cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A100641 Triangle read by rows: denominators of Cotesian numbers C(n,k) (0 <= k <= k).

Original entry on oeis.org

1, 2, 2, 6, 3, 6, 8, 8, 8, 8, 90, 45, 15, 45, 90, 288, 96, 144, 144, 96, 288, 840, 35, 280, 105, 280, 35, 840, 17280, 17280, 640, 17280, 17280, 640, 17280, 17280, 28350, 14175, 14175, 14175, 2835, 14175, 14175, 14175, 28350, 89600, 89600, 2240, 5600, 44800, 44800, 5600
Offset: 0

Views

Author

N. J. A. Sloane, Dec 04 2004

Keywords

Examples

			Triangle A100640/A100641 begins:
[1],
[1/2, 1/2],
[1/6, 2/3, 1/6],
[1/8, 3/8, 3/8, 1/8],
[7/90, 16/45, 2/15, 16/45, 7/90],
[19/288, 25/96, 25/144, 25/144, 25/96, 19/288],
[41/840, 9/35, 9/280, 34/105, 9/280, 9/35, 41/840],
[751/17280, 3577/17280, 49/640, 2989/17280, 2989/17280, 49/640, 3577/17280, 751/17280],
...

References

  • Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 513.
  • L. M. Milne-Thompson, Calculus of Finite Differences, MacMillan, 1951, p. 170.

Crossrefs

Cf. A100640-A100648, A100620, A100621, A002177, A002176.

Programs

  • Maple
    (This defines the Cotesian numbers C(n,i)) with(combinat); C:=proc(n,i) if i=0 or i=n then RETURN( (1/n!)*add(n^a*stirling1(n,a)/(a+1),a=1..n+1) ); fi; (1/n!)*binomial(n,i)* add( add( n^(a+b)*stirling1(i,a)*stirling1(n-i,b)/((b+1)*binomial(a+b+1,b+1)), b=1..n-i+1), a=1..i+1); end;
# Another program:
T:=proc(n,k) (-1)^(n-k)*(n/(n-1))*binomial(n-1,k-1)* integrate(expand(binomial(t-1,n))/(t-k), t=1..n); end;
[[1], seq( [seq(T(n,k),k=1..n)], n=2..14)];
  • Mathematica
    a[n_, i_] /; i == 0 || i == n = 1/n!*Sum[n^a StirlingS1[n, a]/(a + 1), {a, 1, n + 1}]; a[n_, i_] = 1/n!*Binomial[n, i] Sum[n^(a + b)*StirlingS1[i, a]*StirlingS1[n - i, b]/((b + 1)*Binomial[a + b + 1, b + 1]), {b, 1, n}, {a, 1, i + 1}]; Table[a[n, i], {n, 0, 10}, {i, 0, n}] // Flatten // Denominator // Take[#, 52] &
(* Jean-François Alcover, May 17 2011, after Maple prog. *)

A100640 Triangle read by rows: numerators of Cotesian numbers C(n,k) (0 <= k <= n).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 3, 3, 1, 7, 16, 2, 16, 7, 19, 25, 25, 25, 25, 19, 41, 9, 9, 34, 9, 9, 41, 751, 3577, 49, 2989, 2989, 49, 3577, 751, 989, 2944, -464, 5248, -454, 5248, -464, 2944, 989, 2857, 15741, 27, 1209, 2889, 2889, 1209, 27, 15741, 2857, 16067, 26575, -16175, 5675
Offset: 0

Views

Author

N. J. A. Sloane, Dec 04 2004

Keywords

References

  • Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 513.
  • L. M. Milne-Thompson, Calculus of Finite Differences, MacMillan, 1951, p. 170.

Crossrefs

Cf. A100641-A100648, A100620, A100621, A002177, A002176.

