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 1-4 of 4 results.

A179420 E.g.f. A(x) satisfies: A(A(x)) = x*A'(x) with A(0)=0, A'(0)=1.

Original entry on oeis.org

0, 1, 2, 12, 132, 2200, 50280, 1482768, 54171376, 2381590944, 123292821600, 7390709937600, 506182300962624, 39180896544097152, 3396777800819754624, 327323946734658720000, 34831825328790915321600
Offset: 0

Views

Author

Paul D. Hanna, Jul 13 2010

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2200*x^5/5! +...
E.g.f. satisfies: A(A(x)) = x*A'(x) where:
A'(x) = 1 + 2*x + 12*x^2/2! + 132*x^3/3! + 2200*x^4/4! +...
A(A(x)) = x + 4*x^2/2! + 36*x^3/3! + 528*x^4/4! + 11000*x^5/5! +...
Related expansions begin:
A*Dx(A)/2! = 2*x^2/2! + 15*x^3/3! + 180*x^4/4! + 3150*x^5/5! +...
A*Dx(A*Dx(A))/3! = 6*x^3/3! + 104*x^4/4! + 2140*x^5/5! +...
A*Dx(A*Dx(A*Dx(A)))/4! = 24*x^4/4! + 770*x^5/5! + 24600*x^6/6! +...
A*Dx(A*Dx(A*Dx(A*Dx(A))))/5! = 120*x^5/5! + 6264*x^6/6! +...
which generate iterations of A=A(x) as illustrated by:
A(A(x))/x = 1 + 2*A + 2^2*A*Dx(A)/2! + 2^3*A*Dx(A*Dx(A))/3! +...
A(A(A(x)))/x = 1 + 3*A + 3^2*A*Dx(A)/2! + 3^3*A*Dx(A*Dx(A))/3! +...
A_{-1}(x)/x = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! +-...(inverse).
Illustrate a main property of the iterations A_n(x) of A(x) by:
A(x) = A(A(x)) * A(x)/[x*d/dx A(x)];
A(x) = A_3(x) * A_2(x)/[x*d/dx A_2(x)];
A(x) = A_4(x) * A_3(x)/[x*d/dx A_3(x)]; ...
which can be shown consistent by the chain rule of differentiation.
...
The RIORDAN ARRAY (A(x)/x, A(x)) begins:
. 1;
. 1, 1;
. 4/2!, 2, 1;
. 33/3!, 10/2!, 3, 1;
. 440/4!, 90/3!, 18/2!, 4, 1;
. 8380/5!, 1240/4!, 177/3!, 28/2!, 5, 1;
. 211824/6!, 23800/5!, 2544/4!, 300/3!, 40/2!, 6, 1;
. 6771422/7!, 598788/6!, 49680/5!, 4520/4!, 465/3!, 54/2!, 7, 1; ...
where the e.g.f. of column k = A(x)^(k+1)/x for k>=0.
...
The MATRIX LOG of the above Riordan array (A(x)/x, A(x)) begins:
. 0;
. 1, 0;
. 2/2!, 2, 0;
. 12/3!, 4/2!, 3, 0;
. 132/4!, 24/3!, 6/2!, 4, 0;
. 2200/5!, 264/4!, 36/3!, 8/2!, 5, 0;
. 50280/6!, 4400/5!, 396/4!, 48/3!, 10/2!, 6, 0;
. 1482768/7!, 100560/6!, 6600/5!, 528/4!, 60/3!, 12/2!, 7, 0; ...
where the e.g.f. of column k = (k+1)*A(x) for k>=0.

Links

Crossrefs

Cf. A193202, A179421, A179422, A179423, A179424, variant: A179320.
a(n)/n! = A221019(n)/A221020(n).

