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.

A011915 a(n) = floor(n*(n-1)*(n-2)*(n-3)/5).

Original entry on oeis.org

0, 0, 0, 0, 4, 24, 72, 168, 336, 604, 1008, 1584, 2376, 3432, 4804, 6552, 8736, 11424, 14688, 18604, 23256, 28728, 35112, 42504, 51004, 60720, 71760, 84240, 98280, 114004, 131544, 151032, 172608, 196416, 222604, 251328, 282744, 317016, 354312
Offset: 0

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Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Sequences of the form floor(24*binomial(n,4)/m): A052762 (m=1), A033486 (m=2), A162668 (m=3), A033487 (m=4), this sequence (m=5), A033488 (m=6), A011917 (m=7), A050534 (m=8), A011919 (m=9), 2*A011930 (m=10), A011921 (m=11), A034827 (m=12), A011923 (m=13), A011924 (m=14), A011925 (m=15), A011926 (m=16), A011927 (m=17), A011928 (m=18), A011929 (m=19), A011930 (m=20), A011931 (m=21), A011932 (m=22), A011933 (m=23), A000332 (m=24), A011935 (m=25),A011936 (m=26), A011937 (m=27), A011938 (m=28), A011939 (m=29), A011940 (m=30), A011941 (m=31), A011942 (m=32), A011795 (m=120).

Programs

  • Magma
    [Floor(n*(n-1)*(n-2)*(n-3)/5): n in [0..60]]; // Vincenzo Librandi, Jun 19 2012
  • Mathematica
    Table[Floor[n(n-1)(n-2)(n-3)/5], {n,60}] (* Stefan Steinerberger, Apr 10 2006 *)
CoefficientList[Series[4*x^4*(1+2*x+2*x^3+x^4)/((1-x)^4*(1+x^5)),{x,0,60}],x] (* Vincenzo Librandi, Jun 19 2012 *)
  • SageMath
    [24*binomial(n,4)//5 for n in range(61)] # G. C. Greubel, Oct 20 2024

Formula

a(n) = +4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) +a(n-5) -4*a(n-6) +6*a(n-7) -4*a(n-8) +a(n-9).
G.f.: 4*x^4*(1+2*x+2*x^3+x^4) / ( (1-x)^5*(1+x+x^2+x^3+x^4) ). - R. J. Mathar, Apr 15 2010
a(n) = 4*A011930(n). - G. C. Greubel, Oct 20 2024

Extensions

More terms from Stefan Steinerberger, Apr 10 2006
Zero added in front by R. J. Mathar, Apr 15 2010

A130810 If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n) is equal to the number of 4-subsets of X containing none of X_i, (i=1,...,n).

Original entry on oeis.org

16, 80, 240, 560, 1120, 2016, 3360, 5280, 7920, 11440, 16016, 21840, 29120, 38080, 48960, 62016, 77520, 95760, 117040, 141680, 170016, 202400, 239200, 280800, 327600, 380016, 438480, 503440, 575360, 654720, 742016, 837760, 942480, 1056720
Offset: 4

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Author

Milan Janjic, Jul 16 2007

Keywords

Comments

Number of n permutations (n>=4) of 3 objects u,v,z, with repetition allowed, containing n-4 u's. Example: if n=4 then n-4 =(0) zero u, a(1)=16 because we have vvvv zzzz vvvz zzzv vvzv zzvz vzvv zvzz zvvv vzzz vvzz zzvv vzvz zvzv zvvz vzzv. - Zerinvary Lajos, Aug 05 2008
a(n) is the number of 3-dimensional elements in an n-cross polytope where n>=4. - Patrick J. McNab, Jul 06 2015

Links

Crossrefs

Cf. A000332, A038207, A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325, A140354, A046092, A130809. Equals twice A162668.

