A059366 -id:A059366 - cp's OEIS frontend

A001756 a(n) = A059366(n,n-3) = A059366(n,3) for n >= 3, where the triangle A059366 arises from the expansion of a trigonometric integral.

15, 60, 450, 4500, 55125, 793800, 13097700, 243243000, 5016886875, 113716102500, 2808787731750, 75071235739500, 2158298027510625, 66409170077250000, 2177272076104125000, 75769068248423550000, 2789248824895091934375, 108288483790044745687500

Offset: 3

N. J. A. Sloane

nonn

Previous name was: Expansion of an integral.

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 166-167.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 3..100
Louis Comtet, Fonctions génératrices et calcul de certaines intégrales, Publikacije Elektrotechnickog faculteta - Serija Matematika i Fizika, No. 181/196 (1967), 77-87; see p. 85.

Cf. A001147, A007531, A059366.

RecurrenceTable[{a[3]==15,a[n]==a[n-1]n (2n-7)/(n-3)},a,{n,20}] (* Harvey P. Dale, Nov 08 2011 *)
Join[{c = 15}, Table[c = c*n*(2*n - 7)/(n - 3), {n, 4, 20}]] (* T. D. Noe, Aug 10 2012 *)

a(n) = 5*A007531(n)*A001147(n-2)/(2*(2*n-5)). - Philippe Deléham, Jun 26 2006
a(3) = 15, a(n) = a(n-1)*n*(2*n-7)/(n-3). - Philippe Deléham, Sep 19 2006
From Petros Hadjicostas, May 12 2020: (Start)
a(n) = n! * Sum_{k=0..n-3} (-1)^k * 2^(2*k-n) * binomial(n-3, k) * binomial(2*n-2*k, n-3) * binomial(n-2*k+3, n-k) for n >= 3. [Special case of a formula by Comtet, but corrected]
a(n) = 20 * binomial(2*n-6, n-3) * n!/2^n for n >= 3. [Special case of a formula due to Reinhard Zumkeller]
a(n) = binomial(-1/2, 3) * binomial(-1/2, n-3) * (-1)^n * n! * 2^n for n >= 3. (End)
a(n) ~ sqrt(2)*(5/16)*(2*n/e)^n. - Peter Luschny, May 13 2020

More terms from Philippe Deléham, Sep 19 2006
Corrected and extended by Harvey P. Dale, Nov 08 2011
New name by Petros Hadjicostas, May 12 2020

A001757 Expansion of an integral: central elements of rows of triangle in A059366.

Original entry on oeis.org

1, 1, 2, 9, 54, 450, 4500, 55125, 771750, 12502350, 225042300, 4538353050, 99843767100, 2410513805700, 62673358948200, 1762688220418125, 52880646612543750, 1698056319002793750, 57733914846094987500, 2084194325944029048750

Offset: 0

N. J. A. Sloane

For information about the trigonometric integral whose expansion involves the triangle A059366, see my comments and examples there. - Petros Hadjicostas, May 13 2020

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 166-167.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Cf. A000079, A000142, A000984, A059366, A283678.

a(n) = A000984(floor(n/2))*A000984(ceiling(n/2))*A000142(n)/A000079(n). [Corrected by Petros Hadjicostas, May 13 2020]
From Petros Hadjicostas, May 13 2020: (Start)
a(n) = A059366(n, floor(n/2)) = A059366(n, ceiling(n/2)).
a(2*n) = A283678(n). (End)

More terms from Larry Reeves (larryr(AT)acm.org), Feb 08 2001
Corrected and extended by Sean A. Irvine, Nov 19 2012
a(0) = 1 prepended by Petros Hadjicostas, May 13 2020

A001194 a(n) = A059366(n,n-2) = A059366(n,2) for n >= 2, where the triangle A059366 arises in the expansion of a trigonometric integral.

Original entry on oeis.org

3, 9, 54, 450, 4725, 59535, 873180, 14594580, 273648375, 5685805125, 129636356850, 3217338674550, 86331921100425, 2490343877896875, 76844896803675000, 2525635608280785000, 88081541838792376875, 3248654513701342370625

Offset: 2

N. J. A. Sloane

nonn

Old name was: Expansion of an integral.

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 166-167.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinis Cirpons, James East, and James D. Mitchell, Transformation representations of diagram monoids, arXiv:2411.14693 [math.RA], 2024. See p. 3.
Louis Comtet, Fonctions génératrices et calcul de certaines intégrales, Publikacije Elektrotechnickog faculteta - Serija Matematika i Fizika, No. 181/196 (1967), 77-87; see p. 85.

