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.

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

A102773 a(n) = Sum_{i=0..n} binomial(n,i)^2*i!*4^i.

Original entry on oeis.org

1, 5, 49, 709, 13505, 318181, 8916145, 289283429, 10656031489, 439039941445, 19995858681521, 997184081617285, 54026137182982849, 3159127731435043109, 198258247783634075185, 13289190424904891606821, 947419111092028780186625
Offset: 0

Views

Author

Miklos Kristof, Mar 16 2005

Keywords

Links

Crossrefs

Cf. A002720, A025167, A289147.

Programs

  • Maple
    seq(sum('binomial(k,i)^2*i!*4^i', 'i'=0..k),k=0..30);
  • Mathematica
    f[n_] := Sum[k!*4^k*Binomial[n, k]^2, {k, 0, n}]; Table[ f[n], {n, 0, 16}] (* or *)
Range[0, 16]! CoefficientList[ Series[1/(1 - 4x)*Exp[x/(1 - 4x)], {x, 0, 16}], x] (* Robert G. Wilson v, Mar 16 2005 *)
  • PARI
    a(n)=my(t=1); sum(i=1,n, t*=i; binomial(n,i)^2*t<<(2*i), 1) \\ Charles R Greathouse IV, Oct 23 2023

Formula

E.g.f.: (1/(1-4x))*exp(x/(1-4x)).
a(n) = (8*n-3)*a(n-1) - 16*(n-1)^2*a(n-2). - Vaclav Kotesovec, Oct 09 2013
a(n) ~ n^(n+1/4) * exp(sqrt(n)-n-1/8) * 4^n * (1 + 37/(96*sqrt(n))). - Vaclav Kotesovec, Oct 09 2013
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(4*x) * BesselI(0,2*sqrt(x)). - Ilya Gutkovskiy, Jul 17 2020

Extensions

More terms from Robert G. Wilson v, Mar 16 2005

A160611 Numerator of Laguerre(n, -4).

Original entry on oeis.org

1, 5, 17, 143, 355, 4043, 5177, 367271, 713723, 2410003, 109669391, 85569361, 11122330591, 245535162239, 52108328723, 70514170732823, 1753034045867, 3087820148584967, 3365163124738543, 15216530369586809, 9955926989110451149, 63735241273696485041
Offset: 0

Views

Author

N. J. A. Sloane, Nov 14 2009

Keywords

Links

Crossrefs

For denominators see A160612.
Cf. A289147.

Programs

  • Magma
    [Numerator((&+[Binomial(n,k)*(4^k/Factorial(k)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, May 12 2018
  • Mathematica
    Numerator[Table[LaguerreL[n, -4], {n, 0, 50}]] (* G. C. Greubel, May 12 2018 *)
  • PARI
    for(n=0,30, print1(numerator(sum(k=0,n, binomial(n,k)*(4^k/k!))), ", ")) \\ G. C. Greubel, May 12 2018
  • PARI
    a(n) = numerator(pollaguerre(n, 0, -4)); \\ Michel Marcus, Feb 05 2021

Formula

a:= n-> numer(add(binomial(n, i)*4^i/i!, i=0..n)):
seq(a(n), n=0..25); # Alois P. Heinz, Jun 27 2017

A160612 Denominator of Laguerre(n, -4).

Original entry on oeis.org

1, 1, 1, 3, 3, 15, 9, 315, 315, 567, 14175, 6237, 467775, 6081075, 773955, 638512875, 9823275, 10854718875, 7514805375, 21837140325, 9280784638125, 38979295480125, 2143861251406875, 3792985290950625, 1183411410776595, 336196423516078125, 9615217712559834375
Offset: 0

Views

Author

N. J. A. Sloane, Nov 14 2009

Keywords

Links

Crossrefs

For numerators see A160611.
Cf. A289147.

Programs

  • Magma
    [Denominator((&+[Binomial(n,k)*(4^k/Factorial(k)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, May 12 2018
  • Maple
    a:= n-> denom(add(binomial(n, i)*4^i/i!, i=0..n)):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 27 2017
  • Mathematica
    Denominator[Table[LaguerreL[n, -4], {n, 0, 50}]] (* G. C. Greubel, May 12 2018 *)
  • PARI
    for(n=0,30, print1(denominator(sum(k=0,n, binomial(n,k)*(4^k/k!))), ", ")) \\ G. C. Greubel, May 12 2018
  • PARI
    a(n) = denominator(pollaguerre(n, 0, -4)); \\ 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 1-5 of 5 results.

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

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