A140456 -id:A140456 - cp's OEIS frontend

A152736 Triangle read by rows: M*Q, where M = an infinite lower triangular matrix with A140456 in every column: (1, 1, 1, 3, 7, 23, 71, ...) and Q = a matrix with A000085 as the main diagonal the rest zeros.

1, 1, 1, 1, 1, 2, 3, 1, 2, 4, 7, 3, 2, 4, 10, 23, 7, 6, 4, 10, 26, 71, 23, 14, 12, 10, 26, 76, 255, 71, 46, 28, 30, 26, 76, 232, 911, 255, 142, 92, 70, 78, 76, 232, 764, 3535, 911, 510, 284, 230, 182, 228, 232, 764, 2620

Offset: 1

Gary W. Adamson, Dec 12 2008

An eigentriangle.
Row sums = A000085 starting with offset 1.
Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle:
      1;
      1,    1;
      1,    1,    2;
      3,    1,    2,    4;
      7,    3,    2,    4,  10;
     23,    7,    6,    4,  10,  26;
     71,   23,   14,   12,  10,  26,  76;
    255,   71,   46,   28,  30,  26,  76, 232;
    911,  255,  142,   92,  70,  78,  76, 232, 764;
   3535,  911,  510,  284, 230, 182, 228, 232, 764, 2620;
  13903, 3535, 1822, 1020, 710, 598, 532, 696, 764, 2620, 9496;
  ...
Row r = (3, 1, 2, 4) = (3*1, 1*1, 1*2, 1*4) = termwise products of (3, 1, 1, 1) and (1, 1, 2, 4), where A000085 = (1, 1, 2, 4, 10, 26, 76, ...).

Cf. A000085, A140456.

A188144 Binomial transform A140456(n+1) (indecomposable involutions).

Original entry on oeis.org

1, 2, 6, 20, 74, 292, 1218, 5308, 24034, 112484, 542346, 2686268, 13639226, 70863652, 376208706, 2038335580, 11259474754, 63353211332, 362819139978, 2113410084188, 12513610048154, 75274067489284, 459782361574146, 2850369932150908, 17926893505949986, 114337654086861092

Offset: 0

Groux Roland, Mar 22 2011

nonn

a(n) is also the INVERTi transform of A005425(n+1) (self-inverse partial permutations) starting at n=2.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A005425, A140456.

b:= proc(n) b(n):= `if`(n<2, n+1,  2*b(n-1) + (n-1)*b(n-2)) end:
g:= proc(n) g(n):= `if`(n<1, -1, -add(g(n-i)*b(i), i=1..n)) end:
a:= n-> g(n+2):
seq(a(n), n=0..28);  # Alois P. Heinz, Mar 19 2020

nmax = 18; A140456 = CoefficientList[ Series[1 - 1/Total[ CoefficientList[ Series[Exp[x^2/2 + x], {x, 0, nmax + 2}], x]*Range[0, nmax + 2]!* x^Range[0, nmax + 2]], {x, 0, nmax + 2}], x]; a[n_] := Sum[ Binomial[n, k]*A140456[[k + 3]], {k, 0, n}]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Jul 03 2013 *)

seq(n)={Vec(1 - 2*x - 1/serlaplace(exp( 2*x + x^2/2 + O(x^3*x^n) )))} \\ Andrew Howroyd, Jan 06 2020

a(n) is the moment of order n for the probability density function: sqrt(2/Pi^3)*exp((x-2)^2/2)/(1+(erfi((x-2)/sqrt(2)))^2) over the interval -infinity..infinity, with erfi the imaginary error function.

G.f.: A(x) = (1 - 2*x - G(0))/x^2; G(k) = 1 - 2*x - x^2*(k+1)/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 26 2012

Terms a(19) and beyond from Andrew Howroyd, Jan 06 2020

A258306 A(n,k) is the sum over all Motzkin paths of length n of products over all peaks p of (x_p+k*y_p)/y_p, where x_p and y_p are the coordinates of peak p; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 7, 14, 1, 1, 5, 9, 23, 43, 1, 1, 6, 11, 34, 71, 141, 1, 1, 7, 13, 47, 105, 255, 490, 1, 1, 8, 15, 62, 145, 411, 911, 1785, 1, 1, 9, 17, 79, 191, 615, 1496, 3535, 6789, 1, 1, 10, 19, 98, 243, 873, 2269, 6169, 13903, 26809

Offset: 0

Alois P. Heinz, May 25 2015

			Square array A(n,k) begins:
:   1,   1,   1,   1,   1,    1,    1, ...
:   1,   1,   1,   1,   1,    1,    1, ...
:   2,   3,   4,   5,   6,    7,    8, ...
:   5,   7,   9,  11,  13,   15,   17, ...
:  14,  23,  34,  47,  62,   79,   98, ...
:  43,  71, 105, 145, 191,  243,  301, ...
: 141, 255, 411, 615, 873, 1191, 1575, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Wikipedia, Motzkin number

