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 1-7 of 7 results.

A102761 Same as A000179, except that a(0) = 2.

Original entry on oeis.org

2, -1, 0, 1, 2, 13, 80, 579, 4738, 43387, 439792, 4890741, 59216642, 775596313, 10927434464, 164806435783, 2649391469058, 45226435601207, 817056406224416, 15574618910994665, 312400218671253762, 6577618644576902053, 145051250421230224304, 3343382818203784146955, 80399425364623070680706, 2013619745874493923699123
Offset: 0

Views

Author

N. J. A. Sloane, Apr 04 2010, following a suggestion from Vladimir Shevelev

Keywords

Comments

For any integer n>=0, 2 * Integral_{t=-2..2} T_n(t/2)*exp(-t)*dt = 4 * Integral_{z=-1..1} T_n(z)*exp(-2*z)*dz = a(n)*exp(2) - A300484(n)*exp(-2). - Max Alekseyev, Mar 08 2018

References

  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 197.

Crossrefs

Row m=2 in A300481.
A000179, A102761, and A335700 are all essentially the same sequence but with different conventions for the initial terms a(0) and a(1). - N. J. A. Sloane, Aug 06 2020

Programs

  • PARI
    { A102761(n) = subst( serlaplace( 2*polchebyshev(n, 1, (x-2)/2)), x, 1); } \\ Max Alekseyev, Mar 06 2018

Formula

a(n) = Sum_{i=0..n} A127672(n,i) * A000023(i). - Max Alekseyev, Mar 06 2018
a(n) = A300481(2,n) = A300480(-2,n). - Max Alekseyev, Mar 06 2018
a(n) = A335391(0,n) (Touchard). - William P. Orrick, Aug 29 2020

Extensions

Changed a(0)=2 (making the sequence more consistent with existing formulae) by Max Alekseyev, Mar 06 2018

A300480 Rectangular array read by antidiagonals: a(m,n) = 2 * Integral_{t>=0} T_n((t+m)/2)*exp(-t)*dt, m>=0, n>=0, where T_n(x) is n-th Chebyshev polynomial of first kind.

Original entry on oeis.org

2, 2, 1, 2, 2, 0, 2, 3, 3, 3, 2, 4, 8, 10, 18, 2, 5, 15, 29, 47, 95, 2, 6, 24, 66, 130, 256, 592, 2, 7, 35, 127, 327, 697, 1610, 4277, 2, 8, 48, 218, 722, 1838, 4376, 11628, 35010, 2, 9, 63, 345, 1423, 4459, 11770, 31607, 95167, 320589, 2, 10, 80, 514, 2562, 9820, 30248, 85634, 258690
Offset: 0

Views

Author

Max Alekseyev, Mar 06 2018

Keywords

Comments

a(m,n) is a polynomial in m of degree n.
For any integers m>=0, n>=0, 2 * Integral_{t=-m..m} T_n(t/2)*exp(-t)*dt = 4 * Integral_{z=-m/2..m/2} T_n(z)*exp(-2*z)*dz = A300481(m,n)*exp(m) - a(m,n)*exp(-m).

Examples

			Array starts with:
m=0: 2,  1,   0,    3,    18,     95,     592, ...
m=1: 2,  2,   3,   10,    47,    256,    1610, ...
m=2: 2,  3,   8,   29,   130,    697,    4376, ...
m=3: 2,  4,  15,   66,   327,   1838,   11770, ...
m=4: 2,  5,  24,  127,   722,   4459,   30248, ...
...
		

Crossrefs

Values for m<=0 are given in A300481.
Rows: A300482 (m=0), A300483 (m=1), A300484 (m=2), A300485 (m=-1), A102761 (m=-2).
Columns: A007395 (n=0), A000027 (n=1), A005563 (n=2), A084380 (n=3).
Cf. A000179 (almost row m=-2), A127672, A156995.

Programs

  • PARI
    { A300480(m,n) = if(n==0,return(2)); subst( serlaplace( 2*polchebyshev(n,1,(x+m)/2)), x, 1); }

Formula

a(m,n) = Sum_{i=0..n} A127672(n,i) * i! * Sum_{j=0..i} m^j/j!.
a(m,n) = Sum_{i=0..n} A127672(n,i) * A080955(m,i) = Sum_{i=0..n} A127672(n,i) * A089258(i,m).

