A105114 -id:A105114 - cp's OEIS frontend

A114320 Triangle T(n,k) = number of permutations of n elements with k 2-cycles.

1, 1, 1, 1, 3, 3, 15, 6, 3, 75, 30, 15, 435, 225, 45, 15, 3045, 1575, 315, 105, 24465, 12180, 3150, 420, 105, 220185, 109620, 28350, 3780, 945, 2200905, 1100925, 274050, 47250, 4725, 945, 24209955, 12110175, 3014550, 519750, 51975, 10395, 290529855

Offset: 0

Vladeta Jovovic, Feb 05 2006

Row n has 1+floor(n/2) terms. Row sums yield the factorials (A000142). Sum(k*T(n,k),k>0)=n!/2 for n>=2. - Emeric Deutsch, Feb 17 2006

			T(3,1) = 3 because we have (1)(23), (12)(3) and (13)(2).
Triangle begins:
    1;
    1;
    1,   1;
    3,   3;
   15,   6,   3;
   75,  30,  15;
  435, 225,  45,  15;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A008290, A000266, A000090, A088436, A000138, A060725, A060726, A086659, A105422, A105114.

G:= exp((y-1)*x^2/2)/(1-x): Gser:= simplify(series(G,x=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:= n!*coeff(Gser,x^n) od: for n from 0 to 12 do seq(coeff(y*P[n], y^j), j=1..1+floor(n/2)) od;  # yields sequence in triangular form - Emeric Deutsch, Feb 17 2006

d = Exp[-x^2/2!]/(1 - x);f[list_] := Select[list, # > 0 &]; Flatten[Map[f, Transpose[Table[Range[0, 10]!CoefficientList[Series[x^(2 k)/(2^k k!) d, {x, 0, 10}], x], {k, 0, 5}]]]]  (* Geoffrey Critzer, Nov 29 2011 *)

E.g.f.: exp((y-1)*x^2/2)/(1-x). More generally, e.g.f. for number of permutations of n elements with k m-cycles is exp((y-1)*x^m/m)/(1-x).

T(n,k) = n!/(2^k*k!) * Sum_{j=0..floor(n/2)-k} (-1/2)^j/j!. - Alois P. Heinz, Nov 30 2011

More terms from Emeric Deutsch, Feb 17 2006

A218796 Triangular array read by rows: T(n,k) is the number of compositions of n that have exactly k 3's; n>=0, 0<=k<=floor(n/3).

Original entry on oeis.org

1, 1, 2, 3, 1, 6, 2, 11, 5, 21, 10, 1, 39, 22, 3, 73, 46, 9, 136, 97, 22, 1, 254, 200, 54, 4, 474, 410, 126, 14, 885, 832, 290, 40, 1, 1652, 1679, 651, 109, 5, 3084, 3368, 1440, 280, 20, 5757, 6725, 3138, 698, 65, 1, 10747, 13370, 6762, 1688, 195, 6, 20062, 26483, 14424, 3994, 546, 27

Offset: 0

Geoffrey Critzer, Nov 05 2012

Row Sums = 2^(n-1) for n>0.

			1;
1;
2;
3,       1;
6,       2;
11,      5;
21,     10,    1;
39,     22,    3;
73,     46,    9;
136,    97,   22,   1;
254,   200,   54,   4;
474,   410,  126,  14;
885,   832,  290,  40,   1;
1652, 1679,  651, 109,   5;
3084, 3368, 1440, 280,  20;
5757, 6725, 3138, 698,  65,  1;

Alois P. Heinz, Rows n = 0..250, flattened
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 168.

Cf. A105114, A045623, A011782.

Column k=0 gives: A049856(n+2).

T:= proc(n) option remember; local j; if n=0 then 1
      else []; for j to n do zip((x, y)-> x+y, %,
      [`if`(j=3, 0, [][]), T(n-j)], 0) od; %[] fi
    end:
seq (T(n), n=0..25);  # Alois P. Heinz, Nov 05 2012

nn=15;f[list_]:=Select[list,#>0&];Map[f,CoefficientList[Series[1/(1-(x/(1-x)-x^3+y x^3)),{x,0,nn}],{x,y}]]//Grid

O.g.f.: 1/(1-(x/(1-x)-x^3+y*x^3)) and generally for the number of compositions with k parts of size r we have: 1/(1-(x/(1-x)-x^r+y*x^r)).

A120924 Triangle read by rows: T(n,k) is the number of ternary words of length n on {0,1,2}, having k isolated 0's (n >= 0, k >= 0).

Original entry on oeis.org

1, 2, 1, 5, 4, 13, 12, 2, 33, 36, 12, 83, 108, 48, 4, 209, 316, 172, 32, 527, 904, 588, 160, 8, 1329, 2548, 1932, 672, 80, 3351, 7104, 6140, 2592, 480, 16, 8449, 19628, 19020, 9440, 2320, 192, 21303, 53816, 57756, 32896, 10000, 1344, 32, 53713, 146596

Offset: 0

Emeric Deutsch, Jul 16 2006

Row n has 1+ceiling(n/2) terms.
Row sums are the powers of 3 (A000244).
T(n,0) = A120925(n).
Sum_{k=0..ceiling(n/2)} k*T(n,k) = A120926(n).

