A102741 -id:A102741 - cp's OEIS frontend

A349454 Number T(n,k) of endofunctions on [n] with exactly k fixed points, all of which are isolated; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 1, 0, 1, 8, 3, 0, 1, 81, 32, 6, 0, 1, 1024, 405, 80, 10, 0, 1, 15625, 6144, 1215, 160, 15, 0, 1, 279936, 109375, 21504, 2835, 280, 21, 0, 1, 5764801, 2239488, 437500, 57344, 5670, 448, 28, 0, 1, 134217728, 51883209, 10077696, 1312500, 129024, 10206, 672, 36, 0, 1

Offset: 0

Alois P. Heinz, Dec 30 2021

			Triangle T(n,k) begins:
        1;
        0,       1;
        1,       0,      1;
        8,       3,      0,     1;
       81,      32,      6,     0,    1;
     1024,     405,     80,    10,    0,   1;
    15625,    6144,   1215,   160,   15,   0,  1;
   279936,  109375,  21504,  2835,  280,  21,  0, 1;
  5764801, 2239488, 437500, 57344, 5670, 448, 28, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Column k=0 gives A065440.
Row sums give A204042.
Main diagonal and first lower diagonal give A000012, A000004.
T(n+1,n-1) gives A000217.
T(n+3,n) gives A130809.
T(n+3,n-1) gives A102741 for n>=1.
Cf. A055134, A217701, A350212, A350454.

T:= (n, k)-> binomial(n, k)*(n-k-1)^(n-k):
seq(seq(T(n, k), k=0..n), n=0..10);

T(n,k) = binomial(n,k) * (n-k-1)^(n-k).
From Mélika Tebni, Apr 02 2023: (Start)
E.g.f. of column k: -x / (LambertW(-x)*(1+LambertW(-x)))*x^k / k!.
Sum_{k=0..n} k^k*T(n,k) = A217701(n). (End)

A134171 a(n) = (9/2)(n-1)(n-2)*(n-3).

Original entry on oeis.org

0, 0, 0, 27, 108, 270, 540, 945, 1512, 2268, 3240, 4455, 5940, 7722, 9828, 12285, 15120, 18360, 22032, 26163, 30780, 35910, 41580, 47817, 54648, 62100, 70200, 78975, 88452, 98658, 109620, 121365, 133920, 147312, 161568, 176715, 192780, 209790, 227772, 246753

Offset: 1

N. J. A. Sloane, Jan 30 2008

Number of n permutations (n>=3) of 4 objects u, v, z, x with repetition allowed, containing n-3=0 u's. Example: if n=3 then n-3 =zero u, a()=27 because we have vzx, vxz, zvx, zxv, xvz, xzv, vvv, zzz, xxx, vvx, vxv, xvv, xxv, xvx, vxx, vvz, vzv, zvv, zzv, zvz, vzz, xzz, zxz, zzx, xxz, xzx, zxx. A027465 formatted as a triangular array: diagonal: 27, 108, 270, 540, 945, 1512. - Zerinvary Lajos, Aug 06 2008

G. C. Greubel, Table of n, a(n) for n = 1..1000
D. Zvonkine, Home Page.
D. Zvonkine, Counting ramified coverings and intersection theory on Hurwitz spaces II (local structure of Hurwitz spaces and combinatorial results), Moscow Mathematical Journal, Vol. 7, No. 1 (2007), 135-162.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A008585, A027465, A027468. - Zerinvary Lajos, Aug 06 2008

Cf. A000292, A102741, A113335.

[(9/2)*(n-1)*(n-2)*(n-3) : n in [1..50]]; // Wesley Ivan Hurt, May 29 2016

seq(27*binomial(n-1, 3), n=1..30); # Zerinvary Lajos, May 18 2008

LinearRecurrence[{4,-6,4,-1}, {0,0,0,27}, 50] (* G. C. Greubel, May 29 2016 *)

a(n) = 27 * binomial(n-1,3). - Zerinvary Lajos, Aug 06 2008
From Chai Wah Wu, May 29 2016: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.
G.f.: 27*x^4/(1-x)^4. (End)
E.g.f.: 27 + (9/2*(x^3-3*x^2+6*x-6))*exp(x). - G. C. Greubel, May 17 2021
a(n) = 27 * A000292(n-3) for n >= 3. - Alois P. Heinz, May 17 2021
From Amiram Eldar, Sep 24 2022: (Start)
Sum_{n>=4} 1/a(n) = 1/18.
Sum_{n>=4} (-1)^n/a(n) = 4*log(2)/9 - 5/18. (End)

A113335 a(n) = 3^5 * binomial(n+4, 5).

Original entry on oeis.org

243, 1458, 5103, 13608, 30618, 61236, 112266, 192456, 312741, 486486, 729729, 1061424, 1503684, 2082024, 2825604, 3767472, 4944807, 6399162, 8176707, 10328472, 12910590, 15984540, 19617390, 23882040, 28857465, 34628958, 41288373, 48934368, 57672648, 67616208

Offset: 1

Zerinvary Lajos, Aug 06 2008

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A027465.

Sequences of the form 3^m*binomial(n+m-1, m): A008585 (m=1), A027468 (m=2), A134171 (m=3), A102741 (m=4), this sequence (m=5).

[3^5*Binomial(n+4,5): n in [1..30]]; // G. C. Greubel, May 17 2021

Maple
```
seq(binomial(n+4,5)*3^5, n=1..27);
```

With[{c=3^5},Table[c Binomial[n+4,5],{n,30}]]  (* Harvey P. Dale, Apr 11 2011 *)

[3^5*binomial(n+4,5) for n in (1..30)] # G. C. Greubel, May 17 2021

a(n) = 3^5 * binomial(n+4, 5), n >= 1.
From G. C. Greubel, May 17 2021: (Start)
G.f.: 243*x/(1-x)^6.
E.g.f.: (81/40)*x*(120 + 240*x + 120*x^2 + 20*x^3 + x^4)*exp(x). (End)
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=1} 1/a(n) = 5/972.
Sum_{n>=1} (-1)^(n+1)/a(n) = 80*log(2)/243 - 655/2916. (End)

cp's OEIS Frontend

A349454 Number T(n,k) of endofunctions on [n] with exactly k fixed points, all of which are isolated; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A134171 a(n) = (9/2)(n-1)(n-2)*(n-3).

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Magma

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A113335 a(n) = 3^5 * binomial(n+4, 5).

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cp's OEIS Frontend

A349454 Number T(n,k) of endofunctions on [n] with exactly k fixed points, all of which are isolated; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A134171 a(n) = (9/2)*(n-1)*(n-2)*(n-3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A113335 a(n) = 3^5 * binomial(n+4, 5).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

A134171 a(n) = (9/2)(n-1)(n-2)*(n-3).