Programs

  • Maple
    (This defines the Cotesian numbers C(n,i)) with(combinat); C:=proc(n,i) if i=0 or i=n then RETURN( (1/n!)*add(n^a*stirling1(n,a)/(a+1),a=1..n+1) ); fi; (1/n!)*binomial(n,i)* add( add( n^(a+b)*stirling1(i,a)*stirling1(n-i,b)/((b+1)*binomial(a+b+1,b+1)), b=1..n-i+1), a=1..i+1); end;
# Another program:
T:=proc(n, k) (-1)^(n-k)*(n/(n-1))*binomial(n-1, k-1)* integrate(expand(binomial(t-1, n))/(t-k), t=1..n); end;
[[1], seq( [seq(T(n, k), k=1..n)], n=2..14)];
  • Mathematica
    a[n_, i_] /; i == 0 || i == n = 1/n! Sum[n^a*StirlingS1[n, a]/(a + 1), {a, 1, n + 1}]; a[n_, i_] = 1/n!*Binomial[n, i]*Sum[ n^(a + b)*StirlingS1[i, a]*StirlingS1[n - i, b]/((b + 1)*Binomial[a + b + 1, b + 1]), {b, 1, n}, {a, 1, i + 1}]; Table[a[n, i], {n, 0, 10}, {i, 0, n}] // Flatten // Numerator //  Take[#, 59]&
(* Jean-François Alcover, May 17 2011, after Maple prog. *)

A321122 a(n) = n-th row common denominator of A321121.

Original entry on oeis.org

4, 2, 3, 8, 36, 96, 44, 360, 492, 448, 1836, 5016, 2284, 18720, 25572, 23288, 95436, 260736, 118724, 973080, 1329252, 1210528, 4960836, 13553256, 6171364, 50581440, 69095532, 62924168, 257868036, 704508576, 320792204, 2629261800, 3591638412, 3270846208
Offset: 0

Views

Author

Franck Maminirina Ramaharo, Nov 21 2018

Keywords

References

  • Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, The Theory of Splines and Their Applications, Academic Press, 1967. See p. 47, Table 2.5.3.

Links

  • Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, Chapter II. The Cubic Spline, Mathematics in Science and Engineering Volume 38 (1967), pp. 9-74.

Crossrefs

Cf. A321121 (Numerators).
Cf. A002176, A093736, A100621, A100646, A100648, A321119.

Programs

  • Mathematica
    s = -2 + Sqrt[3];
e[n_] := s*(2 + s)*(-1 + s^n)/(2*(1 - s)*(-s + s^n));
f[n_, k_] := 6*s^(1 - k)*(s^(2*k) + s^n)/((1 - s)*(-s + s^n));
w[n_, k_] := If[k == 0 || k == n, 1/4 + e[n]/6, If[k == 1 || k == n - 1, 2 - (1 + 1/6)*e[n], 1 + f[n, k]/4]];
a[n_] := LCM @@ Table[Denominator[FullSimplify[w[n, k]]], {k, 0, n}];
Join[{4, 3, 2}, Table[a[n], {n, 3, 50}]]
  • Maxima
    s : -2 + sqrt(3)$
e(n) := s*(2 + s)*(-1 + s^n)/(2*(1 - s)*(-s + s^n))$
f(n, k) := 6*s^(1 - k)*(s^(2*k) + s^n)/((1 - s)*(-s + s^n))$
w(n, k) := if k = 0 or k = n then 1/4 + e(n)/6 else if k = 1 or k = n - 1 then 2 - (1 + 1/6)*e(n) else 1 + f(n, k)/4$
a(n) := lcm(makelist(denom(fullratsimp(w(n, k))), k, 0, n))$
append([4, 2, 3], makelist(a(n), n, 3, 50));

Formula

Let s = -2 + sqrt(3), and define e(n) = s*(2 + s)*(-1 + s^n)/(2*(1 - s)*(-s + s^n)), f(n,k) = 6*s^(1 - k)*(s^(2*k) + s^n)/((1 - s)*(-s + s^n)), and w(n,0) = 1/4 + e(n)/6, w(n,1) = 2 - (1 + 1/6)*e(n), w(n,k) = 1 + f(n,k)/4 for 2 <= k <= n - 2. Then a(n) = LCM of denominators of {w(n,k), 0 <= k <= n} for n >= 3.
a(n) = 52*a(n-6) - a(n-12) for n >= 15 (conjectured).
G.f.: (4 + 2*x + 3*x^2 + 8*x^3 + 36*x^4 + 96*x^5 - 164*x^6 + 256*x^7 + 336*x^8 + 32*x^9 - 36*x^10 + 24*x^11 + 2*x^13 - 9*x^14)/(1 - 52*x^6 + x^12) (conjectured).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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