A078700 -id:A078700 - cp's OEIS frontend

A078698 Number of ways to lace a shoe that has n pairs of eyelets such that each eyelet has at least one direct connection to the opposite side.

1, 2, 20, 396, 14976, 907200, 79315200, 9551001600, 1513528934400, 305106949324800, 76296489615360000, 23175289163980800000, 8404709419090575360000, 3587225703492542791680000, 1779970753996760560435200000, 1016036270188884847558656000000, 661106386935312429191528448000000

Offset: 1

Hugo Pfoertner, Dec 18 2002

nonn

The lace is "directed": reversing the order of eyelets along the path counts as a different solution. It must begin and end at the extreme pair of eyelets,

			a(3) = 20: label the eyelets 1,2,3 from front to back on the left side then 4,5,6 from back to front on the right side. The lacings are: 124356 154326 153426 142536 145236 135246 and the following and their mirror images: 125346 124536 125436 152346 153246 152436 154236.
Examples for n=2,3,4 can be found following the FORTRAN program at given link.

C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 494.

A. Khrabrov and K. Kokhas, Points on a line, shoelace and dominoes, arXiv:1505.06309 [math.CO], (23-May-2015).
Hugo Pfoertner, Fortran program.
Hugo Pfoertner, Results for n=1..4.
Hugo Pfoertner, Results for n=5.
Index entries for sequences related to shoe lacings

Cf. A078700, A078702, A078602.

Fortran
```
c Program provided at Pfoertner link
```

a[n_] := (n-1)!^2 Sum[Binomial[n-k, k]^2, {k, 0, n/2}];
Array[a, 17] (* Jean-François Alcover, Jul 20 2018 *)

Conjecture: a(n) = (n-1)!^2*A051286(n). - Vladeta Jovovic, Sep 14 2005 (correct, see the Khrabrov/Kokhas reference, Joerg Arndt, May 26 2015)

Terms a(9) and beyond (using A051286) from Joerg Arndt, May 26 2015

A005922 a(1)=1; a(n) = n!*Fibonacci(n+2), n > 1.

Original entry on oeis.org

1, 6, 30, 192, 1560, 15120, 171360, 2217600, 32296320, 522547200, 9300614400, 180583603200, 3798482688000, 86044973414400, 2088355965696000, 54064489070592000, 1487129136869376000, 43312058119249920000

Offset: 1

N. J. A. Sloane

From solution to a difference equation.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
P. R. J. Asveld & N. J. A. Sloane, Correspondence, 1987
P. R. J. Asveld, Another family of Fibonacci-like sequences, Fib. Quart., 25 (1987), 361-364.

Cf. A005921, A078700.

[1] cat [Factorial(n)*Fibonacci(n+2): n in [2..20]]; // Vincenzo Librandi, Jul 10 2012

Join[{1},Table[n!Fibonacci[n+2],{n,2,20}]] (* Harvey P. Dale, May 26 2024 *)

a(n)=if(n>1, n!*fibonacci(n+2), 1) \\ Charles R Greathouse IV, Jun 30 2017

More terms and better description from Vladeta Jovovic, Jan 23 2005

A078702 Number of ways to lace a shoe that has n pairs of eyelets such that each eyelet has at least one direct connection to the opposite side.

Original entry on oeis.org

1, 2, 13, 213, 7584, 454380, 39665160, 4775586480, 756765576000, 152553490810560, 38148245068953600, 11587644586640707200, 4202354709635579481600, 1793612851748170637184000, 889985376998423302704307200, 508018135094443467957310848000, 330553193467656241628008759296000

Offset: 1

Hugo Pfoertner, Dec 18 2002

nonn

The lace is "undirected": reversing the order of eyelets along the path does not count as a different solution. It must begin and end at the extreme pair of eyelets,

			a(3) = 13: label the eyelets 1,2,3 from front to back on the left side then 4,5,6 from back to front on the right side. The lacings are: 124356 154326 153426 142536 145236 135246 125346 124536 125436 152346 153246 152436 154236.

Index entries for sequences related to shoe lacings

Cf. A078698, A078700, A078675.

a(n) = (A078698(n) + A078700(n))/2.

More terms from Hugo Pfoertner, Mar 11 2025

A139167 Triangle T(n,k) read by rows: the coefficient [x^k] of the polynomial (n-1)! sum_{i=0..n} Fibonacci(i)binomial(x,n-i), read by rows, 0<=k

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 18, 11, 0, 1, 120, 50, 23, -2, 1, 960, 494, 65, 45, -5, 1, 9360, 4344, 1354, -15, 85, -9, 1, 105840, 51876, 10444, 3409, -350, 154, -14, 1, 1370880, 653232, 172444, 13300, 8729, -1232, 266, -20, 1, 19958400, 9654480, 2194380, 483272, -13923, 22449, -3150, 438, -27, 1

Offset: 1

Roger L. Bagula, Jun 05 2008

Row sums are 1, 2, 6, 30, 192, 1560, 15120, 171360, 2217600, 32296320,... (see A078700)

			1;
1, 1;
4, 1, 1;
18, 11, 0, 1;
120, 50, 23, -2, 1;
960, 494, 65, 45, -5, 1;
9360, 4344, 1354, -15,85, -9, 1;
105840, 51876, 10444, 3409, -350, 154, -14, 1;
1370880, 653232, 172444, 13300, 8729, -1232, 266, -20, 1;
19958400, 9654480, 2194380, 483272, -13923, 22449, -3150, 438, -27, 1;

Brendan Hassett, Introduction to algebraic Geometry,Cambridge University Press. New York,2007, page 229

Cf. A000045.