Table[3*n*(n-1)*(2*n-4)!/(2^(n-1)*(n-2)!),{n,2,20}] (* Vaclav Kotesovec, Jan 05 2014 *)

a(n) = (2*n - 1)*a(n-1) - 3*(n - 1)*(2*n - 7)!! for n > 3. - Sean A. Irvine, Mar 23 2012
a(n) = 3*n*(n-1)*(2*n-4)!/(2^(n-1)*(n-2)!) for n >= 2. - Vaclav Kotesovec, Jan 05 2014
a(n) = binomial(-1/2, 2) * binomial(-1/2, n-2) * (-1)^n * n! * 2^n for n >= 2. - Petros Hadjicostas, May 13 2020
a(n) ~ sqrt(2)*(3/8)*(2*n/e)^n. - Peter Luschny, May 14 2020

More terms from Sean A. Irvine, Mar 22 2012

New name by Petros Hadjicostas, May 13 2020

A001193 a(n) = (n+1)(2n)!/(2^nn!) = (n+1)(2n-1)!!.

Original entry on oeis.org

1, 2, 9, 60, 525, 5670, 72765, 1081080, 18243225, 344594250, 7202019825, 164991726900, 4111043861925, 110681950128750, 3201870700153125, 99044533658070000, 3262279327362680625, 113987877673731311250, 4211218814057295665625, 164015890652757831187500

Offset: 0

N. J. A. Sloane

Solution to y' = A(x), y(0) = 0 satisfies 0 = x^2 + 2*y^2*x - y^2, where A(x) = e.g.f. - Michael Somos, Mar 11 2004

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 166-167.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100

From Johannes W. Meijer, Nov 12 2009: (Start)
Equals the first right hand column of A167591.
Equals the first left hand column of A167594. (End)
Cf. A059366.

f:= x-> x/sqrt(1-2*x): a:= n-> subs(x=0, (D@@(n+1))(f)(x)):
seq(a(n), n=0..20); # Zerinvary Lajos, Apr 04 2009
# second Maple program:
a:= n-> (n+1)*doublefactorial(2*n-1):
seq(a(n), n=0..23);  # Alois P. Heinz, May 13 2020

Table[(n+1) (2*n-1)!!, {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Apr 14 2011 *)

PARI
```
a(n)=if(n<0, 0, (n+1)*(2*n)!/(2^n*n!))
```

E.g.f.: (1-x)/(1-2*x)^(3/2) = d/dx (x/(1-2*x)^(1/2)).
a(n) = uppermost term in the vector (M(T))^n * [1,0,0,0,...], where T = Transpose and M = the production matrix:
  1, 2;
  1, 2, 3;
  1, 2, 3, 4;
  1, 2, 3, 4, 5;
  ...
- Gary W. Adamson, Jul 08 2011
G.f.: A(x) = 1 + 2*x/(G(0) - 2*x) ; G(k) = 1 + k + x*(k+2)*(2*k+1) - x*(k+1)*(k+3)*(2*k+3)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2011
G.f.: U(0)/2  where U(k) = 1 + (2*k+1)/(1 - x/(x + 1/U(k+1))) (continued fraction). - Sergei N. Gladkovskii, Sep 25 2012
From Peter Bala, Nov 07 2016 and May 14 2020: (Start)
a(n) = (n + 1)*(2*n - 1)/n * a(n-1) with a(0) = 1.
a(n) = 2*a(n-1) + (2*n - 3)*(2*n + 1)*a(n-2) with a(0) = 1, a(1) = 2.
(End)
a(n) = A059366(n+1, n) = A059366(n+1, 1). - Petros Hadjicostas, May 13 2020

Better description from Wouter Meeussen, Mar 08 2001

More terms from James Sellers, May 01 2000

A368235 Triangle read by rows: n-th row polynomial equals the numerator of the rational function (-1)^nf(x) (d/dx)^n (1/f(x)), where f(x) = sqrt(x + x^2).

Original entry on oeis.org

1, 1, 2, 3, 8, 8, 15, 54, 72, 48, 105, 480, 864, 768, 384, 945, 5250, 12000, 14400, 9600, 3840, 10395, 68040, 189000, 288000, 259200, 138240, 46080, 135135, 1018710, 3333960, 6174000, 7056000, 5080320, 2257920, 645120, 2027025, 17297280, 65197440, 142248960, 197568000, 180633600, 108380160, 41287680, 10321920

Offset: 0

Peter Bala, Dec 18 2023

Unsigned row reverse of A123516.

The row polynomials also occur on repeated integration of 1/sqrt(x + x^2). See the example section.

