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.

A277292 G.f. A(x) satisfies: Series_Reversion( A(x) + A(x)^2 ) = A(x) - A(x)^2.

Original entry on oeis.org

1, 1, 4, 21, 122, 758, 4958, 33509, 233810, 1641150, 12364368, 71807506, 1354944972, -33794258600, 2524565441138, -186642439700891, 16196862324254354, -1602823227559245434, 179707702260054046760, -22656977557634759678794, 3191199098536326709613676, -499206960572108744520132444, 86277300996554233583925645468, -16395890677314419248813441481150
Offset: 1

Views

Author

Paul D. Hanna, Oct 12 2016

Keywords

Examples

			G.f.: A(x) = x + x^3 + 4*x^5 + 21*x^7 + 122*x^9 + 758*x^11 + 4958*x^13 + 33509*x^15 + 233810*x^17 + 1641150*x^19 + 12364368*x^21 +...
such that Series_Reversion( A(x) + A(x)^2 ) = A(x) - A(x)^2, where
A(x)^2 = x^2 + 2*x^4 + 9*x^6 + 50*x^8 + 302*x^10 + 1928*x^12 + 12849*x^14 + 88122*x^16 + 621022*x^18 + 4411180*x^20 +...
A(x) + A(x)^2 = x + x^2 + x^3 + 2*x^4 + 4*x^5 + 9*x^6 + 21*x^7 + 50*x^8 + 122*x^9 + 302*x^10 + 758*x^11 + 1928*x^12 + 4958*x^13 + 12849*x^14 + 33509*x^15 + 88122*x^16 + 233810*x^17 + 621022*x^18 + 1641150*x^19 + 4411180*x^20 +...
Also,
A( A(x) + A(x)^2 ) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 +...
which equals the Catalan series (A000108).

Links

Crossrefs

Cf. A318008, A277293, A277294, A000108, A179270, A277292.

Programs

  • PARI
    {a(n) = my(Oxn=x*O(x^(2*n)), A = x +Oxn); for(i=1, 2*n, A = A + (x - subst(A+A^2, x, A-A^2 ))/2); polcoeff(A, 2*n-1)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n) * x^(2*n-1) satisfies:
(1) A( A(x) + A(x)^2 ) = C(x),
(2) C( A(x) - A(x)^2 ) = A(x),
(3) A( A(x) - A(x)^2 ) = -C(-x),
(4) A( A(x-x^2) + A(x-x^2)^2 ) = x,
where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers, A000108.

A276909 E.g.f. A(x) satisfies: Series_Reversion( A(x)*exp(A(x)) ) = A(x)*exp(-A(x)).

Original entry on oeis.org

1, 0, 3, 0, 85, 0, 6111, 0, 872649, 0, 195062395, 0, 76208072733, 0, 12330526252695, 0, 125980697776559377, 0, -857710566759117989133, 0, 11428318296234746748941925, 0, -222333914273403535165432496561, 0, 6242434914385931957857138485252825, 0, -244888574110309970555770302512462694549, 0, 13082369513456349871152908238665975845490989, 0, -930879791318792717095933863751868808486774883065, 0
Offset: 1

Views

Author

Paul D. Hanna, Sep 26 2016

Keywords

Comments

It appears that a(6*k+5) = 1 (mod 3) for k>=0 with a(n) = 0 (mod 3) elsewhere.
Apart from signs, essentially the same as A276910.
E.g.f. A(x) equals the series reversion of the e.g.f. of A276908.

Examples

			E.g.f.: A(x) = x + 3*x^3/3! + 85*x^5/5! + 6111*x^7/7! + 872649*x^9/9! + 195062395*x^11/11! + 76208072733*x^13/13! + 12330526252695*x^15/15! + 125980697776559377*x^17/17! - 857710566759117989133*x^19/19! + 11428318296234746748941925*x^21/21! - 222333914273403535165432496561*x^23/23! + 6242434914385931957857138485252825*x^25/25! +...
such that Series_Reversion( A(x)*exp(A(x)) ) = A(x)*exp(-A(x)).
RELATED SERIES.
A(x)*exp(A(x)) = x + 2*x^2/2! + 6*x^3/3! + 28*x^4/4! + 180*x^5/5! + 1446*x^6/6! + 13888*x^7/7! + 156472*x^8/8! + 2034000*x^9/9! + 29724490*x^10/10! + 476806176*x^11/11! + 8502508884*x^12/12! + 174802753216*x^13/13! + 3768345692398*x^14/14! + 63300353418240*x^15/15! + 1386349221087856*x^16/16! + 149879079531401472*x^17/17! +...+ A276911(n)*x^n/n! +...
exp(A(x)) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 116*x^5/5! + 661*x^6/6! + 8632*x^7/7! + 70617*x^8/8! + 1247248*x^9/9! + 13329001*x^10/10! + 285675776*x^11/11! + 3782734693*x^12/12! + 107823153088*x^13/13! + 1685127882621*x^14/14! + 28683829833856*x^15/15! + 574020572798641*x^16/16! + 133507199865641216*x^17/17! +...+ A276912(n)*x^n/n! +...
Also,  A( A(x)*exp(A(x)) ) = -LambertW(-x), which begins:
A( A(x)*exp(A(x)) ) = x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 625*x^5/5! + 7776*x^6/6! + 117649*x^7/7! + 2097152*x^8/8! +...+ n^(n-1)*x^n/n! +...

