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 1-10 of 10 results.

A291821 Central terms of triangle A291820.

Original entry on oeis.org

1, 2, 38, 1245, 55566, 3062271, 197983540, 14567100211, 1196032764639, 108105913233804, 10651377348228706, 1135394214204504346, 130161542660895451934, 15969338705916554563833, 2088140238440781625249392, 289961586742613652475061397, 42623636996226195002733858964, 6613842247421357561153219091827, 1080515428870235624428208631160409, 185421585676394321869412169323524031
Offset: 1

Views

Author

Paul D. Hanna, Sep 01 2017

Keywords

Comments

The g.f. F(x,y) of A291820 satisfies: F(x - x*y*F(x,y), y) = x + x*(1-y)*F(x,y).

Links

Crossrefs

Cf. A291820.

Programs

  • PARI
    /* As central terms of triangle A291820 */
{A291820(n, k) = my(A=x); for(i=1, n, A = x + subst(x*A, x, y*A + (1-y)*x +x*O(x^n)) ); polcoeff(polcoeff(A, n, x), k, y)}
for(n=1, 20, print1(A291820(2*n-1, n-1), ", "));

Formula

a(n) = A291820(2*n-1, n-1) for n>=1.

A291822 A diagonal of triangle A291820.

Original entry on oeis.org

1, 7, 38, 189, 904, 4242, 19723, 91366, 423124, 1963169, 9138416, 42718416, 200656644, 947423923, 4497458118, 21465533955, 103001236168, 496832195860, 2408570061810, 11732479621260, 57410235742920, 282124153996425, 1391949415580256, 6893204658852960, 34254642268407820, 170769192927927532, 853864581906454264, 4281167768111675732, 21519922572920909984
Offset: 1

Views

Author

Paul D. Hanna, Sep 01 2017

Keywords

Comments

An adjacent diagonal of triangle A291820 equals the Catalan numbers (A000108).

Links

Crossrefs

Cf. A291820, A291821, A277297.

Programs

  • PARI
    /* As a diagonal of triangle A291820 */
{A291820(n, k) = my(A=x); for(i=1, n, A = x + subst(x*A, x, y*A + (1-y)*x +x*O(x^n)) ); polcoeff(polcoeff(A, n, x), k, y)}
for(n=1, 20, print1(A291820(n+2, n-1), ", "));

Formula

a(n) = A291820(n+2, n-1) for n >= 1.
a(n) = A277297(n) / 2 for n >= 1. - Paul D. Hanna, Jul 24 2023

A088714 G.f. satisfies A(x) = 1 + x*A(x)^2*A(x*A(x)).

Original entry on oeis.org

1, 1, 3, 13, 69, 419, 2809, 20353, 157199, 1281993, 10963825, 97828031, 907177801, 8716049417, 86553001779, 886573220093, 9351927111901, 101447092428243, 1130357986741545, 12923637003161409, 151479552582252239
Offset: 0

