A166334 -id:A166334 - cp's OEIS frontend

A193229 A double factorial triangle.

1, 1, 1, 3, 3, 2, 15, 15, 12, 6, 105, 105, 90, 60, 24, 945, 945, 840, 630, 360, 120, 10395, 10395, 9450, 7560, 5040, 2520, 720, 135135, 135135, 124740, 103950, 75600, 45360, 20160, 5040, 2027025, 2027025, 1891890, 1621620, 1247400, 831600, 453600, 181440, 40320

Offset: 0

Gary W. Adamson, Jul 18 2011

The double factorial triangle coefficients are T(n,k), n >= 0 and 0 <= k <= n.
The T(n,0) equal the double factorial numbers A001147(n) = (2*n-1)!!.
The T(n,n) equal the factorial numbers A000142(n) = n!.
The row sums equal the double factorial numbers A000165(n) = (2*n)!!.
The Kn21(n) sums, see A180662 for the definition of these and other triangle sums, equal A130905(n) while the Kn2p(n) sums equal A130905(n+2*p-2) - (n+2*p-2)!*A010844(p-2)/A000165(p-2), p >= 2. - Johannes W. Meijer, Jul 21 2011

			The first few rows of matrix M[i,j] are:
  1, 1, 0, 0, 0, 0, ...
  2, 2, 2, 0, 0, 0, ...
  3, 3, 3, 3, 0, 0, ...
  4, 4, 4, 4, 4, 0, ...
  5, 5, 5, 5, 5, 5, ...
The first few rows of triangle T(n,k) are:
       1;
       1,      1;
       3,      3,      2;
      15,     15,     12,      6;
     105,    105,     90,     60,    24;
     945,    945,    840,    630,   360,   120;
   10395,  10395,   9450,   7560,  5040,  2520,   720;
  135135, 135135, 124740, 103950, 75600, 45360, 20160, 5040;

Seiichi Manyama, Rows n = 0..139, flattened

Cf. A000142, A000165, A001147, A102625.
Cf. A180662, A130905, A010844.
T(2*n,n) gives A166334.

nmax:=7: M := Matrix(1..nmax+1,1..nmax+1): for i from 1 to nmax do for j from 1 to i+1 do M[i,j] := i od: od: for n from 0 to nmax do B := M^n: for k from 0 to n do T(n,k) := B[1,k+1] od: od: for n from 0 to nmax do seq(T(n,k),k=0..n) od: seq(seq(T(n,k),k=0..n),n=0..nmax); # Johannes W. Meijer, Jul 21 2011

row(n)=(matrix(n,n,i,j,(i>j-2)*i)^(n-1))[1,]  \\ M. F. Hasler, Jul 24 2011

T(n,k) = the (k+1)-th term in the top row of M^n, where M is an infinite square production matrix; M[i,j] = i, i >= 1 and 1 <= j <= i+1, and M[i,j] = 0, i >= 1 and j >= i+2, see the examples.

It appears that T(n,k) = (2*n-k)!/(2^(n-k)*(n-k)!) with conjectural e.g.f. 1/(x*(1-2*z) + (1-x)*sqrt(1-2*z)) = 1 + (1+x)*z + (3+3*x+2*x^2)*z^2/2! + .... Cf. A102625. - Peter Bala, Jul 09 2012

Corrected, edited and extended by Johannes W. Meijer, Jul 21 2011

More terms from Seiichi Manyama, Apr 06 2019

A349426 Irregular triangle read by rows: T(n,k) is the number of arrangements of n labeled children with exactly k nontrivial rounds; n >= 3, 1 <= k <= floor(n/3).

Original entry on oeis.org

3, 8, 30, 144, 90, 840, 840, 5760, 7280, 45360, 66528, 7560, 403200, 657720, 151200, 3991680, 7064640, 2356200, 43545600, 82285632, 34890240, 1247400, 518918400, 1035365760, 521080560, 43243200, 6706022400, 14013679680, 8034586560, 1059458400

Offset: 3

Steven Finch, Nov 17 2021

A nontrivial round means the same as a ring or circle consisting of more than one child.

			Triangle starts:
[3]           3;
[4]           8;
[5]          30;
[6]         144,          90;
[7]         840,         840;
[8]        5760,        7280;
[9]       45360,       66528,       7560;
[10]     403200,      657720,     151200;
[11]    3991680,     7064640,    2356200;
[12]   43545600,    82285632,   34890240,    1247400;
[13]  518918400,  1035365760,  521080560,   43243200;
[14] 6706022400, 14013679680, 8034586560, 1059458400;
...
For n = 6, there are 144 ways to make one round and 90 ways to make two rounds.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)

Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.

Row sums give A066165 (variant of Stanley's children's game).
Column 1 gives A001048.
Right border element of row n is A166334(n/3) for each n divisible by 3.
Cf. A066166, A349280 (correspond to Stanley's original game).

f[k_, n_] := n! SeriesCoefficient[(1 - x)^(-x t) Exp[-x^2 t], {x, 0, n}, {t, 0, k}]
Table[f[k, n], {n, 2, 14}, {k, 1, Floor[n/3]}]

E.g.f.: (1 - x)^(-x*t) * exp(-x^2*t).

cp's OEIS Frontend

A193229 A double factorial triangle.

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