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 1-6 of 6 results.

A160624 Denominator of Laguerre(n, 2).

Original entry on oeis.org

1, 1, 1, 3, 3, 15, 45, 315, 315, 2835, 14175, 155925, 467775, 6081075, 42567525, 638512875, 58046625, 10854718875, 8881133625, 1856156927625, 9280784638125, 194896477400625, 2143861251406875, 3792985290950625, 147926426347074375
Offset: 0

Views

Author

N. J. A. Sloane, Nov 14 2009

Keywords

Examples

			1, -1, -1, -1/3, 1/3, 11/15, 37/45, 209/315, 113/315, 23/2835, -4381/14175, -84389/155925, -310517/467775, ... = A160623/A160624.

Links

Crossrefs

For numerators see A160623. Different from A049606.
Cf. A295382.

Programs

  • Magma
    [Denominator((&+[Binomial(n,k)*((-2)^k/Factorial(k)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, May 06 2018
  • Maple
    seq(denom(orthopoly[L](n,2)), n=0 .. 100); # Robert Israel, Jul 23 2015
  • Mathematica
    Denominator[LaguerreL[Range[0,30],2]] (* Vincenzo Librandi, May 24 2012 *)
  • PARI
    for(n=0,30, print1(denominator(sum(k=0,n, binomial(n,k)*((-2)^k/k!))), ", ")) \\ G. C. Greubel, May 06 2018
  • PARI
    a(n) = denominator(pollaguerre(n, 0, 2)); \\ Michel Marcus, Feb 05 2021

Formula

Denominators of coefficients in expansion of exp(-2*x/(1 - x))/(1 - x). - Ilya Gutkovskiy, Aug 29 2018

A160623 Numerator of Laguerre(n, 2).

Original entry on oeis.org

1, -1, -1, -1, 1, 11, 37, 209, 113, 23, -4381, -84389, -310517, -4103887, -25059901, -274436401, -13182829, -104362273, 1748375381, 690031209911, 4647089032189, 112233351264271, 1276100569319881, 2131681036523177, 71497025649480187, 1365106755339875117
Offset: 0

Views

Author

N. J. A. Sloane, Nov 14 2009

Keywords

Examples

			1, -1, -1, -1/3, 1/3, 11/15, 37/45, 209/315, 113/315, 23/2835, -4381/14175, -84389/155925, -310517/467775, ... = A160623/A160624.

Links

Crossrefs

For denominators see A160624.
Cf. A295382.

Programs

  • Magma
    [Numerator((&+[Binomial(n,k)*((-2)^k/Factorial(k)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, May 06 2018
  • Mathematica
    Numerator[LaguerreL[Range[0,30],2]] (* Harvey P. Dale, May 16 2012 *)
  • PARI
    for(n=0,30, print1(numerator(sum(k=0,n, binomial(n,k)*((-2)^k/k!))), ", ")) \\ G. C. Greubel, May 06 2018

Formula

Numerators of coefficients in expansion of exp(-2*x/(1 - x))/(1 - x). - Ilya Gutkovskiy, Aug 29 2018

A331333 Interpolating the factorial and the powers of 2. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 2, 2, 8, 4, 6, 36, 36, 8, 24, 192, 288, 128, 16, 120, 1200, 2400, 1600, 400, 32, 720, 8640, 21600, 19200, 7200, 1152, 64, 5040, 70560, 211680, 235200, 117600, 28224, 3136, 128, 40320, 645120, 2257920, 3010560, 1881600, 602112, 100352, 8192, 256
Offset: 0

Views

Author

Peter Luschny, Jan 19 2020

Keywords

Examples

			Triangle starts:
  [0] 1
  [1] 1,     2
  [2] 2,     8,      4
  [3] 6,     36,     36,      8
  [4] 24,    192,    288,     128,     16
  [5] 120,   1200,   2400,    1600,    400,     32
  [6] 720,   8640,   21600,   19200,   7200,    1152,   64
  [7] 5040,  70560,  211680,  235200,  117600,  28224,  3136,   128
  [8] 40320, 645120, 2257920, 3010560, 1881600, 602112, 100352, 8192, 256

Crossrefs

T(n, 0) = A000142(n), T(n, n) = A000079(n).
Row sums: A087912, alternating row sums: A295382, antidiagonal sums: A222467, positive half sums: A129683, negative half sums: A331334.
Cf. A021009.