Programs

  • Maple
    a:= n-> binomial(2*n,4) +binomial(n,2) -n*binomial(2*n-2,2);
seq(binomial(n, n-4)*2^4, n=4..37); # Zerinvary Lajos, Dec 07 2007
  • Mathematica
    a[n_] := 16 * Binomial[n, 4]; Array[a, 34, 4] (* Amiram Eldar, Jul 25 2025 *)
  • PARI
    a(n) = 16 * binomial(n, 4); \\ Amiram Eldar, Jul 25 2025

Formula

a(n) = binomial(2*n,4) + binomial(n,2) - n*binomial(2*n-2,2).
a(n) = binomial(n,4)*16. - Zerinvary Lajos, Dec 07 2007
G.f.: 16*x^4/(1-x)^5. - Colin Barker, Apr 14 2012
a(n) = 2*n*(n-1)*(n-2)*(n-3)/3 = 2*A162668(n-3). - Robert Israel, Jul 06 2015
a(n) = 16 * A000332(n). - Alois P. Heinz, Oct 26 2020
E.g.f.: 2*exp(x)*x^4/3. - Stefano Spezia, Jul 17 2025
From Amiram Eldar, Jul 25 2025: (Start)
Sum_{n>=4} 1/a(n) = 1/12.
Sum_{n>=4} (-1)^n/a(n) = 2*log(2) - 4/3. (End)

A294033 Triangle read by rows, expansion of exp(x*z)*z*(tanh(z) + sech(z)), T(n, k) for n >= 1 and 0 <= k <= n-1.

Original entry on oeis.org

1, 2, 2, -3, 6, 3, -8, -12, 12, 4, 25, -40, -30, 20, 5, 96, 150, -120, -60, 30, 6, -427, 672, 525, -280, -105, 42, 7, -2176, -3416, 2688, 1400, -560, -168, 56, 8, 12465, -19584, -15372, 8064, 3150, -1008, -252, 72, 9, 79360, 124650, -97920, -51240, 20160, 6300, -1680, -360, 90, 10, -555731, 872960, 685575, -359040, -140910, 44352, 11550, -2640, -495, 110, 11
Offset: 1

Views

Author

Peter Luschny, Oct 24 2017

Keywords

Examples

			Triangle starts:
  [1][   1]
  [2][   2,   2]
  [3][  -3,   6,    3]
  [4][  -8, -12,   12,    4]
  [5][  25, -40,  -30,   20,    5]
  [6][  96, 150, -120,  -60,   30,  6]
  [7][-427, 672,  525, -280, -105, 42, 7]

Crossrefs

T(n, 0) = signed A065619. Row sums of abs(T(n,k)) = A231179.
Diagonals A000027, A002378, A027480, A162668.
A003506 (m=1), this seq. (m=2), A294034 (m=3).
Cf. A000111, A247453.

Programs

  • Maple
    gf := exp(x*z)*z*(tanh(z)+sech(z)):
s := n -> n!*coeff(series(gf,z,n+2),z,n):
C := n -> PolynomialTools:-CoefficientList(s(n),x):
ListTools:-FlattenOnce([seq(C(n), n=1..7)]);
# Alternatively:
T := (n, k) -> `if`(n = k+1, n,
(k+1)*binomial(n,k+1)*2^(n-k-1)*(euler(n-k-1, 1/2)+euler(n-k-1, 1))):
for n from 1 to 7 do seq(T(n,k), k=0..n-1) od;
  • Mathematica
    L[0] := 1; L[n_] := (-1)^Binomial[n, 2] 2 Abs[PolyLog[-n, -I]];
p[n_] := n Sum[Binomial[n - 1, k - 1] L[k - 1] x^(n - k), {k, 0, n}];
Table[CoefficientList[p[n], x], {n, 1, 11}] // Flatten

Formula

T(n, k) = (k+1)*binomial(n,k+1)*2^(n-k-1)*(Euler(n-k-1, 1/2) + Euler(n-k-1, 1)) for 0 <= k <= n-2.
T(n, k) is the coefficient of x^k of the polynomial p(n) = n*Sum_{k=1..n} binomial(n-1, k-1)*L(k-1)*x^(n-k) and L(n) = (-1)^binomial(n,2)*A000111(n). In particular n divides T(n, k).
Showing 1-3 of 3 results.

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

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