangle (see below) needs the exponential convolution (LambertW(-z)/(-z))^2 = Sum_{n>=0} c(2; n)*z^n/n!, where c(2; n) = Sum_{m=0..n} |A137352(n+1, m)|*2^m = A007334(n+2).
The row sums give n! = A000142(n).

			The triangle T begins:
n\k    0       1        2         3         4          5          6         7
0:     1
1:    -3       4
2:     9     -32       25
3:   -27     192     -375       216
4:    81   -1024     3750     -5184      2401
5:  -243    5120   -31250     77760    -84035      32768
6:   729  -24576   234375   -933120   1764735   -1572864     531441
7: -2187  114688 -1640625   9797760 -28824005   44040192  -33480783  10000000
...
n = 8:  6561 -524288 10937500 -94058496 403536070 -939524096 1205308188 -800000000 2143588,
n = 9: -19683 2359296 -70312500 846526464 -5084554482 16911433728 -32543321076 36000000000 -21221529219 5159780352.

Paolo Xausa, Table of n, a(n) for n = 0..5049 (rows 0..100 of the triangle, flattened)
Wolfdieter Lang, On a Certain Family of Sidi Polynomials, May 2023.
Avram Sidi, Numerical Quadrature and Nonlinear Sequence Transformations; Unified Rules for Efficient Computation of Integrals with Algebraic and Logarithmic Endpoint Singularities, Math. Comp., 35 (1980), 851-874.

Cf. A000142 (row sums), A075513, A154715, A258773.

Columns k = 0..6 involve (see above): A002697, A007334, A018215, A081135, A081144, A128964, A137352, A139641, A141413, A173155, A173191.

A362353row[n_]:=Table[(-1)^(n-k)Binomial[n,k](k+3)^n,{k,0,n}];Array[A362353row,10,0] (* Paolo Xausa, Jul 30 2023 *)

T(n, k) = (-1)^(n-k)*binomial(n, k)*(k + 3)^n, for n >= 0, k = 0, 1, ..., n.
O.g.f. of column k: (x*(k + 3))^k/(1 - (k + 3)*x)^(k+1), for k >= 0.
E.g.f. of column k: exp(-(k + 3)*x)*((k + 3)*x)^k/k!, for k >= 0.
E.g.f. of the triangle, that is, the e.g.f. of its row polynomials {PS(2;n,y)}_{n>=0}): ES(2;y,x) = exp(-3*x)*(1/3)*(d/dz)(W(-z)/(-z))^2, after replacing z by x*y*exp(-x), where W is the Lambert W-function for the principal branch. This becomes ES(2;y,x) = exp(-3*x)*exp(3*(-W(-z)))/(1 - (-W(-z)), with z = x*y*exp(-x).

a(41)-a(44) from Paolo Xausa, Jul 31 2023

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

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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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A362353 Triangle read by rows: T(n,k) = (-1)^(n-k)*binomial(n, k)*(k+3)^n, for n >= 0, and k = 0,1, ..., n. Coefficients of certain Sidi polynomials.

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A362353 Triangle read by rows: T(n,k) = (-1)^(n-k)binomial(n, k)(k+3)^n, for n >= 0, and k = 0,1, ..., n. Coefficients of certain Sidi polynomials.