			Triangle begins
 n\k |     0       1        2        3        4        5       6
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  0  |     1
  1  |     1       2
  2  |     3       8        8
  3  |    15      54       72       48
  4  |   105     480      864      768      384
  5  |   945    5250    12000    14400     9600     3840
  6  | 10395   68040   189000   288000   259200   138240   46080
 ...
Repeated integration of 1/f(x), where f(x) = sqrt(x + x^2):
Let I denote the integral operator h(x) -> Integral_{t = 0..x} h(t) dt.
Let g(x) = I(1/f(x)) = log(2*x + 1 + 2*f(x)). Then
(2^1 * 1!^2) * I^(2)(1/f(x)) = (2*x + 1)*g(x) - 2*f(x).
(2^2 * 2!^2) * I^(3)(1/f(x)) = (8*x^2 + 8*x + 3)*g(x) - 6*(2*x + 1)*f(x).
(2^3 * 3!^2) * I^(4)(1/f(x)) = (48*x^3 + 72*x^2 + 54*x + 15)*g(x) - 2*(44*x^2 + 44*x + 15)*f(x).
(2^4 * 4!^2) * I^(5)(1/f(x)) = (384*x^4 + 768*x^3 + 864*x^2 + 480*x + 105)*g(x) - 10*(2*x + 1)*(40*x^2 + 40*x + 21)*f(x).

Cf. A001147 (column 1), 2*A161120 (column 2), A000165 (main diagonal) A014479 (first subdiagonal), 3*A286725 (second subdiagonal).

Cf. A059366, A098461, A123516, A156919, A177145, A286724, A331817.

# sequence in triangular form
T := (n, k) -> n! * 2^(2*k-n) * binomial(n, k)*binomial(2*n-2*k, n-k):
for n from 0 to 8 do seq(T(n, k), k = 0..n) od;

T(n, k) = n! * 2^(2*k-n) * binomial(n, k) * binomial(2*n-2*k, n-k).
k*T(n, k) = (2*n^2)*T(n-1, k-1) for k >= 1 with T(n, 0) = (2*n - 1)!! = A001147(n).
T(n, 1) = 2*A161120(n).
T(n, n) = 2^n * n! = A000165(n); T(n+1, n) = 2^n * n! * (n+1)^2 = A014479(n);
T(n+2, n) = 3 * 2^(n-1)*(n+2)!*binomial(n+2, 2) = 3 * A286725(n).
More generally, T(n+r, n) = (2*r - 1)!! * A286724(n+r, r).
E.g.f.: Sum_{k >= 0} (1/2^k)*binomial(2*k, k)*t^k/(1 - 2*t*x)^(k+1) = 1 + (1 + 2*x)*t + (3 + 8*x + 8*x^2)*t^2/2! + (15 + 54*x + 72*x^2 + 48*x^3)*t^3/3! + ....
n-th row polynomial R(n, x) = (-2)^n*(x + x^2)^(n+1/2)*(d/dx)^n (1/sqrt(x + x^2)).
Recurrence for row polynomials:
R(n+1, x) = (2*x + 1)*(2*n + 1)*R(n, x) - 4*x*(x + 1)*n^2*R(n-1, x), with R(0, x) = 1.
R'(n, x) = 2*n^2 * R(n-1, x) for n >= 1.
Functional equation: R(n, -1 - x) = (-1)^n * R(n, x).
Conjecture: the zeros of the polynomial R(n, -x) lie on the vertical line Re(x) = 1/2 in the complex plane.
(-1)^n * x^n * R(n, (- 1 - x)/x) equals the n-th row polynomial of A123516.
(1 - x)^n * R(n, x/(1 - x)) equals the n-th row polynomial of A059366.
Let D denote the operator (1/x)*d/dx. Then D^(n+1)( arcsinh(x) ) = (-1)^n*R(n, x^2)/(x*sqrt(1 + x^2))^(2*n+1).
R(n, 1/2) = A331817(n); R(n, -1/2) = A177145(n+1);
(2^n) * R(n, 1/4) = A098461(n).
Alternating row sums R(n, -1) = (-1)^n * A001147(n).

cp's OEIS Frontend

A001756 a(n) = A059366(n,n-3) = A059366(n,3) for n >= 3, where the triangle A059366 arises from the expansion of a trigonometric integral.

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A001757 Expansion of an integral: central elements of rows of triangle in A059366.

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A001194 a(n) = A059366(n,n-2) = A059366(n,2) for n >= 2, where the triangle A059366 arises in the expansion of a trigonometric integral.

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Mathematica

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A001193 a(n) = (n+1)*(2*n)!/(2^n*n!) = (n+1)*(2n-1)!!.

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A368235 Triangle read by rows: n-th row polynomial equals the numerator of the rational function (-1)^n*f(x) * (d/dx)^n (1/f(x)), where f(x) = sqrt(x + x^2).

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Programs

Maple

Formula

A001193 a(n) = (n+1)(2n)!/(2^nn!) = (n+1)(2n-1)!!.

A368235 Triangle read by rows: n-th row polynomial equals the numerator of the rational function (-1)^nf(x) (d/dx)^n (1/f(x)), where f(x) = sqrt(x + x^2).