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 1-3 of 3 results.

A034430 Convolution of A001147 (double factorial numbers) with itself.

Original entry on oeis.org

1, 2, 7, 36, 249, 2190, 23535, 299880, 4426065, 74294010, 1397669175, 29123671500, 665718201225, 16560190196550, 445300709428575, 12869793995058000, 397815487883438625, 13095523164781307250, 457362512442763302375, 16890682269050394304500
Offset: 0

Views

Author

Jim FitzSimons (cherry(AT)neta.com)

Keywords

Comments

Old name was "Expand arctan(sqrt(x)*sqrt(x+2))/(sqrt(x)*sqrt(x+2)) and multiply n-th term by 1.3.5...(2n+1)".

Links

Crossrefs

Cf. A000111, A001147, A233481.

Programs

  • Maple
    A034430 := proc(n) option remember; if n=0 then 1 elif n=1 then 2 else
(3*n-1)*A034430(n-1)-(1+2*n^2-3*n)*A034430(n-2) fi end: seq(A034430(n),n=0..19); # Peter Luschny, Dec 14 2013
  • Mathematica
    Range[0, 19]! * CoefficientList[Series[1/(1 - x)/Sqrt[1 - 2*x], {x, 0, 19}], x] (* David Scambler, May 24 2012 *)

Formula

E.g.f.: 1/(1-x)/sqrt(1-2*x). - Vladeta Jovovic, May 11 2003
a(n) = Integral_{x=-infinity..infinity} x^(2*n+1)*exp(-x^2)*erfi(x/sqrt(2)), with erfi the imaginary error function. - Groux Roland, Mar 26 2011
E.g.f.: d/dx(F(x)^(-1)) where (-1) denotes the compositional inverse and F(x) = sin(x)/(1+sin(x)) = x - 2*x^2/2! + 5*x^3/3! - 16*x^4/4! + .... See A000111. - Peter Bala, Jun 24 2012
E.g.f.: E(x) = 1/sqrt(1-2*x)/(1-x) = (1 + x/(U(0)-x))/(1-x), where U(k) = (2*k+1)*x + (k+1) - (k+1)*(2*k+3)*x/U(k+1); (continued fraction Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Jun 27 2012
G.f.: hypergeom([1,1/2],[],2*x)^2. - Mark van Hoeij, May 16 2013
a(n-1)*n = A233481(n) for n >= 1. - Peter Luschny, Dec 14 2013
D-finite with recurrence: a(n) = (3*n-1)*a(n-1)-(2*n-1)*(n-1)*a(n-2) for n >= 2. - Peter Luschny, Dec 14 2013
a(n) ~ 2^(n+3/2) * n^n / exp(n). - Vaclav Kotesovec, Dec 20 2013
a(n) = 2*Pochhammer(1/2, n+1)*hyper2F1([1/2, -n], [3/2], -1). - Peter Luschny, Aug 02 2014
a(n) = -(2*n+1)!! * 2^(-n-1) * Im(Beta(2, n+1, 1/2)). - Vladimir Reshetnikov, Apr 23 2016
Expansion of square of continued fraction 1/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - 5*x/(1 - ...)))))). - Ilya Gutkovskiy, Apr 19 2017

Extensions

Better name from Philippe Deléham, Mar 21 2005

A336598 Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords cross the marked chord.

Original entry on oeis.org

1, 4, 2, 21, 18, 6, 144, 156, 96, 24, 1245, 1500, 1260, 600, 120, 13140, 16470, 16560, 11160, 4320, 720, 164745, 207270, 231210, 194040, 108360, 35280, 5040, 2399040, 2976120, 3507840, 3402000, 2419200, 1149120, 322560, 40320
Offset: 1

Views

Author

Donovan Young, Jul 29 2020

Keywords

Examples

			Triangle begins:
     1;
     4,    2;
    21,   18,    6;
   144,  156,   96,  24;
  1245, 1500, 1260, 600, 120;
...
For n = 2 and k = 1, let the four vertices be {1,2,3,4}. The marked chord can be either (1,3), and so crossed once by (2,4), or (2,4), and so crossed once by (1,3). Hence T(2,1) = 2.

Links

Crossrefs

Row sums are n*A001147(n) for n > 0.
First column is A233481(n) for n > 0.
Leading diagonal is A000142(n) for n > 0.
Sub-leading diagonal is n*A000142(n) for n > 1.
Cf. A336599, A336600, A336601.

Programs

  • Mathematica
    CoefficientList[Normal[Series[x/Sqrt[1-2*x]/(1-x(1+y)),{x,0,10}]]/.{x^n_.->x^n*n!},{x,y}]
  • PARI
    T(n)={[Vecrev(p) | p<-Vec(serlaplace(x/sqrt(1 - 2*x + O(x^n))/(1 - x*(1 + y))))]}
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jul 29 2020

Formula

T(n,k) = n*T(n-1,k) + n*T(n-1,k-1), with T(n,0) = A233481(n) for n > 0.
E.g.f.: x/sqrt(1 - 2*x)/(1 - x*(1 + y)).

A321627 The Riordan square of the double factorial of odd numbers. Triangle T(n, k), 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 15, 21, 7, 1, 105, 144, 48, 10, 1, 945, 1245, 372, 84, 13, 1, 10395, 13140, 3357, 726, 129, 16, 1, 135135, 164745, 35415, 6873, 1233, 183, 19, 1, 2027025, 2399040, 434520, 73116, 12306, 1920, 246, 22, 1
Offset: 0

Views

Author

Peter Luschny, Dec 07 2018

Keywords

Comments

The Riordan square is defined in A321620.
Triangle, read by rows, given by [1, 2, 3, 4, 5, 6, 7, …] DELTA [1, 0, 0, 0, 0, 0, 0, 0, …] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 17 2020

Examples

			Triangle starts:
[0][     1]
[1][     1,      1]
[2][     3,      4,     1]
[3][    15,     21,     7,    1]
[4][   105,    144,    48,   10,    1]
[5][   945,   1245,   372,   84,   13,   1]
[6][ 10395,  13140,  3357,  726,  129,  16,  1]
[7][135135, 164745, 35415, 6873, 1233, 183, 19, 1]

Crossrefs

First column are the double factorial of odd numbers A001147.
Second column is number of singletons in pair-partitions A233481.
Row sums are A321628, alternating row sums are A000007.
Cf. A321620.

Programs

  • Maple
    # The function RiordanSquare is defined in A321620.
cf := proc(dim) local k, m; m := 1;
for k from dim by -1 to 1 do m := 1 - k*x/m od;
1/m end: RiordanSquare(cf(9), 9);
  • Mathematica
    (* The function RiordanSquare is defined in A321620. *)
cf[dim_] := Module[{k, m=1}, For[k=dim, k >= 1, k--, m = 1 - k*x/m]; 1/m];
RiordanSquare[cf[9], 9] (* Jean-François Alcover, Jun 15 2019, from Maple *)
Showing 1-3 of 3 results.

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