A300481 Rectangular array read by antidiagonals: a(m,n) = 2 * Integral_{t>=0} T_n((t-m)/2)*exp(-t)*dt, m>=0, n>=0, where T_n(x) is n-th Chebyshev polynomial of first kind.

Original entry on oeis.org

2, 2, 1, 2, 0, 0, 2, -1, -1, 3, 2, -2, 0, 2, 18, 2, -3, 3, 1, 7, 95, 2, -4, 8, -6, 2, 34, 592, 2, -5, 15, -25, 15, 13, 218, 4277, 2, -6, 24, -62, 82, -28, 80, 1574, 35010, 2, -7, 35, -123, 263, -269, 106, 579, 12879, 320589
Offset: 0

Views

Author

Max Alekseyev, Mar 06 2018

Keywords

Comments

Although negative values of m are not present here or in A300480, the two arrays are connected with the formula: a(m,n) = A300480(-m,n). Thus, they essentially represent two "halves" of the same array indexed by integers m.
a(m,n) is a polynomial in m of degree n.
For any integers m>=0, n>=0, 2 * Integral_{t=-m..m} T_n(t/2)*exp(-t)*dt = 4 * Integral_{z=-m/2..m/2} T_n(z)*exp(-2*z)*dz = a(m,n)*exp(m) - A300480(m,n)*exp(-m).

Examples

			Array starts with:
m=0: 2,  1,  0,    3,   18,     95,    592, ...
m=1: 2,  0, -1,    2,    7,     34,    218, ...
m=2: 2, -1,  0,    1,    2,     13,     80, ...
m=3: 2, -2,  3,   -6,   15,    -28,    106, ...
m=4: 2, -3,  8,  -25,   82,   -269,    920, ...
...
		

Crossrefs

Values for m<=0 are given in A300480.
Rows: A300482 (m=0), A300485 (m=1), A102761 (m=2), A300483 (m=-1), A300484 (m=-2).
Columns (up to signs and offset): A007395 (n=0), A000027 (n=1), A005563 (n=2).
Cf. A000179 (almost row m=2), A127672, A156995.

Programs

Formula

a(m,n) = A300480(-m,n) = Sum_{i=0..n} A127672(n,i) * i! * Sum_{j=0..i} (-m)^j/j!.
a(m,n) = Sum_{i=0..n} A127672(n,i) * A292977(i,m).

A300482 a(n) = 2 * Integral_{t>=0} T_n(t/2) * exp(-t) * dt, n>=0, where T_n(x) is n-th Chebyshev polynomial of first kind.

Original entry on oeis.org

2, 1, 0, 3, 18, 95, 592, 4277, 35010, 320589, 3249648, 36137959, 437555090, 5730924667, 80743426272, 1217763999465, 19576502192898, 334180669811993, 6037275621582880, 115081732852805771, 2308342741080096402
Offset: 0

Views

Author

Max Alekseyev, Mar 06 2018

Keywords

Crossrefs

Row m=0 in A300480 and A300481.

Programs

  • PARI
    { A300482(n) = if(n==0, return(2)); subst( serlaplace( 2*polchebyshev(n, 1, x/2)), x, 1); }

Formula

a(n) = Sum_{i=0..n} A127672(n,i) * i!.

A300483 a(n) = 2 * Integral_{t>=0} T_n((t+1)/2) * exp(-t) * dt, n>=0, where T_n(x) is n-th Chebyshev polynomial of first kind.

Original entry on oeis.org

2, 2, 3, 10, 47, 256, 1610, 11628, 95167, 871450, 8833459, 98233158, 1189398050, 15578268382, 219483388403, 3310225751098, 53214450175743, 908397242172212, 16411016615547530, 312824583201360248, 6274726126933368879, 132115002152296986730, 2913432246090160413827
Offset: 0

Views

Author

Max Alekseyev, Mar 06 2018

Keywords

Comments

For any integer n>=0, 2 * Integral_{t=-1..1} T_n(t/2)*exp(-t)*dt = 4 * Integral_{z=-1/2..1/2} T_n(z)*exp(-2*z)*dz = A300485(n)*exp(1) - a(n)*exp(-1).

Crossrefs

Row m=1 in A300480.