Views

Author

Paul D. Hanna, Oct 12 2003, May 22 2008

Keywords

Comments

Equals row sums of triangle A291820.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 69*x^4 + 419*x^5 + 2809*x^6 +...
The g.f. A(x) satisfies:
x*A(x) = x + x^2*A(x) + d/dx x^4*A(x)^2/2! + d^2/dx^2 x^6*A(x)^3/3! + d^3/dx^3 x^8*A(x)^4/4! +...
The logarithm of the g.f. is given by:
log(A(x)) = x*A(x) + d/dx x^3*A(x)^2/2! + d^2/dx^2 x^5*A(x)^3/3! + d^3/dx^3 x^7*A(x)^4/4! + d^4/dx^4 x^9*A(x)^5/5! +...
From _Paul D. Hanna_, Apr 16 2007: (Start)
G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + x*A*B;
B = A + x*B*C;
C = B + x*C*D;
D = C + x*D*E;
E = D + x*E*F ; ...
where variables B,C,D,E,..., are formed from successive iterations of x*A(x):
B = A(x)*A(x*A(x)), C = B*A(x*B), D = C*A(x*C), E = D*A(x*D), ...;
more explicilty,
B = 1 + 2*x + 8*x^2 + 42*x^3 + 258*x^4 + 1764*x^5 + 13070*x^6 +...,
C = 1 + 3*x + 15*x^2 + 93*x^3 + 655*x^4 + 5039*x^5 + 41453*x^6 +...,
D = 1 + 4*x + 24*x^2 + 172*x^3 + 1372*x^4 + 11796*x^5 +...,
E = 1 + 5*x + 35*x^2 + 285*x^3 + 2545*x^4 + 24255*x^5 +...,
... (End)
Related expansions:
A(x*A(x)) = 1 + x + 4*x^2 + 22*x^3 + 142*x^4 + 1016*x^5 + 7838*x^6 + 64174*x^7 + 552112*x^8 +...
A(x)^2 = 1 + 2*x + 7*x^2 + 32*x^3 + 173*x^4 + 1054*x^5 + 7039*x^6 + 50632*x^7 + 387613*x^8 +...
d/dx x^4*A(x)^2/2! = 2*x^3 + 5*x^4 + 21*x^5 + 112*x^6 + 692*x^7 + 4743*x^8 +...
d^2/dx^2 x^6*A(x)^3/3! = 5*x^4 + 21*x^5 + 112*x^6 + 696*x^7 + 4815*x^8 +...
d^3/dx^3 x^8*A(x)^4/4! = 14*x^5 + 84*x^6 + 540*x^7 + 3795*x^8 +...
d^4/dx^4 x^10*A(x)^5/5! = 42*x^6 + 330*x^7 + 2475*x^8 + 19305*x^9 +...
...
d^(n-1)/dx^(n-1) x^(2*n)*A(x)^n/n! = A000108(n)*x^(n+1) +...

Crossrefs

Cf. A291820, A088713, A212910, A212919.
Apart from signs, same as A067145. - Philippe Deléham, Jun 18 2006
Cf. A002449, A030266, A087949, A088717, A091713, A120971.

Programs

  • Mathematica
    m = 21; A[] = 1; Do[A[x] = 1 + x A[x]^2 A[x A[x]] + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 06 2019 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, n++; A = x + O(x^2); for(i=2, n, A = x / (1 - subst(A, x, A))); polcoeff(A, n))}; /* Michael Somos, May 21 2005 */
  • PARI
    {a(n)=local(A); if(n<0, 0, A=1+x+O(x^2); for(i=1,n, A=1/(1-x*A*subst(A,x,x*A)));polcoeff(A,n))}
  • PARI
    {a(n)=local(A); if(n<0, 0, A=1+x+O(x^2);for(i=0,n, A=(1/x)*serreverse(x-x^2*A));polcoeff(A,n))}
  • PARI
    {a(n,m=1)=if(n==0,1,if(m==0,0^n,sum(k=0,n,m*binomial(n+k+m,k)/(n+k+m)*a(n-k,k))))} \\ Paul D. Hanna, Jul 09 2009
  • PARI
    /* n-th Derivative: */
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
/* G.f.: [Paul D. Hanna, Dec 18 2010] */
{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n,A=exp(sum(m=1, n, Dx(m-1, x^(2*m-1)*A^m/m!))+x*O(x^n))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* n-th Derivative: */
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
/* G.f.: [Paul D. Hanna, May 31 2012] */
{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n,A=1+(1/x)*sum(m=1, n+1, Dx(m-1, x^(2*m)*A^m/m!))+x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. satisfies:
(1) A(x) = (1/x)*Series_Reversion(x - x^2*A(x)).
(2) A(x) = 1 + (1/x)*Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n)*A(x)^n/n!.
(3) A(x) = exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1)*A(x)^n/n! ).
(4) A(x) = 1/(1 - x*A(x)*A(x*A(x))).
(5) A(x) = f(x*A(x)) = (1-1/f(x))/x where f(x) is the g.f. of A088713.
Given g.f. A(x), then B(x) = x*A(x) satisfies 0 = f(x, B(x), B(B(x))) where f(a0, a1, a2) = a0 - a1 + a1*a2. - Michael Somos, May 21 2005
From Paul D. Hanna, Jul 09 2009: (Start)
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n with a(0,m)=1, then
a(n,m) = Sum_{k=0..n} m*C(n+k+m,k)/(n+k+m) * a(n-k,k).
(End)
a(n) = Sum_{k=0..n} A291820(n+1,k). - Paul D. Hanna, Sep 01 2017

