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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A322631 a(n) = 2*binomial(7*n-1,2*n)/(7*n-1).

Original entry on oeis.org

5, 110, 3876, 164450, 7713420, 385300240, 20096692635, 1081790956890, 59647783837425, 3351648108957720, 191230475831922200, 11049110585626417200, 645189590847792998601, 38014810319396501088720, 2257261555792984515847380, 134939208350635886836436490
Offset: 1

Views

Author

Hugo Pfoertner, Dec 21 2018

Keywords

Comments

In 2012, Nakamigawa and Tokushige stated: Let A[x,y] = number of lattice paths starting at (0,0) that stay in y < 2*x/5 + 2/5 and B[x,y] = number of lattice paths starting at (0,0) that stay in y < 2*x/5 + 1/5, then a(t) = A[5*t-1,2*t-1] + B[5*t-1,2*t-1]. Their theorem was mentioned by D. Knuth in Problem 4 "Lattice Paths of Slope 2/5" in his lecture "Problems That Philippe (Flajolet) Would Have Loved". Knuth reported the empirical observation that A[5*t-1,2*t-1]/B[5*t-1,2*t-1] = a - b/t + O(t^-2), with constants a~=1.63026 and b~=0.159. Knuth's conjecture was proved by C. Banderier and M. Wallner, who also found the exact values of a and b. Numerical values of a and b are provided in A322632 and A322633.

Examples

			  A[i,0] = B[i,0] = 1.
  A[i,j] = if 5*j < 2*i + 2 then A[i-1,j] + A[i,j-1] , else 0.
  \i 1   2   3   4   5   6   7   8   9  10  11  12   13   14
  j --------------------------------------------------------
  0| 1   1   1   1   1   1   1   1   1   1   1   1    1    1
  1| 0   1   2   3   4   5   6   7   8   9  10  11   12   13
  2| 0   0   0   0   4   9  15  22  30  39  49  60   72   85
  3| 0   0   0   0   0   0  15  37  67 106 155 215  287  372
  4| 0   0   0   0   0   0   0   0   0 106 261 476  763 1135
  5| 0   0   0   0   0   0   0   0   0   0   0 476 1239 2374
.
  B[i,j] = if 5*j < 2*i + 1 then B[i-1,j] + B[i,j-1], else 0.
  \i 1   2   3   4   5   6   7   8   9  10  11  12   13   14
  j --------------------------------------------------------
  0| 1   1   1   1   1   1   1   1   1   1   1   1    1    1
  1| 0   0   1   2   3   4   5   6   7   8   9  10   11   12
  2| 0   0   0   0   3   7  12  18  25  33  42  52   63   75
  3| 0   0   0   0   0   0   0  18  43  76 118 170  233  308
  4| 0   0   0   0   0   0   0   0   0  76 194 364  597  905
  5| 0   0   0   0   0   0   0   0   0   0   0   0  597 1502
.
  A+B:
  \i 1   2   3   4   5   6   7   8   9  10  11  12   13   14
  j --------------------------------------------------------
  0| 2   2   2   2   2   2   2   2   2   2   2   2    2    2
  1| 0   1   3   5   7   9  11  13  15  17  19  21   23   25
  2| 0   0   0   0   7  16  27  40  55  72  91 112  135  160
  3| 0   0   0   0   0   0  15  55 110 182 273 385  520  680
  4| 0   0   0   0   0   0   0   0   0 182 455 840 1360 2040
  5| 0   0   0   0   0   0   0   0   0   0   0 476 1836 3876
.
  t = 1: a(1) = 5 because
  A[5*1-1,2*1-1] = A[4,1] = 3, B[4,1] = 2,  A[4,1]+B[4,1] = 5;
  t = 2: a(2) = 110 because
  A[5*2-1,2*2-1] = A[9,3] = 67, B[9,3] = 43,  A[9,3]+B[9,3] = 110;
  t = 3: a(3) = 3876 because
  A[5*3-1,2*3-1] = A[14,5] = 2374, B[14,5] = 1502,  A[14,5]+B[14,5] = 3876.

Links

Crossrefs

Cf. A274052, A322632, A322633.

Programs

  • Maple
    a:=n->2*binomial(7*n-1,2*n)/(7*n-1): seq(a(n),n=1..20); # Muniru A Asiru, Dec 21 2018
  • PARI
    for(t=1,16,print1(binomial(7*t-1,2*t)*(2/(7*t-1)),", "))

Formula

From Robert Israel, Dec 23 2018: (Start)
7*(7*n + 4)*(7*n + 1)*(7*n + 5)*(7*n + 2)*(7*n - 1)*(7*n + 3)*a(n) - 10*(5*n + 1)*(5*n + 2)*(2*n + 1)*(5*n + 3)*(5*n + 4)*(n + 1)*a(n + 1) = 0.
G.f.: 5*x*hypergeom([6/7, 1, 8/7, 9/7, 10/7, 11/7, 12/7], [6/5, 7/5, 3/2, 8/5, 9/5, 2], (823543*x)*1/12500)
a(n) ~ sqrt(35/Pi)*(823543/12500)^n/(49*n^(3/2)). (End)

A322632 Decimal expansion of the real solution to 23*x^5 - 41*x^4 + 10*x^3 - 6*x^2 - x - 1 = 0. Constant occurring in the asymptotic behavior of the number of lattice paths of slope 2/5, observed by D. Knuth.

Original entry on oeis.org

1, 6, 3, 0, 2, 5, 7, 6, 6, 2, 9, 9, 0, 3, 5, 0, 1, 4, 0, 4, 2, 4, 8, 0, 1, 8, 4, 9, 3, 1, 5, 9, 8, 6, 3, 0, 0, 5, 1, 4, 5, 8, 4, 4, 2, 6, 6, 9, 0, 1, 4, 9, 4, 0, 5, 8, 4, 9, 8, 5, 0, 2, 6, 5, 9, 5, 2, 5, 6, 8, 9, 1, 2, 9, 8, 6, 8, 5, 0, 4, 7, 9, 8, 3, 4, 1, 3, 2, 4, 1
Offset: 1

Views

Author

Hugo Pfoertner, Dec 21 2018

Keywords

Comments

In his 2014 lecture in Paris "Problems That Philippe (Flajolet) Would Have Loved" D. Knuth discussed as Problem 4 "Lattice Paths of Slope 2/5" and reported as an empirical observation that A[5*t-1,2*t-1]/B[5*t-1,2*t-1] = a - b/t + O(t^-2), with constants a~=1.63026 and b~=0.159. For the meaning of A and B see A322631. The exact values of a (this sequence) and b (A322633) were found in 2016 by Banderier and Wallner.

Examples

			1.6302576629903501404248018493159863005145844266901494058498502659525689...

Links

Crossrefs

Cf. A322631, A322633.

Programs

  • Maple
    evalf[100](solve(23*x^5-41*x^4+10*x^3-6*x^2-x-1=0,x)[1]); # Muniru A Asiru, Dec 21 2018
  • Mathematica
    RealDigits[Root[23#^5 - 41#^4 + 10#^3 - 6#^2 - # - 1&, 1], 10, 100][[1]] (* Jean-François Alcover, Dec 30 2018 *)
  • PARI
    solve(x=1,2,23*x^5-41*x^4+10*x^3-6*x^2-x-1)
Showing 1-2 of 2 results.

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