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 1-4 of 4 results.

A289192 A(n,k) = n! * Laguerre(n,-k); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 7, 6, 1, 4, 14, 34, 24, 1, 5, 23, 86, 209, 120, 1, 6, 34, 168, 648, 1546, 720, 1, 7, 47, 286, 1473, 5752, 13327, 5040, 1, 8, 62, 446, 2840, 14988, 58576, 130922, 40320, 1, 9, 79, 654, 4929, 32344, 173007, 671568, 1441729, 362880
Offset: 0

Views

Author

Alois P. Heinz, Jun 28 2017

Keywords

Examples

			Square array A(n,k) begins:
:   1,    1,    1,     1,     1,     1, ...
:   1,    2,    3,     4,     5,     6, ...
:   2,    7,   14,    23,    34,    47, ...
:   6,   34,   86,   168,   286,   446, ...
:  24,  209,  648,  1473,  2840,  4929, ...
: 120, 1546, 5752, 14988, 32344, 61870, ...

Links

Crossrefs

Columns k=0-10 give: A000142, A002720, A087912, A277382, A289147, A289211, A289212, A289213, A289214, A289215, A289216.
Rows n=0-2 give: A000012, A000027(k+1), A008865(k+2).
Main diagonal gives A277373.
Cf. A253286, A341014.

Programs

  • Maple
    A:= (n,k)-> n! * add(binomial(n, i)*k^i/i!, i=0..n):
seq(seq(A(n, d-n), n=0..d), d=0..12);
  • Mathematica
    A[n_, k_] := n! * LaguerreL[n, -k];
Table[A[n - k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 05 2019 *)
  • PARI
    {T(n, k) = if(n<2, k*n+1, (2*n+k-1)*T(n-1, k)-(n-1)^2*T(n-2, k))} \\ Seiichi Manyama, Feb 03 2021
  • PARI
    T(n, k) = n!*pollaguerre(n, 0, -k); \\ Michel Marcus, Feb 05 2021
  • Python
    from sympy import binomial, factorial as f
def A(n, k): return f(n)*sum(binomial(n, i)*k**i/f(i) for i in range(n + 1))
for n in range(13): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, Jun 28 2017

Formula

A(n,k) = n! * Sum_{i=0..n} k^i/i! * binomial(n,i).
E.g.f. of column k: exp(k*x/(1-x))/(1-x).
A(n, k) = (-1)^n*KummerU(-n, 1, -k). - Peter Luschny, Feb 12 2020
A(n, k) = (2*n+k-1)*A(n-1, k) - (n-1)^2*A(n-2, k) for n > 1. - Seiichi Manyama, Feb 03 2021

A160603 Numerator of Laguerre(n, -8).

Original entry on oeis.org

1, 9, 49, 619, 737, 35111, 305917, 1930753, 14779003, 46162429, 139399997, 92993347501, 606843495583, 5559358281401, 240588034396789, 660664176177209, 682418523335551, 471285453584720627, 732894385728160361
Offset: 0

Views

Author

N. J. A. Sloane, Nov 14 2009

Keywords

Links

Crossrefs

For denominators see A160604.
Cf. A289214.

Programs

  • Magma
    [Numerator((&+[Binomial(n,k)*(8^k/Factorial(k)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, May 13 2018
  • Mathematica
    Numerator[Table[LaguerreL[n, -8], {n, 0, 50}]] (* G. C. Greubel, May 13 2018 *)
  • PARI
    for(n=0,30, print1(numerator(sum(k=0,n, binomial(n,k)*(8^k/k!))), ", ")) \\ G. C. Greubel, May 13 2018
  • PARI
    a(n) = numerator(pollaguerre(n, 0, -8)); \\ Michel Marcus, Feb 05 2021

A160604 Denominator of Laguerre(n, -8).

Original entry on oeis.org

1, 1, 1, 3, 1, 15, 45, 105, 315, 405, 525, 155925, 467775, 2027025, 42567525, 58046625, 30405375, 10854718875, 8881133625, 206239658625, 9280784638125, 17717861581875, 714620417135625, 7044115540336875, 147926426347074375
Offset: 0

Views

Author

N. J. A. Sloane, Nov 14 2009

Keywords

Links

Crossrefs

For numerators see A160603.
Cf. A289214.

