A300484 -id:A300484 - cp's OEIS frontend

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

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

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

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

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

Vladimir Shevelev, Spectrum of permanent's values and its extremal magnitudes in Λ_n^3 and Λ_n(α,β,γ), arXiv:1104.4051 [math.CO], 2011.

Row m=2 in A300481.
Cf. A000023, A000179, A000186, A300484.
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

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

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

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

Max Alekseyev, Mar 06 2018

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).

			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, ...
...

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.

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

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

Max Alekseyev, Mar 06 2018

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).

			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, ...
...

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.

{ A300481(m,n) = A300480(-m,n); } \\ see code in A300480

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

Max Alekseyev, Mar 06 2018

nonn

Row m=0 in A300480 and A300481.

Cf. A102761, A300483, A300484, A300485.

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

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

Max Alekseyev, Mar 06 2018

nonn

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).

Robert Israel, Table of n, a(n) for n = 0..449

Row m=1 in A300480.

Cf. A102761, A300482, A300484, A300485.

seq(2*int(orthopoly[T](n,(t+1)/2)*exp(-t),t=0..infinity),n=0..50); # Robert Israel, Mar 06 2018

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 *)

{ A300483(n) = if(n==0, return(2)); subst( serlaplace( 2*polchebyshev(n, 1, (x+1)/2)), x, 1); }

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

Max Alekseyev, Mar 06 2018

sign

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).

Row m=1 in A300481.

Cf. A102761, A300482, A300483, A300484.

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

a(n) = Sum_{i=0..n} A127672(n,i) * A000166(i).

a(n) = A300481(1,n) = A300480(-1,n).

A156995 Triangle T(n, k) = 2nbinomial(2n-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

Roger L. Bagula, Feb 20 2009

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).

			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;
  ...

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

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Row sums are A300484.

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

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 *)

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

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

Edited and changed T(0,0) = 2 (to make formula continuous and constant along the diagonal k = n) by Max Alekseyev, Mar 06 2018

cp's OEIS Frontend

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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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.

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