			T(2,0)=5 because we have 00,11,12,21 and 22; T(2,1)=4 because we have 01,02,10 and 20; T(3,2)=2 because we have 010 and 020.
Triangle starts:
   1;
   2,   1;
   5,   4;
  13,  12,  2;
  33,  36, 12;
  83, 108, 48, 4;

Cf. A000244, A105114, A120925, A120926.

G:=(1-(1-t)*z*(1-z))/(1-3*z+2*(1-t)*z^2*(1-z)): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 13 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 0 to 13 do seq(coeff(P[n],t,j),j=0..ceil(n/2)) od; # yields sequence in triangular form

G.f. = G(t,z) = (1-(1-t)z(1-z))/(1 - 3z + 2(1-t)z^2*(1-z)).

A120925 Number of ternary words on {0,1,2} having no isolated 0's.

Original entry on oeis.org

1, 2, 5, 13, 33, 83, 209, 527, 1329, 3351, 8449, 21303, 53713, 135431, 341473, 860983, 2170865, 5473575, 13800961, 34797463, 87737617, 221219847, 557779233, 1406373239, 3546000945, 8940814823, 22543189057, 56839939415, 143315069777

Offset: 0

Emeric Deutsch, Jul 16 2006

Column 0 of A120924.

			a(2)=5 because we have 00,11,12,21 and 22.

Michael De Vlieger, Table of n, a(n) for n = 0..2490
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 11.
Maksym Druchok, Volodymyr Krasnov, Taras Krokhmalskii, and Oleg Derzhko, One-dimensionally confined ammonia molecules: A theoretical study, arXiv:2307.06186 [cond-mat.stat-mech], 2023. See p. 5.
Index entries for linear recurrences with constant coefficients, signature (3,-2,2).

Cf. A105114, A120924.

a[0]:=1:a[1]:=2:a[2]:=5: for n from 3 to 32 do a[n]:=3*a[n-1]-2*a[n-2]+2*a[n-3] od: seq(a[n],n=0..32);

nn=20;a=x^2/(1-x);CoefficientList[Series[(a+1)/(1-(2x a)/(1-2x))/(1-2x),{x,0,nn}],x]  (* Geoffrey Critzer, Jan 13 2013 *)
LinearRecurrence[{3,-2,2},{1,2,5},30] (* Harvey P. Dale, Nov 16 2024 *)

a(n) = 3a(n-1)-2a(n-2)+2a(n-3); a(0)=1, a(1)=2,a(2)=5.

G.f.: (1-z+z^2)/(1-3z+2z^2-2z^3).

A296559 Triangle read by rows: T(n,k) is the number of compositions of n having k parts equal to 1 or 2 (0<=k<=n).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 2, 1, 3, 1, 1, 4, 3, 3, 4, 1, 2, 4, 9, 5, 6, 5, 1, 3, 7, 12, 16, 9, 10, 6, 1, 4, 13, 18, 28, 26, 16, 15, 7, 1, 6, 19, 36, 42, 55, 41, 27, 21, 8, 1, 9, 29, 60, 82, 90, 97, 64, 43, 28, 9, 1, 13, 47, 94, 152, 170, 177, 160, 99, 65, 36, 10, 1, 19, 73, 158, 252, 335, 333, 323, 253, 151, 94, 45, 11, 1

Offset: 0

Emeric Deutsch, Dec 15 2017

Sum of entries in row n = 2^{n-1} = A011782(n) (n>=1).

Sum(kT(n,k), k>=0) = (3n+5)*2^{n-4} = A106472(n-1) (n>=3).

			T(3,2) = 2 because we have [1,2],[2,1].
T(6,3) = 5 because we have [2,2,2],[1,1,1,3],[1,1,3,1],[1,3,1,1],[3,1,1,1].
Triangle begins:
  1,
  0, 1,
  0, 1, 1,
  1, 0, 2, 1,
  1, 2, 1, 3, 1,
  1, 4, 3, 3, 4, 1,
  2, 4, 9, 5, 6, 5, 1,
  3, 7, 12, 16, 9, 10, 6, 1,
  4, 13, 18, 28, 26, 16, 15, 7, 1,
  ...

Cf. A011782, A105114, A105422, A106472.

g := (1-x)/(1-(1+t)*x-(1-t)*x^3): gser := simplify(series(g, x = 0, 17)): for n from 0 to 15 do p[n] := sort(expand(coeff(gser, x, n))) end do: for n from 0 to 15 do seq(coeff(p[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form

nmax = 12;
s = Series[(1-x)/(1 - (1+t) x - (1-t) x^3), {x, 0, nmax}, {t, 0, nmax}];
T[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {t, 0, k}];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 16 2017 *)

G.f.: G(t,x) = (1-x)/(1 - (1 + t)x - (1 - t)x^3).

cp's OEIS Frontend

A114320 Triangle T(n,k) = number of permutations of n elements with k 2-cycles.

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A218796 Triangular array read by rows: T(n,k) is the number of compositions of n that have exactly k 3's; n>=0, 0<=k<=floor(n/3).

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A120924 Triangle read by rows: T(n,k) is the number of ternary words of length n on {0,1,2}, having k isolated 0's (n >= 0, k >= 0).

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A120925 Number of ternary words on {0,1,2} having no isolated 0's.

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A296559 Triangle read by rows: T(n,k) is the number of compositions of n having k parts equal to 1 or 2 (0<=k<=n).

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