Programs

  • Maple
    A331333 := proc(n, k) local S; S := proc(n, k) option remember;
`if`(k = 0, 1, `if`(k > n, 0, 2*S(n-1, k-1)/k + S(n-1, k))) end: n!*S(n, k) end:
seq(seq(A331333(n, k), k=0..n), n=0..8);

Formula

T(n, k) = n!*S(n, k) where S(n, k) is recursively defined by:
if k = 0 then 1 else if k > n then 0 else 2*S(n-1, k-1)/k + S(n-1, k).
From Peter Bala, Jan 19 2020: (Start)
T(n,k) = 2^k*(n!/k!)*binomial(n,k).
E.g.f.: exp((2*x*t)/(1 - x))/(1 - x) = 1 + (1 + 2*t)*x + (2 + 8*t + 4*t^2)*x^2/2! + .... Cf. A021009. (End)

A295381 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-k*x/(1 - x))/(1 - x).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, -1, -1, 6, 1, -2, -2, -4, 24, 1, -3, -1, -2, -15, 120, 1, -4, 2, 6, 8, -56, 720, 1, -5, 7, 14, 33, 88, -185, 5040, 1, -6, 14, 16, 24, 102, 592, -204, 40320, 1, -7, 23, 6, -31, -104, -9, 3344, 6209, 362880, 1, -8, 34, -22, -120, -380, -1328, -3762, 14464, 112400, 3628800
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 21 2017

Keywords

Examples

			E.g.f. of column k: A_k(x) = 1 + (1 - k)*x/1! + (k^2 - 4*k + 2)*x^2/2! + (-k^3 + 9*k^2 - 18*k + 6)*x^3/3! + (k^4 - 16*k^3 + 72*k^2 - 96*k + 24)*x^4/4! + ...
Square array begins:
    1,   1,   1,    1,    1,    1, ...
    1,   0,  -1,   -2    -3,   -4, ...
    2,  -1,  -2,   -1,    2,    7, ...
    6,  -4,  -2,    6,   14,   16, ...
   24, -15,   8,   33,   24,  -31, ...
  120, -56,  88,  102, -104, -380, ...

Links

Crossrefs

Columns k=0..2 give A000142, A009940, A295382.
Main diagonal gives A277423.
Cf. A289192.

Programs

  • Mathematica
    Table[Function[k, n! SeriesCoefficient[Exp[-k x/(1 - x)]/(1 - x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, n! LaguerreL[n, k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, n! Hypergeometric1F1[-n, 1, k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

Formula

E.g.f. of column k: exp(-k*x/(1 - x))/(1 - x).
A(n,k) = n!*Laguerre(n,k).

A354941 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k! * (-2)^(n-k).

Original entry on oeis.org

1, -1, -10, -2, 488, 4088, -9968, -730480, -9751936, -11540096, 2480655104, 62522038016, 680469314560, -8292439149568, -606011029669888, -15765339965278208, -183530875864317952, 4164677242501038080, 318357069130977181696, 10359690304436314505216, 176911847384965046337536
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 12 2022

Keywords

Crossrefs

Cf. A000023, A274246, A295382, A343840, A354942.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k]^3 k! (-2)^(n - k), {k, 0, n}], {n, 0, 20}]
nmax = 20; CoefficientList[Series[BesselI[0, 2 Sqrt[x]] Sum[(-2)^k x^k/k!^3, {k, 0, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^3
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)^3 * k! * (-2)^(n-k)); \\ Michel Marcus, Jun 12 2022

Formula

Sum_{n>=0} a(n) * x^n / n!^3 = BesselI(0,2*sqrt(x)) * Sum_{n>=0} (-2)^n * x^n / n!^3.

A375855 Triangle read by rows: T(n, k) = 2^k * hypergeom([-n, -k], [], -1/2).

Original entry on oeis.org

1, 1, 1, 1, 0, -2, 1, -1, -2, 2, 1, -2, 0, 8, 8, 1, -3, 4, 8, -24, -88, 1, -4, 10, -4, -56, 32, 592, 1, -5, 18, -34, -40, 312, 400, -3344, 1, -6, 28, -88, 96, 512, -1472, -6144, 14464, 1, -7, 40, -172, 448, 32, -4544, 4160, 63616, -2944
Offset: 0

Views

Author

Detlef Meya, Aug 31 2024

Keywords

Examples

			Triangle starts:
[0] 1;
[1] 1,  1;
[2] 1,  0, -2;
[3] 1, -1, -2,    2;
[4] 1, -2,  0,    8,   8;
[5] 1, -3,  4,    8, -24, -88;
[6] 1, -4, 10,   -4, -56,  32,   592;
[7] 1, -5, 18,  -34, -40, 312,   400, -3344;
[8] 1, -6, 28,  -88,  96, 512, -1472, -6144, 14464;
[9] 1, -7, 40, -172, 448,  32, -4544,  4160, 63616, -2944;
...

Links

Crossrefs

Cf. A375854, A000012, A295382 (main diagonal).

Programs

  • Maple
    T := (n, k) -> 2^k * hypergeom([-n, -k], [], -1/2);
for n from 0 to 9 do seq(simplify(T(n, k)), k=0..n) od; # Peter Luschny, Sep 02 2024
  • Mathematica
    T[n_, k_] := (-1)^k*Sum[(-2)^(k - j)*Binomial[n, j]*Binomial[k, j]*j!, {j, 0, k}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
  • Python
    from math import comb, factorial
def A375855(n,k):
    return (-1)**k*sum((-2)**(k-j)*comb(n, j)*comb(k, j)*factorial(j) for j in range(k+1)) # John Tyler Rascoe, Sep 05 2024

Formula

T(n, k) = (-1)^k*Sum_{j=0..k} (-2)^(k - j)*binomial(n, j)*binomial(k, j)*j!.
Showing 1-6 of 6 results.

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

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