Programs

  • Mathematica
    a[n_] := a[n] = Module[{A}, A[x_] = x+x^2+Sum[a[m]*x^m/m!, {m, 3, n-1}]; If[n<3, n!*Coefficient[A[x], x, n], n!*Coefficient[A[A[x]], x, n]/(n-2)] ]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jan 15 2018, translated from PARI *)
  • Maxima
    Co(n, k, F):=if k=1  then F(n) else sum(F(i+1)*Co(n-i-1, k-1, F), i, 0, n-k);
a(n):=if n=0 then 0 else if n<3 then 1 else sum(Co(n,k,a)*a(k),k,2,n-1)/(n-2); /* Vladimir Kruchinin, Jun 29 2011 */
  • PARI
    {a(n)=local(A=x+x^2+sum(m=3,n-1,a(m)*x^m/m!)+x*O(x^n));if(n<3,n!*polcoeff(A,n),n!*polcoeff(subst(A,x,A),n)/(n-2))}

Formula

E.g.f. A(x) equals the e.g.f. of column 0 in the matrix log of the Riordan array (A(x)/x, A(x)).
Let A_n(x) denote the n-th iteration of e.g.f. A(x) with A_0(x)=x,
then A=A(x) satisfies:
A(x)/x = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! +...
A_{-1}(x)/x = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! -+...
A_n(x)/x = 1 + n*A + n^2*A*Dx(A)/2! + n^3*A*Dx(A*Dx(A))/3! + n^4*A*Dx(A*Dx(A*Dx(A)))/4! +...
where Dx(F) = d/dx(x*F).
Further, we have: A(x) = A_{n+1}(x) * A_n(x)/[x*d/dx A_n(x)] which holds for all n.
a(n)=sum(k=2..n-1, R(n-1,k-1)*a(k))/(n-2), n>2, a(1)=1, a(2)=1, where R is the Riordan array (A(x)/x, A(x)). [Vladimir Kruchinin, Jun 29 2011]
E.g.f. satisfies: A(x) = Series_Reversion(-G(-x)) where G(x) is the e.g.f. of A193202 and satisfies: G(G(x)) = x*G'(G(x)). [Paul D. Hanna, Jul 22 2011]

A027614 Related to Clebsch-Gordan formulas.

Original entry on oeis.org

1, 1, 3, 14, 80, 468, 2268, 10224, 313632, 9849600, 21954240, -8894136960, -105857556480, 20609598562560, 650835095904000, -80028503341516800, -5018759207362252800, 503681435808239001600, 56090762228110443724800
Offset: 1

Views

Author

Allan Adler (ara(AT)zurich.ai.mit.edu), Dec 15 1997

Keywords

Links

Crossrefs

Cf. A123521, A179320.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0, 1, If[k==1, 2*(n-1), T[n-2, k-2] + Binomial[2*n-k-1, 2*n -2*k-1] ]]; (* T = A123521 *)
b[n_]:= b[n]= If[n==1, 1, (-1/(2*(n-1)))*Sum[b[n-j+1]*T[n,j], {j,2,2*Floor[n/2]}]];
A027614[n_]:= (-1)^(n+1)*n!*b[n];
Table[A027614[n], {n, 40}] (* G. C. Greubel, Sep 01 2022 *)
  • PARI
    {a(n)=local(A=2*x, B); for(m=2, n, B=(1-x)/(1+x+O(x^(n+3)))*subst(A, x, x/(1-x+O(x^(n+3)))^2); A=A-polcoeff(B, m+1)*x^m/(m-1)/2); (-1)^(n-1)*n!*polcoeff(A, n)/2};
vector(20, n, a(n)) \\ G. C. Greubel, Aug 23 2022
  • SageMath
    @CachedFunction
def T(n,k): # T = A123521
    if (k==0): return 1
    elif (k==1): return 2*(n-1)
    else: return T(n-2, k-2) + binomial(2*n-k-1, 2*n-2*k-1)
@CachedFunction
def b(n):
    if (n==1): return 1
    else: return (-1/(2*(n-1)))*sum(T(n,j)*b(n-j+1) for j in (2..2*floor(n/2)))
def A027614(n): return (-1)^(n+1)*factorial(n)*b(n)
[A027614(n) for n in (1..40)] # G. C. Greubel, Sep 01 2022

Formula

a(n) = (-1)^(n-1)*A179320(n)/2. - G. C. Greubel, Aug 23 2022
a(n) = (-1)^(n+1) * n! * b(n), where b(n) = (-1/(2*(n-1))) * Sum_{j=2..2*floor(n/2)} A123521(n, j)*b(n-j+1), b(1) = 1. - G. C. Greubel, Sep 01 2022

A179330 E.g.f. satisfies: A(x) = (1+x)/(1+3*x) * A(x*(1+x)^2).

