cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 21-27 of 27 results.

A124008 Number of permutations of n distinct letters (ABCD...) each of which appears thrice with n-4 fixed points.

Original entry on oeis.org

9, 189, 1431, 5355, 14310, 31374, 60354, 105786, 172935, 267795, 397089, 568269
Offset: 0

Views

Author

Zerinvary Lajos, Nov 01 2006

Keywords

Examples

			1
0, 0, 0, 1
1, 0, "9", 0, 9, 0, 1
56, 216, 378, 435, 324, "189", 54", 27, 0, 1
13833, 49464, 84510, 90944, 69039, 38448, 16476, 5184, "1431", 216, 54, 0, 1
6699824, 23123880, 38358540, 40563765, 30573900, 17399178, 7723640, 2729295, 776520, 180100, 33372, "5355", 540, 90, 0, 1
etc...

Crossrefs

Cf. A059058, A027468, A059073, A000459.

Programs

  • Maple
    p := (x, k)->k!^2*sum(x^j/((k-j)!^2*j!), j=0..k); R := (x, n, k)->p(x, k)^n; f := (t, n, k)->sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k); for n from 0 to 6 do seq(coeff(f(t, n, 3), t, m)/3!^n, m=0..3*n); od;

A124009 Number of permutations of n distinct letters (ABCD...) each of which appears thrice with one fixed point.

Original entry on oeis.org

0, 0, 216, 49464, 23123880, 19180338840, 25791442770240, 52614269909090064, 154809621283047068016, 631429039396055199165840, 3457808596178310768284115720, 24763433580060911383347280813320
Offset: 0

Views

Author

Zerinvary Lajos, Nov 01 2006

Keywords

Examples

			1
0, "0", 0, 1
1, "0", 9, 0, 9, 0, 1
56, "216", 378, 435, 324, 189, 54", 27, 0, 1
13833, "49464", 84510, 90944, 69039, 38448, 16476, 5184, 1431, 216, 54, 0, 1
6699824, "23123880", 38358540, 40563765, 30573900, 17399178, 7723640, 2729295, 776520, 180100, 33372, 5355, 540, 90, 0, 1
etc...

Crossrefs

Cf. A059058, A027468, A059073, A000459.

Programs

  • Maple
    p := (x, k)->k!^2*sum(x^j/((k-j)!^2*j!), j=0..k); R := (x, n, k)->p(x, k)^n; f := (t, n, k)->sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k); for n from 0 to 6 do seq(coeff(f(t, n, 3), t, m)/3!^n, m=0..3*n); od;

A124042 Number of permutations of n distinct letters (ABCD...) each of which appears thrice with two fixed points.

Original entry on oeis.org

0, 9, 378, 84510, 38358540, 31234760055, 41467520432646, 83805898840005132, 244832935610272588920, 993012060508835944545045, 5413243051841698780829328690, 38622438042365626607874252846474
Offset: 0

Views

Author

Zerinvary Lajos, Nov 02 2006

Keywords

Examples

			1
0, 0, "0", 1
1, 0, "9", 0, 9, 0, 1
56, 216, "378", 435, 324, 189, 54", 27, 0, 1
13833, 49464, "84510", 90944, 69039, 38448, 16476, 5184, 1431, 216, 54, 0, 1
6699824, 23123880, "38358540", 40563765, 30573900, 17399178, 7723640, 2729295, 776520, 180100, 33372, 5355, 540, 90, 0, 1
etc...

Crossrefs

Cf. A059058, A027468, A059073, A000459.

Programs

  • Maple
    p := (x, k)->k!^2*sum(x^j/((k-j)!^2*j!), j=0..k); R := (x, n, k)->p(x, k)^n; f := (t, n, k)->sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k); for n from 0 to 6 do seq(coeff(f(t, n, 3), t, m)/3!^n, m=0..3*n); od;

A124043 Number of permutations of n distinct letters (ABCD...) each of which appears thrice with three fixed points.

Original entry on oeis.org

1, 0, 435, 90944, 40563765, 32659846104, 43036380310735, 86514409614060000, 251739515511526387401, 1017865281673593548065520, 5534999211214597734889370091, 39411238922605740572075832485280
Offset: 0

Views

Author

Zerinvary Lajos, Nov 02 2006

Keywords

Examples

			1
0, 0, 0, "1"
1, 0, 9, "0", 9, 0, 1
56, 216, 378, "435", 324, 189, 54", 27, 0, 1
13833, 49464, 84510, "90944", 69039, 38448, 16476, 5184, 1431, 216, 54, 0, 1
6699824, 23123880, 38358540, "40563765", 30573900, 17399178, 7723640, 2729295, 776520, 180100, 33372, 5355, 540, 90, 0, 1
etc...

Crossrefs

Cf. A059058, A027468, A059073, A000459.

Programs

  • Maple
    p := (x, k)->k!^2*sum(x^j/((k-j)!^2*j!), j=0..k); R := (x, n, k)->p(x, k)^n; f := (t, n, k)->sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k); for n from 0 to 6 do seq(coeff(f(t, n, 3), t, m)/3!^n, m=0..3*n); od;

A162940 a(n) = binomial(n+1,2)*6^2.