Programs

  • Magma
    [Denominator((&+[Binomial(n,k)*(8^k/Factorial(k)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, May 13 2018
  • Mathematica
    Denominator[Table[LaguerreL[n, -8], {n, 0, 50}]] (* G. C. Greubel, May 13 2018 *)
  • PARI
    for(n=0,30, print1(denominator(sum(k=0,n, binomial(n,k)*(8^k/k!))), ", ")) \\ G. C. Greubel, May 13 2018
  • PARI
    a(n) = denominator(pollaguerre(n, 0, -8)); \\ Michel Marcus, Feb 05 2021

A343847 T(n, k) = (n - k)! * [x^(n-k)] exp(k*x/(1 - x))/(1 - x). Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 7, 3, 1, 24, 34, 14, 4, 1, 120, 209, 86, 23, 5, 1, 720, 1546, 648, 168, 34, 6, 1, 5040, 13327, 5752, 1473, 286, 47, 7, 1, 40320, 130922, 58576, 14988, 2840, 446, 62, 8, 1, 362880, 1441729, 671568, 173007, 32344, 4929, 654, 79, 9, 1
Offset: 0

Views

Author

Peter Luschny, May 07 2021

Keywords

Examples

			Triangle starts:
0:     1;
1:     1,      1;
2:     2,      2,     1;
3:     6,      7,     3,     1;
4:    24,     34,    14,     4,    1;
5:   120,    209,    86,    23,    5,   1;
6:   720,   1546,   648,   168,   34,   6,  1;
7:  5040,  13327,  5752,  1473,  286,  47,  7,  1;
8: 40320, 130922, 58576, 14988, 2840, 446, 62,  8,  1;
.
Array whose upward read antidiagonals are the rows of the triangle.
n\k   0       1       2        3        4         5        6
-----------------------------------------------------------------
0:    1,      1,      1,       1,       1,        1,        1, ...
1:    1,      2,      3,       4,       5,        6,        7, ...
2:    2,      7,     14,      23,      34,       47,       62, ...
3:    6,     34,     86,     168,     286,      446,      654, ...
4:   24,    209,    648,    1473,    2840,     4929,     7944, ...
5:  120,   1546,   5752,   14988,   32344,    61870,   108696, ...
6:  720,  13327,  58576,  173007,  414160,   866695,  1649232, ...
7: 5040, 130922, 671568, 2228544, 5876336, 13373190, 27422352, ...

Crossrefs

Row sums: A343848. T(2*n, n) = A277373(n). Variant: A289192.
Columns: A000142, A002720, A087912, A277382, A289147, A289211, A289212, A289213, A289214, A289215, A289216.
Cf. A021009 (Laguerre polynomials), A344048.

Programs

  • Maple
    T := proc(n, k) option remember;
if n = k then return 1 elif n = k+1 then return k+1 fi;
(2*n-k-1)*T(n-1, k) - (n-k-1)^2*T(n-2, k) end:
seq(print(seq(T(n ,k), k = 0..n)), n = 0..7);
  • Mathematica
    T[n_, k_] := (-1)^(n - k) HypergeometricU[k - n, 1, -k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
(* Alternative: *)
TL[n_, k_] := (n - k)! LaguerreL[n - k, -k];
Table[TL[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
  • PARI
    T(n, k) = (n - k)!*sum(j=0, n - k, binomial(n - k, j) * k^j / j!)
for(n=0, 9, for(k=0, n, print(T(n, k))))
  • SageMath
    # Columns of the array.
def column(k, len):
    R. = PowerSeriesRing(QQ, default_prec=len)
    f = exp(k * x / (1 - x)) / (1 - x)
    return f.egf_to_ogf().list()
for col in (0..6): print(column(col, 20))

Formula

T(n, k) = (-1)^(n - k)*U(k - n, 1, -k), where U is the Kummer U function.
T(n, k) = (n - k)! * L(n - k, -k), where L is the Laguerre polynomial function.
T(n, k) = (n - k)! * Sum_{j = 0..n - k} binomial(n - k, j) k^j / j!.
T(n, k) = (2*n-k-1)*T(n-1, k) - (n-k-1)^2*T(n-2, k) for n - k >= 2.
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.