Original entry on oeis.org

0, 2, -6, 42, -468, 7080, -133128, 2938824, -73169568, 1997384832, -58814501760, 1868053207680, -65311214042880, 2585560450337280, -115344597684718080, 5424254194395456000, -244310147229735014400, 10256126830544041574400
Offset: 0

Views

Author

Paul D. Hanna, Jul 21 2010

Keywords

Examples

			E.g.f.: A(x) = 2*x - 6*x^2/2! + 42*x^3/3! - 468*x^4/4! + 7080*x^5/5! - 133128*x^6/6! + 2938824*x^7/7! - 73169568*x^8/8! + 1997384832*x^9/9! - 58814501760*x^10/10! + 1868053207680*x^11/11! - 65311214042880*x^12/12! +...
...
A(x*(1+x)^2) = 2*x + 2*x^2/2! - 18*x^3/3! + 108*x^4/4! - 480*x^5/5! - 2808*x^6/6! + 162792*x^7/7! - 3940128*x^8/8! + 57267648*x^9/9! + 534366720*x^10/10! - 78703384320*x^11/11! + 2883142045440*x^12/12! +...
...
where A(x*(1+x)^2) = (1+3*x)/(1+x) * A(x).
...
Related expansions begin:
. A = 2*x - 6*x^2/2! + 42*x^3/3! - 468*x^4/4! + 7080*x^5/5! +...
. A*Dx(A)/2! = 8*x^2/2! - 90*x^3/3! + 1332*x^4/4! - 25200*x^5/5! +...
. A*Dx(A*Dx(A))/3! = 48*x^3/3! - 1248*x^4/4! + 32760*x^5/5! -+...
. A*Dx(A*Dx(A*Dx(A)))/4! = 384*x^4/4! - 18480*x^5/5! + 770400*x^6/6! -+...
. A*Dx(A*Dx(A*Dx(A*Dx(A))))/5! = 3840*x^5/5! - 300672*x^6/6! +-...
...
Sums of which generate the square of the g.f. of A001764:
. G001764(-x)^2 = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! -+...
. G001764(-x)^2 = 1 - 2*x + 7*x^2 - 30*x^3 + 143*x^4 - 728*x^5 +...+ A006013(n)*(-x)^n +...
...
The Riordan array ((1+x)^2, x*(1+x)^2) (cf. A116088) begins:
1;
2, 1;
1, 4, 1;
0, 6, 6, 1;
0, 4, 15, 8, 1;
0, 1, 20, 28, 10, 1;
0, 0, 15, 56, 45, 12, 1; ...
The matrix log of Riordan array ((1+x)^2, x*(1+x)^2) begins:
0;
2, 0;
-6/2!, 4, 0;
42/3!, -12/2!, 6, 0;
-468/4!, 84/3!, -18/2!, 8, 0;
7080/5!, -936/4!, 126/3!, -24/2!, 10, 0;
-133128/6!, 14160/5!, -1404/4!, 168/3!, -30/2!, 12, 0; ...
where the g.f. of the leftmost column equals the e.g.f. of this sequence.

Crossrefs

Cf. A179331, variants: A179320, A179420.

Programs

  • PARI
    /* E.g.f. satisfies: A(x) = (1+x)/(1+3*x)*A(x*(1+x)^2): */
{a(n)=local(A=2*x, B); for(m=2, n, B=(1+x)/(1+3*x+O(x^(n+3)))*subst(A,x,x*(1+x)^2+O(x^(n+3))); A=A-polcoeff(B, m+1)*x^m/(m-1)/2); n!*polcoeff(A, n)}
  • PARI
    /* (1+x)^2 = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! +...: */
{a(n)=local(A=0+sum(m=1,n-1,a(m)*x^m/m!),D=1,R=0);R=-((1+x)^2+x*O(x^n))+1+sum(m=1,n,(D=A*deriv(x*D+x*O(x^n)))/m!);-n!*polcoeff(R,n)}
  • PARI
    /* First column of the matrix log of triangle A116088: */
{a(n)=local(M=matrix(n+1, n+1, r, c, if(r>=c, polcoeff(((1+x)^2+x*O(x^n))^c,r-c))), LOG, ID=M^0); LOG=sum(m=1, n+1, -(ID-M)^m/m); n!*LOG[n+1, 1]}