B := proc(x,k)
    mul( (x-i+1)/i,i=1..k) ;
end proc:
A139167 := proc(n,k)
    local f,i ;
    f := 0 ;
    for i from 0 to n do
        f := f+combinat[fibonacci](i)*B(x,n-i) ;
    end do;
    %*(n-1)! ;
    coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, May 08 2013

Clear[a, p, x] a[0] = 0; a[1] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2]; p[x, 0] = a[0]; p[x_, n_] := p[x, n] = Sum[a[i]*Binomial[x, n - i], {i, 0, n}]; Table[If[n > 0, ExpandAll[(n - 1)!*p[x, n]], 0], {n, 0, 10}]; a = Table[CoefficientList[If[n > 0, ExpandAll[(n - 1)!*p[x, n]], 0], x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[If[n > 0, ExpandAll[(n - 1)!*p[x, n]], 0], x]], {n, 0, 10}]

Edited by R. J. Mathar, May 08 2013

A322044 Triangle read by rows: numerators of coefficients (highest degree first) of polynomials interpolating Fibonacci numbers.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 3, 14, 30, 1, 2, 23, 94, 192, 1, 0, 35, 180, 744, 1560, 1, -3, 55, 255, 1744, 7308, 15120, 1, -7, 91, 245, 3304, 19922, 82284, 171360, 1, -12, 154, 0, 5929, 40572, 255996, 1068240, 2217600, 1, -18, 258, -756, 11361, 64638, 602972, 3746376, 15533568, 32296320

Offset: 0

N. J. A. Sloane, Dec 07 2018

Row n has denominator n!.

			Triangle begins:
  1;
  1,  2;
  1,  3,  6;
  1,  3, 14,  30;
  1,  2, 23,  94,  192;
  1,  0, 35, 180,  744, 1560;
  1, -3, 55, 255, 1744, 7308, 15120;
  ...

Brian Hopkins and Aram Tangboonduangjit, Fibonacci-producing rational polynomials, Fib. Q., 56:4 (2018), 303-312.

Alois P. Heinz, Rows n = 0..140, flattened (first 17 rows from Brian Hopkins)

Main diagonal gives A078700(n+1).
Second column is negation of A167544.
Cf. A000045, A000142.

F:= proc(n) option remember; (<<0|1>, <1|1>>^n)[1, 2] end:
T:= n-> (p-> seq(coeff(p, x, n-j), j=0..n))(n!*expand(add(
      F(i+n+2)*binomial(x, i)*binomial(n-x, n-i), i=0..n))):
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 24 2019

F[n_] := F[n] = MatrixPower[{{0, 1}, {1, 1}}, n][[1, 2]];
T[n_] := Function[p, Table[Coefficient[p, x, n - j], {j, 0, n}]][n! * FunctionExpand[Sum[F[i + n + 2] Binomial[x, i] Binomial[n - x, n - i], {i, 0, n}]]];
T /@ Range[0, 10] // Flatten (* Jean-François Alcover, May 29 2020, after Alois P. Heinz *)

The degree n polynomial is defined to be the interpolating polynomial of (0, F(n+2)), (1, F(n+3)), ..., (n,F(2n+2)) where F(n) is the n-th Fibonacci number. Theorem 2.1 of the paper proves the alternative form Sum_{i=0..n} F(i+n+2) * binomial(x,i) * binomial(n-x,n-i). - Brian Hopkins, Feb 24 2019

Edited by Brian Hopkins, Feb 24 2019

cp's OEIS Frontend

A078698 Number of ways to lace a shoe that has n pairs of eyelets such that each eyelet has at least one direct connection to the opposite side.

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A005922 a(1)=1; a(n) = n!*Fibonacci(n+2), n > 1.

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A078702 Number of ways to lace a shoe that has n pairs of eyelets such that each eyelet has at least one direct connection to the opposite side.

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A139167 Triangle T(n,k) read by rows: the coefficient [x^k] of the polynomial (n-1)! *sum_{i=0..n} Fibonacci(i)*binomial(x,n-i), read by rows, 0<=k

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A322044 Triangle read by rows: numerators of coefficients (highest degree first) of polynomials interpolating Fibonacci numbers.

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Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A139167 Triangle T(n,k) read by rows: the coefficient [x^k] of the polynomial (n-1)! sum_{i=0..n} Fibonacci(i)binomial(x,n-i), read by rows, 0<=k