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 1-5 of 5 results.

A107711 Triangle read by rows: T(0,0)=1, T(n,m) = binomial(n,m) * gcd(n,m)/n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 5, 10, 5, 1, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 1, 7, 7, 35, 7, 7, 1, 1, 1, 1, 4, 28, 14, 14, 28, 4, 1, 1, 1, 1, 9, 12, 42, 126, 42, 12, 9, 1, 1, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 1, 1, 11, 55, 165, 66, 462, 66, 165, 55, 11, 1, 1
Offset: 0

Views

Author

Leroy Quet, Jun 10 2005

Keywords

Comments

T(0,0) is an indeterminate, but 1 seems a logical value to assign it. T(n,0) = T(n,1) = T(n,n-1) = T(n,n) = 1.
T(2n,n) = A001700(n-1) (n>=1). - Emeric Deutsch, Jun 13 2005

Examples

			T(6,2)=5 because binomial(6,2)*gcd(6,2)/6 = 15*2/6 = 5.
The triangle T(n,m) begins:
n\m 0  1  2   3   4    5   6   7  8  9  10...
0:  1
1:  1  1
2:  1  1  1
3:  1  1  1   1
4:  1  1  3   1   1
5:  1  1  2   2   1    1
6:  1  1  5  10   5    1   1
7:  1  1  3   5   5    3   1   1
8:  1  1  7   7  35    7   7   1  1
9:  1  1  4  28  14   14  28   4  1  1
10: 1  1  9  12  42  126  42  12  9  1   1
n\m 0  1  2   3   4    5   6   7  8  9  10...
... reformatted - _Wolfdieter Lang_, Feb 23 2014

Links

Crossrefs

Row sums A159554.
Cf. A007318, A001700, A000108, A050169, A234040, A026741, A234042.
Cf. A109004.

Programs

  • Haskell
    a107711 n k = a107711_tabl !! n !! k
a107711_row n = a107711_tabl !! n
a107711_tabl = [1] : zipWith (map . flip div) [1..]
               (tail $ zipWith (zipWith (*)) a007318_tabl a109004_tabl)
-- Reinhard Zumkeller, Feb 28 2014
  • Maple
    a:=proc(n,k) if n=0 and k=0 then 1 elif k<=n then binomial(n,k)*gcd(n,k)/n else 0 fi end: for n from 0 to 13 do seq(a(n,k),k=0..n) od; # yields sequence in triangular form. - Emeric Deutsch, Jun 13 2005
  • Mathematica
    T[0, 0] = 1; T[n_, m_] := Binomial[n, m] * GCD[n, m]/n;
Table[T[n, m], {n, 1, 13}, {m, 1, n}] // Flatten (* Jean-François Alcover, Nov 16 2017 *)

Formula

From Wolfdieter Lang, Feb 28 2014 (Start)
T(n, m) = T(n-1,m)*(n-1)*gcd(n,m)/((n-m)*gcd(n-1,m)), n > m >= 1, T(n, 0) = 1, T(n, n) = 1, otherwise 0.
T(n, m) = binomial(n-1,m-1)*gcd(n,m)/m for n >= m >= 1, T(n,0) = 1, otherwise 0 (from iteration of the preceding recurrence).
T(n, m) = T(n-1, m-1)*(n-1)*gcd(n,m)/(m*gcd(n-1,m-1)) for n >= m >= 2, T(n, 0) = 1, T(n, 1) = 0, otherwise 0 (from the preceding formula).
T(2*n, n) = A001700(n-1) (n>=1) (see the Emeric Deutsch comment above), T(2*n, n-1) = A234040(n), T(2*n+1,n) = A000108(n), n >= 0 (Catalan numbers).
Column sequences: T(n+2, 2) = A026741(n+1), T(n+3, 3) = A234041(n), T(n+4, 4) = A208950(n+2), T(n+5, 5) = A234042, n >= 0. (End)

Extensions

More terms from Emeric Deutsch, Jun 13 2005

A234043 a(n) = binomial(5*(n+1),4)/5, with n >= 0.

Original entry on oeis.org

1, 42, 273, 969, 2530, 5481, 10472, 18278, 29799, 46060, 68211, 97527, 135408, 183379, 243090, 316316, 404957, 511038, 636709, 784245, 956046, 1154637, 1382668, 1642914, 1938275, 2271776, 2646567, 3065923, 3533244, 4052055, 4626006, 5258872, 5954553, 6717074
Offset: 0

Views

Author

Wolfdieter Lang, Feb 24 2014

Keywords

Comments

Used as one of the 5-section parts of A234042.
The Fuss-Catalan numbers are Cat(d,k) = (1/(k*(d-1)+1))*binomial(k*d,k) and enumerate the (d+1)-gon partitions of a (k*(d-1)+2)-gon (cf. Whieldon and Schuetz link). a(n) = Cat(n,5) (Offset=1), so enumerates the (n+1)-gon partitions of a (5*(n-1)+2)-gon. Analogous series are A000326 (k=3) and A100157 (k=4). - Tom Copeland, Oct 05 2014

Links

Crossrefs

Cf. A100157, A000326, A001622, A151989, A234042.

