A001564 -id:A001564 - cp's OEIS frontend

A082038 A square array of quadratic-factorial numbers, read by antidiagonals.

1, 1, 1, 1, 3, 2, 1, 7, 14, 6, 1, 13, 42, 78, 24, 1, 21, 86, 258, 504, 120, 1, 31, 146, 546, 1752, 3720, 720, 1, 43, 222, 942, 3768, 13320, 30960, 5040, 1, 57, 314, 1446, 6552, 28920, 113040, 287280, 40320, 1, 73, 422, 2058, 10104, 50520, 246960, 1063440

Offset: 0

Paul Barry, Apr 02 2003

Rows include A000142, A001564, A082035, A082036.

			Rows begin
1 1 2 6 24 ...
1 3 14 78 504 ...
1 7 42 258 1752 ...
1 13 86 546 3768 ...
1 21 146 942 6552 ...

Cf. A082037, A082039.

Square array defined by T(n, k)=((kn)^2+kn+1)n!

A082035 a(n) = (4n^2+2n+1) * n!.

Original entry on oeis.org

1, 7, 42, 258, 1752, 13320, 113040, 1063440, 11007360, 124467840, 1527724800, 20237817600, 287879961600, 4377595622400, 70875950745600, 1217444836608000, 22115388911616000, 423623726862336000, 8534364149735424000

Offset: 0

Paul Barry, Apr 02 2003

A row of the number array A082038.

Cf. A001564, A082036.

Table[(4n^2+2n+1)n!,{n,0,20}] (* Harvey P. Dale, Jul 15 2011 *)

a(n) = 4*A002775(n) + A007680(n).

A180193 Triangle read by rows: T(n,k) is the number of permutations of [n] having k blocks of odd length (0<=k<=n).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 3, 0, 3, 2, 0, 11, 0, 11, 0, 14, 0, 53, 0, 53, 6, 0, 96, 0, 309, 0, 309, 0, 78, 0, 724, 0, 2119, 0, 2119, 24, 0, 852, 0, 6070, 0, 16687, 0, 16687, 0, 504, 0, 9300, 0, 56418, 0, 148329, 0, 148329, 120, 0, 8040, 0, 106170, 0, 577556, 0, 1468457, 0

Offset: 0

Emeric Deutsch, Sep 09 2010

A block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions. For example, the permutation 5412367 has 4 blocks: 5, 4, 123, and 67.

			T(3,1)=3 because we have (123), 23(1), and (3)12 (the blocks of odd length are shown between parentheses). T(4,0)=2 because we have 1234 and 3412.
Triangle starts:
  1;
  0,1;
  1,0,1;
  0,3,0,3;
  2,0,11,0,11;
  0,14,0,53,0,53;

A. N. Myers, Counting permutations by their rigid patterns, J. Combin. Theory, Series A, Vol. 99, No. 2 (2002), pp. 345-357.

Cf. A000166, A000255, A001564, A180194, A180195

d[ -1] := 0: d[0] := 1: for n to 50 do d[n] := n*d[n-1]+(-1)^n end do: T := proc (n, k) if `mod`(n+k, 2) = 1 then 0 else sum(binomial(k+j, j)*binomial((1/2)*n+(1/2)*k-1, k+j-1)*(d[k+j]+d[k+j-1]), j = 0 .. (1/2)*n-(1/2)*k) end if end proc; for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form

T(n,k) = Sum(binomial(k+j,j)*binomial((n+k+2)/2,k+j-1)*[d(k+j)+d(k+j-1)], j=0..(n-k)/2) if n and k are of the same parity; T(n,k)=0 if n and k have opposite parities (0<=k<=n).
T(n,n) = T(n,n-2) = d(n)+d(n-1) = A000255(n-1), where d(i)=A000166(i) are the derangement numbers.
T(2n+1,1) = A001564(n).
Sum(k*T(n,k),k>=0) = A180195(n).
Sum of entries in row n = n! = A000142(n).

A184178 Irregular triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k isolated fixed points.

Original entry on oeis.org

1, 0, 1, 2, 3, 3, 13, 8, 3, 56, 51, 12, 1, 325, 294, 93, 8, 2193, 2068, 687, 90, 2, 17133, 16392, 5862, 888, 45, 151403, 146484, 54861, 9463, 660, 9, 1492804, 1454716, 565044, 106652, 9320, 264, 16236705, 15903261, 6354090, 1285990, 131145, 5565, 44

Offset: 0

Emeric Deutsch, Feb 13 2011

A fixed point j of a permutation is said to be isolated if neither j-1 nor j+1 is a fixed point. For example, 4135267 has only 3 as an isolated fixed point.

Row n has 1+ceiling(n/2) terms if n >= 4.