A291813 G.f. A(x) satisfies: A(x - 2*x*A(x)) = x - x*A(x).

Original entry on oeis.org

1, 1, 5, 35, 297, 2873, 30657, 353727, 4355497, 56709337, 775575269, 11085971235, 164979882033, 2548461481105, 40762085472929, 673751263927071, 11489101983573105, 201838769635965969, 3648620371959258149, 67795012307507004291, 1293607920940368319641, 25326486746707799668105, 508368313083167614599201, 10454499119633293760277151, 220120546753823908307191769, 4742197866143368618862457641
Offset: 1

Views

Author

Paul D. Hanna, Sep 01 2017

Keywords

Examples

			G.f.: A(x) = x + x^2 + 5*x^3 + 35*x^4 + 297*x^5 + 2873*x^6 + 30657*x^7 + 353727*x^8 + 4355497*x^9 + 56709337*x^10 + 775575269*x^11 + 11085971235*x^12 +...
such that  A(x - 2*x*A(x)) = x - x*A(x).
RELATED SERIES.
A(x - 2*x*A(x)) = x - x^2 - x^3 - 5*x^4 - 35*x^5 - 297*x^6 - 2873*x^7 - 30657*x^8 +...
which equals x - x*A(x).
Series_Reversion( x - 2*x*A(x) ) = x + 2*x^2 + 10*x^3 + 70*x^4 + 594*x^5 + 5746*x^6 + 61314*x^7 + 707454*x^8 + 8710994*x^9 + 113418674*x^10 +...
which equals 2*A(x) - x.
A( 2*A(x) - x ) = x + 3*x^2 + 19*x^3 + 159*x^4 + 1561*x^5 + 17087*x^6 + 202975*x^7 + 2574391*x^8 + 34495545*x^9 + 484770627*x^10 + 7107406323*x^11 + 108289787415*x^12 + 1709478736593*x^13 + 27894511442079*x^14 +...
which equals (A(x) - x) / (2*A(x) - x).

Crossrefs

Cf. A291820, A291814, A291815, A291816, A291817, A291818, A291819, A276358, A291743, A291744.

Programs

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

Formula

G.f. A(x) also satisfies:
(1) A(x) = (1/2)*Series_Reversion( x - 2*x*A(x) ) + x/2.
(2) A( 2*A(x) - x ) = (A(x) - x) / (2*A(x) - x).
a(n) = Sum_{k=0..n-1} A291820(n, k) * 2^k.

A291814 G.f. A(x) satisfies: A(x - 3*x*A(x)) = x - 2*x*A(x).

Original entry on oeis.org

1, 1, 7, 67, 769, 10009, 143359, 2218255, 36625657, 639659737, 11741022235, 225390779647, 4508109360985, 93665093491381, 2016669357747667, 44905700922069463, 1032419000661778213, 24472819932819733957, 597384952530618840715, 15000294032677574361955, 387082666821619977435277, 10256260095368150955828565, 278811213889895147327704519, 7770474960716476086765483619
Offset: 1