Original entry on oeis.org

0, 36, 108, 216, 360, 540, 756, 1008, 1296, 1620, 1980, 2376, 2808, 3276, 3780, 4320, 4896, 5508, 6156, 6840, 7560, 8316, 9108, 9936, 10800, 11700, 12636, 13608, 14616, 15660, 16740, 17856, 19008, 20196, 21420, 22680, 23976, 25308, 26676, 28080, 29520, 30996
Offset: 0

Views

Author

Zerinvary Lajos, Jul 18 2009, Jul 19 2009

Keywords

Comments

Number of n permutations (n>=2) of 7 objects s, t, u, v, z, x, y with repetition allowed, containing n-2 u's. Example: If n=2 then n-2 = zero (0) u, a(1)=36 because we have ss, st, sv, sz, sx, sy, ts, tt, tv, tz, tx, ty, vs, vt, vv, vz, vx, vy, zs, zt, zv, zz, zx, zy, xs, xt, xv, xz, xx, xy, ys, yt, yv, yz, yx, yy. If n=3 then n-2 = one (1) u, a(2) = 108, >> ssu, stu, etc. If n=4 then n-2 = two (2) u, a(2)= 216, >> ssuu, stuu, ..., txuu, etc. If n=5 then n-2 = three (3) u, a(3)=360, >> ssuuu, stuuu, ..., txuuu, etc.

Links

Crossrefs

Cf. A000217, A002378, A046092, A027468, A035008, A123296.
Cf. A049598, A163758.

Programs

  • Mathematica
    Table[Binomial[n + 1, 2]*6^2, {n, 0, 58}]
  • PARI
    a(n)=18*n*(n+1) \\ Charles R Greathouse IV, Jun 16 2017

Formula

From Amiram Eldar, Sep 01 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/18.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/9 - 1/18. (End)
From Amiram Eldar, Feb 22 2023: (Start)
a(n) = 18*n*(n+1) = 36*A000217(n) = 18*A002378(n).
Product_{n>=1} (1 - 1/a(n)) = -(18/Pi)*cos(sqrt(11)*Pi/6).
Product_{n>=1} (1 + 1/a(n)) = (18/Pi)*cos(sqrt(7)*Pi/6). (End)
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: 36*x/(1-x)^3.
E.g.f.: 18*x*(2 + x)*exp(x).
a(n) = 3*A049598(n) = 2*A163758(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A162942 a(n) = binomial(n+1,2)*7^2.

Original entry on oeis.org

0, 49, 147, 294, 490, 735, 1029, 1372, 1764, 2205, 2695, 3234, 3822, 4459, 5145, 5880, 6664, 7497, 8379, 9310, 10290, 11319, 12397, 13524, 14700, 15925, 17199, 18522, 19894, 21315, 22785, 24304, 25872, 27489, 29155, 30870, 32634, 34447, 36309
Offset: 0

Views

Author

Zerinvary Lajos, Jul 18 2009

Keywords

Comments

Number of n permutations (n>=2) of 8 objects r, s, t, u, v, z, x, y with repetition allowed, containing n-2 u's.

Examples

			If n=2 then n-2=zero (0) u, a(1) = 49 because we have sr, tr, vr, zr, xr, yr, rs, rt, rv, rz, rx, ry, ss, st, sv, sz, sx, sy, ts, tt, tv, tz, tx, ty, vs, vt, vv, vz, vx, vy, zs, zt, zv, zz, zx, zy, xs, xt, xv, xz, xx, xy, ys, yt, yv, yz, yx, yy. If n=3 then n-2 = one (1) u, a(2) = 147 >> ssu, stu, etc.. Tf n=4 then n-2 = two (2) u, a(2) = 294 >> ssuu, stuu, ..., txuu, etc.. If n=5 then n-2 = three (3) u, a(3) = 490 >> rsuuu, stuuu, ..., rxuuu, etc..

Links

Crossrefs

Cf. A000217, A027468, A027469, A035008, A046092, A123296, A162940.

Programs

  • Mathematica
    Table[Binomial[n + 1, 2]*7^2, {n, 0, 58}]
  • PARI
    a(n)=49*binomial(n+1,2) \\ Charles R Greathouse IV, May 02 2014

Formula

a(n) = A027469(n+2). - R. J. Mathar, Jul 18 2009
G.f.: -49*x/(x-1)^3. - R. J. Mathar, May 02 2014
From Amiram Eldar, Sep 04 2022: (Start)
Sum_{n>=1} 1/a(n) = 2/49.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(2*log(2)-1)/49. (End)
From Elmo R. Oliveira, Dec 27 2024: (Start)
E.g.f.: 49*exp(x)*x*(2 + x)/2.
a(n) = 49*A000217(n) = 49*n*(n+1)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A124070 Number of permutations of n distinct letters (ABCD...) each of which appears thrice with 4 fixed points.

Original entry on oeis.org

9, 324, 69039, 30573900, 24571261710, 32346221908896, 64986793207684866, 189028409383462290696, 764111162168487304691175, 4154377697330090433618612780, 29576798800687086868033152117849
Offset: 0

Views

Author

Zerinvary Lajos, Nov 05 2006

Keywords

Examples

			1
0, 0, 0, 1
1, 0, 9, 0, "9", 0, 1
56, 216, 378, 435, "324", 189, 54", 27, 0, 1
13833, 49464, 84510, 90944, "69039", 38448, 16476, 5184, 1431, 216, 54, 0, 1
6699824, 23123880, 38358540, 40563765, "30573900", 17399178, 7723640, 2729295, 776520, 180100, 33372, 5355, 540, 90, 0, 1
etc...

Crossrefs

Cf. A059058, A027468, A059073, A000459, A124007, A124008, A124009, A124042, A124043.
Previous Showing 21-27 of 27 results.

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