Programs

  • Maple
    seq(2*int(orthopoly[T](n,(t+1)/2)*exp(-t),t=0..infinity),n=0..50); # Robert Israel, Mar 06 2018
  • Mathematica
    a[n_] := 2 Integrate[ChebyshevT[n, (t + 1)/2] Exp[-t], {t, 0, Infinity}];
    Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Feb 28 2019 *)
  • PARI
    { A300483(n) = if(n==0, return(2)); subst( serlaplace( 2*polchebyshev(n, 1, (x+1)/2)), x, 1); }

Formula

a(n) = Sum_{i=0..n} A127672(n,i) * A000522(i).
a(n) = A300480(1,n) = A300481(-1,n).
a(n) ~ exp(1) * n!. - Vaclav Kotesovec, Apr 15 2020

A300485 a(n) = 2 * Integral_{t>=0} T_n((t-1)/2) * exp(-t) * dt, n>=0, where T_n(x) is n-th Chebyshev polynomial of first kind.

Original entry on oeis.org

2, 0, -1, 2, 7, 34, 218, 1574, 12879, 117938, 1195479, 13294412, 160967522, 2108289364, 29703846535, 447990339602, 7201792686815, 122938198060734, 2220989581865882, 42336203570931402, 849191837620701239, 17879821236086808098
Offset: 0

Views

Author

Max Alekseyev, Mar 06 2018

Keywords

Comments

For any integer n>=0, 2 * Integral_{t=-1..1} T_n(t/2)*exp(-t)*dt = 4 * Integral_{z=-1/2..1/2} T_n(z)*exp(-2*z)*dz = a(n)*exp(1) - A300483(n)*exp(-1).

Crossrefs

Row m=1 in A300481.

Programs

  • PARI
    { A300485(n) = if(n==0, return(2)); subst( serlaplace( 2*polchebyshev(n, 1, (x-1)/2)), x, 1); }

Formula

a(n) = Sum_{i=0..n} A127672(n,i) * A000166(i).
a(n) = A300481(1,n) = A300480(-1,n).

A156995 Triangle T(n, k) = 2*n*binomial(2*n-k, k)*(n-k)!/(2*n-k), with T(0, 0) = 2, read by rows.

Original entry on oeis.org

2, 1, 2, 2, 4, 2, 6, 12, 9, 2, 24, 48, 40, 16, 2, 120, 240, 210, 100, 25, 2, 720, 1440, 1296, 672, 210, 36, 2, 5040, 10080, 9240, 5040, 1764, 392, 49, 2, 40320, 80640, 74880, 42240, 15840, 4032, 672, 64, 2, 362880, 725760, 680400, 393120, 154440, 42768
Offset: 0

Views

Author

Roger L. Bagula, Feb 20 2009

Keywords

Comments

For n>=1, o.g.f. of n-th row is a polynomial p(x,n) = Sum_{k=0..n} ( 2*n*(n-k)! * binomial(2*n-k, k)/(2*n-k)) * x^k. These polynomials are hit polynomials for the reduced ménage problem (Riordan 1958).

Examples

			Triangle starts with:
  n=0:     2;
  n=1:     1,     2;
  n=2:     2,     4,     2;
  n=3:     6,    12,     9,     2;
  n=4:    24,    48,    40,    16,     2;
  n=5:   120,   240,   210,   100,    25,    2;
  n=6:   720,  1440,  1296,   672,   210,   36,   2;
  n=7:  5040, 10080,  9240,  5040,  1764,  392,  49,  2;
  n=8: 40320, 80640, 74880, 42240, 15840, 4032, 672, 64, 2;
  ...
		

References

  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 197-199

Crossrefs

Row sums are A300484.

Programs

  • Magma
    A156995:= func< n,k | n eq 0 select 2 else 2*n*Factorial(n-k)*Binomial(2*n-k, k)/(2*n-k) >;
    [A156995(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 14 2021
    
  • Mathematica
    T[n_, k_]:= If[n==0, 2, 2*n*Binomial[2*n-k, k]*(n-k)!/(2*n-k)];
    Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, May 14 2021 *)
  • Sage
    def A156995(n,k): return 2 if (k==n) else 2*n*factorial(n-k)*binomial(2*n-k,k)/(2*n-k)
    flatten([[A156995(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 14 2021

Formula

T(n, k) = 2*n*binomial(2*n-k, k)*(n-k)!/(2*n-k), with T(0, 0) = 2.

Extensions

Edited and changed T(0,0) = 2 (to make formula continuous and constant along the diagonal k = n) by Max Alekseyev, Mar 06 2018
Showing 1-7 of 7 results.