Views

Author

Paul D. Hanna, Sep 02 2017

Keywords

Examples

			G.f.: A(x) = x + x^2 + 7*x^3 + 67*x^4 + 769*x^5 + 10009*x^6 + 143359*x^7 + 2218255*x^8 + 36625657*x^9 + 639659737*x^10 + 11741022235*x^11 + 225390779647*x^12 +...
such that  A(x - 3*x*A(x)) = x - 2*x*A(x).
RELATED SERIES.
A(x - 3*x*A(x)) = x - 2*x^2 - 2*x^3 - 14*x^4 - 134*x^5 - 1538*x^6 - 20018*x^7 +...
which equals x - 2*x*A(x).
Series_Reversion( x - 3*x*A(x) ) = x + 3*x^2 + 21*x^3 + 201*x^4 + 2307*x^5 + 30027*x^6 + 430077*x^7 + 6654765*x^8 +...
which equals 3*A(x) - 2*x.
A( 3*A(x) - 2*x ) = x + 4*x^2 + 34*x^3 + 382*x^4 + 5038*x^5 + 74134*x^6 + 1184650*x^7 + 20224990*x^8 + 364994554*x^9 + 6911857450*x^10 + 136622440786*x^11 + 2807805653098*x^12 +...
which equals (A(x) - x) / (3*A(x) - 2*x).

Crossrefs

Cf. A291820, A291813, A291815, A291816, A291817, A291818, A291819, A276358, A291743, A291744.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = (1/3)*serreverse( x - 3*x*A +x*O(x^n) ) + 2*x/3 ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = (1/3)*Series_Reversion( x - 3*x*A(x) ) + 2*x/3.
(2) A( 3*A(x) - 2*x ) = (A(x) - x) / (3*A(x) - 2*x).
a(n) = Sum_{k=0..n-1} A291820(n, k) * 3^k.

A291815 G.f. A(x) satisfies: A(x - 4*x*A(x)) = x - 3*x*A(x).

Original entry on oeis.org

1, 1, 9, 109, 1569, 25481, 454105, 8730373, 178996865, 3881556561, 88477557289, 2109927671453, 52443846331297, 1354646602217945, 36275862587452281, 1005099719255707829, 28765965099599741953, 849204340574458575777, 25827102287376124267593, 808349897942417046805197, 26011340193853765710238241, 859773626049480606121078057, 29168437337569276216572259097
Offset: 1

Views

Author

Paul D. Hanna, Sep 02 2017

Keywords

Examples

			G.f.: A(x) = x + x^2 + 9*x^3 + 109*x^4 + 1569*x^5 + 25481*x^6 + 454105*x^7 + 8730373*x^8 + 178996865*x^9 + 3881556561*x^10 + 88477557289*x^11 + 2109927671453*x^12 +...
such that  A(x - 4*x*A(x)) = x - 3*x*A(x).
RELATED SERIES.
A(x - 4*x*A(x)) = x - 3*x^2 - 3*x^3 - 27*x^4 - 327*x^5 - 4707*x^6 - 76443*x^7 +...
which equals x - 3*x*A(x).
Series_Reversion( x - 4*x*A(x) ) = x + 4*x^2 + 36*x^3 + 436*x^4 + 6276*x^5 + 101924*x^6 + 1816420*x^7 + 34921492*x^8 +...
which equals 4*A(x) - 3*x.
A( 4*A(x) - 3*x )  = x + 5*x^2 + 53*x^3 + 741*x^4 + 12153*x^5 + 222405*x^6 + 4421501*x^7 + 93949493*x^8 + 2110952881*x^9 + 49786323589*x^10 + 1225967873349*x^11 + 31395927333829*x^12 +...
which equals (A(x) - x) / (4*A(x) - 3*x).

Crossrefs

Cf. A291820, A291813, A291814, A291816, A291817, A291818, A291819, A276358, A291743, A291744.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = (1/4)*serreverse( x - 4*x*A +x*O(x^n) ) + 3*x/4 ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = (1/4)*Series_Reversion( x - 4*x*A(x) ) + 3*x/4.
(2) A( 4*A(x) - 3*x) = (A(x) - x) / (4*A(x) - 3*x).
a(n) = Sum_{k=0..n-1} A291820(n, k) * 4^k.