Links

Crossrefs

Cf. A276910, A276911, A276912, A276913, A179270, A276908.

Programs

  • PARI
    {a(n) = my(A=x +x*O(x^n));
for(i=1,n, A = A + (x - subst(A*exp(A),x,A*exp(-A)))/2); n!*polcoeff(A,n)}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(V=[1], A=x); for(i=1, n\2+1, V = concat(V, [0, 0]); A = sum(m=1, #V, V[m]*x^m/m!) +x*O(x^#V); V[#V] = -(#V)!/2 * polcoeff( subst( A*exp(A), x, A*exp(-A) ), #V) ); V[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies:
(1) A( A(x)*exp(A(x)) ) = -LambertW(-x),
(2) A( A(x)*exp(-A(x)) ) = LambertW(x),
where LambertW( x*exp(x) ) = x.
(3) Series_Reversion( A( x*exp(x) ) ) = A( x*exp(-x) ).

A276910 E.g.f. A(x) satisfies: inverse of function A(x)*exp(i*A(x)) equals the conjugate, A(x)*exp(-i*A(x)), where i=sqrt(-1).

Original entry on oeis.org

1, 0, -3, 0, 85, 0, -6111, 0, 872649, 0, -195062395, 0, 76208072733, 0, -12330526252695, 0, 125980697776559377, 0, 857710566759117989133, 0, 11428318296234746748941925, 0, 222333914273403535165432496561, 0, 6242434914385931957857138485252825, 0, 244888574110309970555770302512462694549, 0, 13082369513456349871152908238665975845490989, 0, 930879791318792717095933863751868808486774883065, 0
Offset: 1

Views

Author

Paul D. Hanna, Sep 22 2016

Keywords

Comments

Apart from signs, essentially the same as A276909.

Examples

			E.g.f.: A(x) = x - 3*x^3/3! + 85*x^5/5! - 6111*x^7/7! + 872649*x^9/9! - 195062395*x^11/11! + 76208072733*x^13/13! - 12330526252695*x^15/15! + 125980697776559377*x^17/17! + 857710566759117989133*x^19/19! + 11428318296234746748941925*x^21/21! + 222333914273403535165432496561*x^23/23! + 6242434914385931957857138485252825*x^25/25! +...
such that Series_Reversion( A(x)*exp(i*A(x)) ) = A(x)*exp(-i*A(x)).
RELATED SERIES.
A(x)*exp(i*A(x)) = x + 2*I*x^2/2! - 6*x^3/3! - 28*I*x^4/4! + 180*x^5/5! + 1446*I*x^6/6! - 13888*x^7/7! - 156472*I*x^8/8! + 2034000*x^9/9! + 29724490*I*x^10/10! - 476806176*x^11/11! - 8502508884*I*x^12/12! + 174802753216*x^13/13! + 3768345692398*I*x^14/14! - 63300353418240*x^15/15! - 1386349221087856*I*x^16/16! + 149879079531401472*x^17/17! +...+ A276911(n)*i^(n-1)*x^n/n! +...
exp(i*A(x)) = 1 + I*x - x^2/2! - 4*I*x^3/3! + 13*x^4/4! + 116*I*x^5/5! - 661*x^6/6! - 8632*I*x^7/7! + 70617*x^8/8! + 1247248*I*x^9/9! - 13329001*x^10/10! - 285675776*I*x^11/11! + 3782734693*x^12/12! + 107823153088*I*x^13/13! - 1685127882621*x^14/14! - 28683829833856*I*x^15/15! + 574020572798641*x^16/16! + 133507199865641216*I*x^17/17! +...+ A276912(n)*i^(n-1)*x^n/n! +...
Also,  A( A(x)*exp(i*A(x)) ) = i*LambertW(-i*x), which begins:
A( A(x)*exp(i*A(x)) ) = x + 2*I*x^2/2! - 9*x^3/3! - 64*I*x^4/4! + 625*x^5/5! + 7776*I*x^6/6! - 117649*x^7/7! - 2097152*I*x^8/8! +...+ -n^(n-1)*(-i)^(n-1)*x^n/n! +...