			T(3,1)=3 because we  have 132, 321, and 213.
T(4,2)=3 because we have 1432, 1324, and 3214.
Triangle starts:
    1;
    0,   1;
    2;
    3,   3;
   13,   8,   3;
   56,  51,  12,   1;
  325, 294,  93,   8;

Cf. A000166, A001564, A184179.

d[0] := 1: d[1] := 1: for n to 50 do d[n] := n*d[n-1]+(-1)^n end do: a := proc (n, k) options operator, arrow: add(d[n-j]*add(binomial(j-k-m-1, m-1)*binomial(n+1-j, k+m)*binomial(k+m, k), m = 0 .. floor((1/2)*j-(1/2)*k)), j = k .. n) end proc: 1; 0, 1; 2; 3, 3; for n from 4 to 11 do seq(a(n, k), k = 0 .. ceil((1/2)*n)) end do; # yields sequence in triangular form

T(n,k) = Sum_{j=k..n} d(n-j)*Sum_{m=0..floor((j-k)/2)} binomial(j-k-m-1, m-1)*binomial(n+1-j, k+m)*binomial(k+m, k), where d(i)=A000166(i) are the derangement numbers.
Sum of entries in row n is n!.
T(n,0) = A184179(n).
Sum_{k>=0} k*T(n,k) = A001564(n-2) (n>=3).

A162968 Number of pairs of consecutive non-fixed points in all permutations of {1,2,...,n}.

Original entry on oeis.org

1, 6, 42, 312, 2520, 22320, 216720, 2298240, 26490240, 330220800, 4430764800, 63707212800, 977642265600, 15953627289600, 275919291648000, 5042392363008000, 97102667870208000, 1965528727658496000, 41724269440229376000, 926935665115299840000

Offset: 2

Emeric Deutsch, Jul 19 2009

			a(3)=6 because in 123, 132, 213, 231, 312, 321 we have 0+1+1+2+2+0 such pairs.

Cf. A001564.

seq(factorial(n-1)*(n^2-3*n+3), n = 2 .. 20);

Table[(n-1)!(n^2-3n+3),{n,2,30}] (* Harvey P. Dale, Mar 28 2012 *)

a(n) = (n-1)! * (n^2 - 3*n + 3) (n>=2).
a(n) = A001564(n-2)*(n-1) for n>=2. - Anton Zakharov, Sep 14 2016
D-finite with recurrence a(n) +(-n-5)*a(n-1) +(4*n-1)*a(n-2) +3*(-n+3)*a(n-3)=0. - R. J. Mathar, Jul 22 2022

A358624 Triangle read by rows. The coefficients of the Hahn polynomials in ascending order of powers. T(n, k) = n! * [x^k] hypergeom([-x, -n, n + 1], [1, 1], 1).

Original entry on oeis.org

1, 1, 2, 2, 6, 6, 6, 22, 30, 20, 24, 100, 170, 140, 70, 120, 548, 1050, 1120, 630, 252, 720, 3528, 7476, 8820, 6720, 2772, 924, 5040, 26136, 59388, 78708, 64680, 37884, 12012, 3432, 40320, 219168, 529896, 748440, 704550, 432432, 204204, 51480, 12870

Offset: 0

Peter Luschny, Nov 26 2022

			[0]     1;
[1]     1,      2;
[2]     2,      6,      6;
[3]     6,     22,     30,     20;
[4]    24,    100,    170,    140,     70;
[5]   120,    548,   1050,   1120,    630,    252;
[6]   720,   3528,   7476,   8820,   6720,   2772,    924;
[7]  5040,  26136,  59388,  78708,  64680,  37884,  12012,  3432;
[8] 40320, 219168, 529896, 748440, 704550, 432432, 204204, 51480, 12870;

A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer-Verlag Berlin Heidelberg, 1991.

Cf. A000142, A000984, A001564 (row sums), A133942 (alternating row sums).

H := (n, x) -> n!*hypergeom([-x, -n, n + 1], [1, 1], 1):
for n from 0 to 8 do seq(coeff(simplify(H(n, x)), x, k), k = 0..n) od;

The general formula for the Hahn polynomials is H(n, x, N, a, b) = (-1)^n*(Pochhammer(N-n, n)*Pochhammer(b+1, n) / n!)*hypergeom([-n, -x, a + b + n + 1], [b + 1, 1 - N], 1). We consider here the case N = a = b = 0.

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A082038 A square array of quadratic-factorial numbers, read by antidiagonals.

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A082035 a(n) = (4n^2+2n+1) * n!.

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A180193 Triangle read by rows: T(n,k) is the number of permutations of [n] having k blocks of odd length (0<=k<=n).

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A184178 Irregular triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k isolated fixed points.

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A162968 Number of pairs of consecutive non-fixed points in all permutations of {1,2,...,n}.

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A358624 Triangle read by rows. The coefficients of the Hahn polynomials in ascending order of powers. T(n, k) = n! * [x^k] hypergeom([-x, -n, n + 1], [1, 1], 1).

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