Programs

  • Magma
    [Binomial(5*(n+1),4)/5: n in [0..40]]; // Vincenzo Librandi, Feb 26 2014
  • Mathematica
    CoefficientList[Series[(1 + 37 x + 73 x^2 + 14 x^3)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 26 2014 *)

Formula

G.f: (1 + 37*x + 73*x^2 + 14*x^3)/(1-x)^5.
a(n) = A234042(5*n+1) for n >= 0.
a(n) = (n+1)*(5*n+2)*(5*n+3)*(5*n+4)/24.
From Amiram Eldar, Sep 20 2022: (Start)
Sum_{n>=0} 1/a(n) = 10*sqrt(5)*log(phi) + 5*log(5) - 2*sqrt(25-38/sqrt(5))*Pi, where phi is the golden ratio (A001622).
Sum_{n>=0} (-1)^n/a(n) = 4*sqrt(5)*log(phi) + 2*sqrt(26-38/sqrt(5))*Pi - 32*log(2). (End)

A238471 a(n) = binomial(5n+6, 4)/5 for n >= 0.

Original entry on oeis.org

3, 66, 364, 1197, 2990, 6293, 11781, 20254, 32637, 49980, 73458, 104371, 144144, 194327, 256595, 332748, 424711, 534534, 664392, 816585, 993538, 1197801, 1432049, 1699082, 2001825, 2343328, 2726766, 3155439, 3632772, 4162315, 4747743, 5392856, 6101579, 6877962
Offset: 0

Views

Author

Wolfdieter Lang, Feb 28 2014

Keywords

Comments

This sequence appears in the 5-section of A234042.

Crossrefs

Cf. A000332, A001622, A234042, A151989, A234043, A238472, A238473.

Programs

  • Mathematica
    a[n_] := Binomial[5*n + 6, 4]/5; Array[a, 40, 0] (* Amiram Eldar, Sep 20 2022 *)

Formula

a(n) = binomial(5*n+6, 4)/5 = (5*n+6)*(5*n+3)*(5*n+4)*(n+1)/4! for n >= 0.
a(n) = A234042(5*n+2) for n >= 0.
a(n) = 3*b(n) + 51*b(n-1) + 64*b(n-2) + 7*b(n-3), with b(n) = binomial(n+4,4) = A000332(n) for n >= 0.
O.g.f.: (3 + 51*x + 64*x^2 + 7*x^3)/(1-x)^5.
Sum_{n>=0} 1/a(n) = 2*sqrt(5+2/sqrt(5))*Pi - 10*sqrt(5)*log(phi) - 15*log(5) + 20, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 20 2022

A238472 a(n) = binomial(5*n+7, 4)/5 for n >= 0.

Original entry on oeis.org

7, 99, 476, 1463, 3510, 7192, 13209, 22386, 35673, 54145, 79002, 111569, 153296, 205758, 270655, 349812, 445179, 558831, 692968, 849915, 1032122, 1242164, 1482741, 1756678, 2066925, 2416557, 2808774, 3246901, 3734388, 4274810, 4871867, 5529384, 6251311, 7041723
Offset: 0

Views

Author

Wolfdieter Lang, Feb 28 2014

Keywords

Comments

This sequence appears in the 5-section of A234042.

Crossrefs

Cf. A000332, A001622, A234042, A151989, A234043, A238471, A238473.

Programs

  • Mathematica
    a[n_] := Binomial[5*n + 7, 4]/5; Array[a, 40, 0] (* Amiram Eldar, Sep 20 2022 *)

Formula

a(n) = binomial(5*n+7, 4)/5 for n >= 0.
a(n) = A234042(5*n+3) for n >= 0.
a(n) = 7*b(n) + 64*b(n-1) + 51*b(n-2) + 3*b(n-3), with b(n) = binomial(n+4,4) = A000332(n) for n >= 0.
O.g.f.: (7 + 64*x + 51*x^2 + 3*x^3)/(1-x)^5.
Sum_{n>=0} 1/a(n) = 2*sqrt(5+2/sqrt(5))*Pi + 10*sqrt(5)*log(phi) + 15*log(5) - 50, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 20 2022

A238473 a(n) = binomial(5*n+8, 4)/5 for n >= 0.

Original entry on oeis.org

14, 143, 612, 1771, 4095, 8184, 14763, 24682, 38916, 58565, 84854, 119133, 162877, 217686, 285285, 367524, 466378, 583947, 722456, 884255, 1071819, 1287748, 1534767, 1815726, 2133600, 2491489, 2892618, 3340337, 3838121, 4389570, 4998409, 5668488, 6403782, 7208391
Offset: 0

Views

Author

Wolfdieter Lang, Feb 28 2014

Keywords

Comments

This sequence appears in the 5-section of A234042.

Crossrefs

Cf. A000332, A001622, A234042, A151989, A234043, A238471, A238472.

Programs

  • Mathematica
    a[n_] := Binomial[5*n + 8, 4]/5; Array[a, 40, 0] (* Amiram Eldar, Sep 20 2022 *)

Formula

a(n) = binomial(5*n+8, 4)/5 = (5*n+8)*(5*n+7)*(5*n+6)*(n+1)/4! for n >= 0.
a(n) = A234042(5*n+2) for n >= 0.
a(n) = 14*b(n) + 73*b(n-1) + 37*b(n-2) + b(n-3), with b(n) = binomial(n+4,4) = A000332(n) for n >= 0.
O.g.f.: (14 + 73*x + 37*x^2 + x^3)/(1-x)^5.
Sum_{n>=0} 1/a(n) = 110/3 - 2*sqrt(25 - 38/sqrt(5))*Pi - 10*sqrt(5)*log(phi) - 5*log(5), where phi is the golden ratio (A001622). - Amiram Eldar, Sep 20 2022
Showing 1-5 of 5 results.

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