A291816 G.f. A(x) satisfies: A(x - 3*x*A(x)) = x - x*A(x).

Original entry on oeis.org

1, 2, 16, 182, 2524, 39992, 699520, 13231034, 266985280, 5694001172, 127481465536, 2981125793144, 72532301230672, 1830526849868000, 47802726801684544, 1289123410465365782, 35841130838977837348, 1025903099063974343984, 30195807234087904770952, 912951678159786641659796, 28327442752528049524839856, 901289532361030971832330544, 29382621186595702051011638128
Offset: 1

Views

Author

Paul D. Hanna, Sep 02 2017

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 16*x^3 + 182*x^4 + 2524*x^5 + 39992*x^6 + 699520*x^7 + 13231034*x^8 + 266985280*x^9 + 5694001172*x^10 +...
such that  A(x - 3*x*A(x)) = x - x*A(x).
RELATED SERIES.
A(x - 3*x*A(x)) = x - x^2 - 2*x^3 - 16*x^4 - 182*x^5 - 2524*x^6 - 39992*x^7 - 699520*x^8 +...
which equals x - x*A(x).
Series_Reversion( x - 3*x*A(x) ) = x + 3*x^2 + 24*x^3 + 273*x^4 + 3786*x^5 + 59988*x^6 + 1049280*x^7 + 19846551*x^8 +...
which equals (3/2)*A(x) - x/2.
A( (3*A(x) - x)/2 )  = x + 5*x^2 + 52*x^3 + 713*x^4 + 11458*x^5 + 205160*x^6 + 3984304*x^7 + 82576109*x^8 + 1807215616*x^9 + 41461917398*x^10 +...
which equals (A(x) - x) / (3*A(x) - x).

Crossrefs

Cf. A291820, A291813, A291814, A291815, A291817, A291818, A291819, A276358, A291743, A291744.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = (2/3)*serreverse( x - 3*x*A +x*O(x^n) ) + x/3 ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = (2/3)*Series_Reversion( x - 3*x*A(x) ) + x/3.
(2) A( (3*A(x) - x)/2 ) = (A(x) - x) / (3*A(x) - x).
a(n) = Sum_{k=0..n-1} A291820(n, k) * 3^k * 2^(n-k-1).

A291817 G.f. A(x) satisfies: A(x - 3*x*A(x)) = x + x*A(x).

Original entry on oeis.org

1, 4, 40, 580, 10312, 209752, 4707952, 114128308, 2946787192, 80268150808, 2290811949904, 68149759850680, 2105074730357968, 67308231895605520, 2222306773263886624, 75615701295449074084, 2647156154616207962920, 95215874465318554556776, 3514739264129342710455184, 133012394550946993742673304, 5156112210399927390763631056, 204570581814658024128260509360
Offset: 1

Views

Author

Paul D. Hanna, Sep 02 2017

Keywords

Examples

			G.f.: A(x) = x + 4*x^2 + 40*x^3 + 580*x^4 + 10312*x^5 + 209752*x^6 + 4707952*x^7 + 114128308*x^8 + 2946787192*x^9 + 80268150808*x^10 +...
such that  A(x - 3*x*A(x)) = x + x*A(x).
RELATED SERIES.
A(x - 3*x*A(x)) = x + x^2 + 4*x^3 + 40*x^4 + 580*x^5 + 10312*x^6 + 209752*x^7 + 4707952*x^8 +...
which equals x + x*A(x).
Series_Reversion( x - 3*x*A(x) ) = x + 3*x^2 + 30*x^3 + 435*x^4 + 7734*x^5 + 157314*x^6 + 3530964*x^7 + 85596231*x^8 +...
which equals (3/4)*A(x) + x/4.
A( (3*A(x) + x)/4 )  = x + 7*x^2 + 94*x^3 + 1651*x^4 + 33886*x^5 + 773458*x^6 + 19117780*x^7 + 503529979*x^8 + 13983485770*x^9 + 406470316978*x^10 +...
which equals (A(x) - x) / (3*A(x) + x).