Links

Crossrefs

Cf. A276909, A276911, A276912, A179270.

Programs

  • PARI
    {a(n) = my(V=[1],A=x); for(i=1,n\2+1, V = concat(V,[0,0]); A = sum(m=1,#V,V[m]*x^m/m!) +x*O(x^#V); V[#V] = -(#V)!/2 * polcoeff( subst( A*exp(I*A), x, A*exp(-I*A) ),#V) );V[n]}
for(n=1,30,print1(a(n),", "))

Formula

E.g.f. A(x) satisfies: A( A(x)*exp(i*A(x)) ) = i*LambertW(-i*x), where LambertW( x*exp(x) ) = x.

A182399 G.f. A(x) satisfies: A(A(x)) - A(A(x))^2 = x + x^2.

Original entry on oeis.org

1, 1, 1, 3, 7, 21, 61, 187, 583, 1837, 5885, 19027, 62167, 204917, 680621, 2275211, 7648519, 25852573, 87812093, 299349795, 1023570647, 3515918501, 12140103149, 41894710427, 143835281351, 501071173901, 1808088546557, 6212411239539, 17720665594455
Offset: 1

Views

Author

Paul D. Hanna, Apr 27 2012

Keywords

Comments

a(33) is the first negative term.
If B(x) = x + 2*x^2 + 8*x^3 + 36*x^4 + 160*x^5 + 736*x^6 + 3648*x^7 + ..., then g.f. A(x) = x + B(x * A(x)). - Michael Somos, Jun 27 2017

Examples

			G.f.: A(x) = x + x^2 + x^3 + 3*x^4 + 7*x^5 + 21*x^6 + 61*x^7 + 187*x^8 +...
Related expansions:
A(A(x)) = x + 2*x^2 + 4*x^3 + 12*x^4 + 40*x^5 + 144*x^6 + 544*x^7 + 2128*x^8 +...
A(A(x))^2 = x^2 + 4*x^3 + 12*x^4 + 40*x^5 + 144*x^6 + 544*x^7 + 2128*x^8 +...
where A(A(x)) - A(A(x))^2 = x + x^2.
Let C(x) satisfy C(x-x^2) = x, where C(x) begins:
C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 +...+ A000108(n-1)*x^n +...
then
A(-C(-x)) = x + x^3 + 4*x^5 + 21*x^7 + 122*x^9 + 758*x^11 + 4958*x^13 +...+ (-1)^(n-1)*A179270(2*n-1)*x^(2*n-1) +...

Links

Crossrefs

Cf. A025227, A179270, A000108.

Programs

  • Maxima
    T(n, m):= if n=m then 1 else ((sum((binomial(k+m,n-k-m)*binomial(2*k+m-1,k+m-1))/(k+m),k,0,n-m))*m -sum(T(n, i) *T(i, m), i, m+1, n-1))/2;
makelist(T(n, 1), n, 1, 10); /* Vladimir Kruchinin, Apr 28 2012 */
  • PARI
    {a(n)=local(A=x+x^2,G);for(i=1,n,G=subst(A,x,A+x*O(x^n));A=A+(x+x^2-G+G^2)/2);polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))
  • PARI
    /* Faster vectorized version: */
{MM=100;A=[1];B=x;C=(1-sqrt(1-4*(x+x^2+x*O(x^MM))))/2; for(n=1,oo,A=concat(A,0);B=x*Ser(A); A[n]=Vec((B+subst(C+x*O(x^n),x,serreverse(B)))/2)[n]; print1(A[n],", "))}
  • PARI
    /* PARI/GP Version of Vladimir Kruchinin's formula: */
{T(n, m)=if(n==m,1, if(n>m, (sum(k=0,n-m,(binomial(k+m,n-k-m)*binomial(2*k+m-1,k+m-1))/(k+m))*m - sum(i=m+1,n-1,T(n, i) *T(i, m)))/2 ))}
{a(n)=T(n,1)}

Formula

G.f. satisfies: A(-A(-x)) = x.
G.f. satisfies: A(A(x)) = (1 - sqrt(1-4*(x+x^2)))/2 is the g.f. of A025227; thus, A(A(x)) = C(x+x^2) where C(x-x^2) = x.
G.f. satisfies: A(-C(-x)) = -I*G(I*x) where C(x-x^2) = x and G(x) is the g.f. of A179270 such that the inverse of function G(x) + I*G(x)^2 equals the complex conjugate: G(x) - I*G(x)^2.
a(n) = T(n,1), with T(n, m) = (sum((binomial(k+m,n-k-m)*binomial(2*k+m-1,k+m-1))/(k+m),k,0,n-m)*m -sum(T(n, i) *T(i, m), i, m+1, n-1))/2, n>m, T(n,n) = 1. - Vladimir Kruchinin, Apr 28 2012
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.