Formula

E.g.f. A=A(x) satisfies:
. (1+x)^2 = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! +...
. (1+x)^2*(1+x*(1+x)^2)^2 = 1 + 2*A + 2^2*A*Dx(A)/2! + 2^3*A*Dx(A*Dx(A))/3! + 2^4*A*Dx(A*Dx(A*Dx(A)))/4! +...
. G001764(-x)^2 = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! -+...; G001764(x) = g.f. of A001764;
where Dx(F) = d/dx(x*F).
INVERSION FORMULA:
More generally, if A(x) = A(G(x)) * G(x)/(x*G'(x)) with G(0)=0, G'(0)=1,
then G(x) can be obtained from A=A(x) by the series:
. G(x)/x = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! +... where Dx(F) = d/dx(x*F).
ITERATION FORMULA:
Let G_n(x) denote the n-th iteration of G(x) = x*(1+x)^2, and A=A(x), then:
. A(x) = A(G_n(x)) * G_n(x)/(x*G_n'(x)) for all n;
. G_n(x)/x = 1 + n*A + n^2*A*Dx(A)/2! + n^3*A*Dx(A*Dx(A))/3! + n^4*A*Dx(A*Dx(A*Dx(A)))/4! +... where Dx(F) = d/dx(x*F).
...
MATRIX LOG OF RIORDAN ARRAY (G(x)/x, G(x)) where G(x) = x*(1+x)^2:
. k*A(x) = e.g.f. of column k of the matrix log of triangle A116088 for k>=0.

A261885 E.g.f. satisfies: A(x) = A( x/(1-x)^3 ) * (1-x)/(1+2*x) with A(0)=0.

Original entry on oeis.org

0, 3, -6, 33, -288, 3240, -41328, 538776, -6772608, 108502848, -3985200000, 160423053120, -1094935449600, -334098799856640, 6171528273039360, 2583214608931891200, -150276838897808179200, -27910708844766496358400, 3428432556793417713254400, 462153194693572097337753600, -107869378361311784037187584000
Offset: 0

Views

Author

Paul D. Hanna, Sep 04 2015

Keywords

Examples

			E.g.f.: A(x) = 3*x - 6*x^2/2! + 33*x^3/3! - 288*x^4/4! + 3240*x^5/5! - 41328*x^6/6! + 538776*x^7/7! - 6772608*x^8/8! + 108502848*x^9/9! - 3985200000*x^10/10! +...
such that
A(x/(1-x)^3) = 3*x + 12*x^2/2! + 33*x^3/3! + 108*x^4/4! + 900*x^5/5! + 2952*x^6/6! - 19152*x^7/7! + 1694592*x^8/8! + 1847232*x^9/9! - 1796670720*x^10/10! +...
where
A(x/(1-x)^3) = (1+2*x)/(1-x)*A(x).

Crossrefs

Cf. A261886, A179320, A179420, A179330.

Programs

  • PARI
    /* E.g.f. satisfies: A(x) = (1-x)/(1+2*x)*A(x/(1-x)^3): */
{a(n)=local(A=3*x, B); for(m=2, n, B=(1-x)/(1+2*x+O(x^(n+3)))*subst(A, x, x/(1-x+O(x^(n+3)))^3); A=A-polcoeff(B, m+1)*x^m/(m-1)/3); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    /* 1/(1-x)^3 = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! +...: */
{a(n)=local(A=0+sum(m=1, n-1, a(m)*x^m/m!), D=1, R=0); R=-1/(1-x+x*O(x^n))^3+1+sum(m=1, n, (D=A*deriv(x*D+x*O(x^n)))/m!); -n!*polcoeff(R, n)}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f. A = A(x) satisfies:
(1) 1/(1-x)^3 = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! +...
(2) G(-x)^3 = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! -+..., where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.
(3) (1-x)^6/((1-x)^3 - x)^3 = 1 + 2*A + 2^2*A*Dx(A)/2! + 2^3*A*Dx(A*Dx(A))/3! + 2^4*A*Dx(A*Dx(A*Dx(A)))/4! +...
where Dx(F) = d/dx(x*F).
Showing 1-4 of 4 results.

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