Columns k=0-1 give: A258312, A140456(n+2).
Main diagonal gives A266386.
Cf. A258307, A258309.

b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (x+k*y)/y, 1)
                  +b(x-1, y, false, k) +b(x-1, y+1, true, k)))
    end:
A:= (n, k)-> b(n, 0, false, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[x_, y_, t_, k_] := b[x, y, t, k] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False, k]*If[t, (x + k*y)/y, 1] + b[x - 1, y, False, k] + b[x - 1, y + 1, True, k]]]; A[n_, k_] :=   b[n, 0, False, k]; Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 23 2017, translated from Maple *)

A(n,k) = Sum_{i=0..min(floor(n/2),k)} C(k,i) * i! * A258307(n,i).

A258309 A(n,k) is the sum over all Motzkin paths of length n of products over all peaks p of (k*x_p+y_p)/y_p, where x_p and y_p are the coordinates of peak p; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 7, 9, 1, 1, 5, 10, 23, 21, 1, 1, 6, 13, 43, 71, 51, 1, 1, 7, 16, 69, 151, 255, 127, 1, 1, 8, 19, 101, 261, 703, 911, 323, 1, 1, 9, 22, 139, 401, 1485, 2983, 3535, 835, 1, 1, 10, 25, 183, 571, 2691, 6973, 14977, 13903, 2188

Offset: 0

Alois P. Heinz, May 25 2015

			Square array A(n,k) begins:
:  1,   1,   1,    1,    1,    1,    1, ...
:  1,   1,   1,    1,    1,    1,    1, ...
:  2,   3,   4,    5,    6,    7,    8, ...
:  4,   7,  10,   13,   16,   19,   22, ...
:  9,  23,  43,   69,  101,  139,  183, ...
: 21,  71, 151,  261,  401,  571,  771, ...
: 51, 255, 703, 1485, 2691, 4411, 6735, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Wikipedia, Motzkin number

Columns k=0-1 give: A001006, A140456(n+2).
Main diagonal gives A261785.
Cf. A258306, A258310.

b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (k*x+y)/y, 1)
                  +b(x-1, y, false, k) +b(x-1, y+1, true, k)))
    end:
A:= (n, k)-> b(n, 0, false, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[x_, y_, t_, k_] := b[x, y, t, k] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False, k]*If[t, (k*x + y)/y, 1] + b[x - 1, y, False, k] + b[x - 1, y + 1, True, k]]];
A[n_, k_] := b[n, 0, False, k];
Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 04 2017, translated from Maple *)

A(n,k) = Sum_{i=0..min(floor(n/2),k)} C(k,i) * i! * A258310(n,i).

A211371 The number of indecomposable n-permutations that have only cycles of length 3 or less.

Original entry on oeis.org

0, 1, 1, 3, 7, 29, 131, 585, 3083, 17089, 97987, 607977, 3926731, 26344001, 185908739, 1358432937, 10279616891, 80819893393, 655374770131, 5482528852761, 47329769940331, 420061520283617, 3832533793409027, 35926633641149865, 345280194806563931

Offset: 0

Geoffrey Critzer, May 11 2012

nonn

			a(4) = 7 because we have: 2431, 3241, 3412, 4132, 4213, 4231, 4321.

Cf. A057693, A003319, A140456.

nn = 20; a = x + x^2/2 + x^3/3; b = Total[Range[0, nn]! CoefficientList[Series[Exp[a], {x, 0, nn}], x]* x^Range[0, nn]]; CoefficientList[Series[1 - 1/b, {x, 0, nn}], x]

G.f.: 1-1/A(x) where A(x) is the o.g.f. for A057693.

cp's OEIS Frontend

A152736 Triangle read by rows: M*Q, where M = an infinite lower triangular matrix with A140456 in every column: (1, 1, 1, 3, 7, 23, 71, ...) and Q = a matrix with A000085 as the main diagonal the rest zeros.

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A188144 Binomial transform A140456(n+1) (indecomposable involutions).

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A258306 A(n,k) is the sum over all Motzkin paths of length n of products over all peaks p of (x_p+k*y_p)/y_p, where x_p and y_p are the coordinates of peak p; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A258309 A(n,k) is the sum over all Motzkin paths of length n of products over all peaks p of (k*x_p+y_p)/y_p, where x_p and y_p are the coordinates of peak p; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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Programs

Maple

Mathematica

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A211371 The number of indecomposable n-permutations that have only cycles of length 3 or less.

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