Crossrefs

Cf. A291820, A291813, A291814, A291815, A291816, A291818, A291819.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = (4/3)*serreverse( x - 3*x*A +x*O(x^n) ) - x/3 ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = (4/3)*Series_Reversion( x - 3*x*A(x) ) - x/3.
(2) A( (3*A(x) + x)/4 ) = (A(x) - x) / (3*A(x) + x).
a(n) = Sum_{k=0..n-1} A291820(n, k) * 3^k * 4^(n-k-1).

A291818 G.f. A(x) satisfies: A(x - 4*x*A(x)) = x - x*A(x).

Original entry on oeis.org

1, 3, 33, 519, 9969, 218907, 5307201, 139123215, 3889995297, 114928234611, 3563543673825, 115375173490839, 3885328265571345, 135675583665864843, 4900856792035006593, 182756242210436579871, 7023982500750575903553, 277842871320960134512611, 11297961688442941015761825, 471773677417286920645721895, 20211597594930636918024401457, 887652829316087359743197592315
Offset: 1

Views

Author

Paul D. Hanna, Sep 02 2017

Keywords

Examples

			G.f.: A(x) = x + 3*x^2 + 33*x^3 + 519*x^4 + 9969*x^5 + 218907*x^6 + 5307201*x^7 + 139123215*x^8 + 3889995297*x^9 + 114928234611*x^10 +...
such that  A(x - 4*x*A(x)) = x - x*A(x).
RELATED SERIES.
A(x - 4*x*A(x)) = x - x^2 - 3*x^3 - 33*x^4 - 519*x^5 - 9969*x^6 - 218907*x^7 - 5307201*x^8 +...
which equals x - x*A(x).
Series_Reversion( x - 4*x*A(x) ) = x + 4*x^2 + 44*x^3 + 692*x^4 + 13292*x^5 + 291876*x^6 + 7076268*x^7 + 185497620*x^8 +...
which equals (4/3)*A(x) - x/3.
A( (4*A(x) - x)/3 )  = x + 7*x^2 + 101*x^3 + 1919*x^4 + 42713*x^5 + 1058967*x^6 + 28469325*x^7 + 816617535*x^8 + 24729787889*x^9 + 784895219495*x^10 +...
which equals (A(x) - x) / (4*A(x) - x).

Crossrefs

Cf. A291820, A291813, A291814, A291815, A291816, A291817, A291819.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = (3/4)*serreverse( x - 4*x*A +x*O(x^n) ) + x/4 ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = (4/3)*Series_Reversion( x - 3*x*A(x) ) - x/3.
(2) A( (4*A(x) - x)/3 ) = (A(x) - x) / (4*A(x) - x).
a(n) = Sum_{k=0..n-1} A291820(n, k) * 4^k * 3^(n-k-1).

A291819 G.f. A(x) satisfies: A(x - x*A(x)) = x + 3*x*A(x).

Original entry on oeis.org

1, 4, 24, 196, 1944, 21944, 272080, 3627412, 51288200, 761782104, 11805102064, 189901153112, 3158767322992, 54165347282960, 955189096759776, 17289056525343716, 320678326091307448, 6087009196570756488, 118109764108446889008, 2340448760238788518488, 47324471620802426563376, 975739573623235107473968, 20500725692629852174532192, 438679922664144046444438488, 9555430871381022848971028208
Offset: 1

Views

Author

Paul D. Hanna, Sep 02 2017

Keywords

Examples

			G.f.: A(x) = x + 4*x^2 + 24*x^3 + 196*x^4 + 1944*x^5 + 21944*x^6 + 272080*x^7 + 3627412*x^8 + 51288200*x^9 + 761782104*x^10 +...
such that  A(x - x*A(x)) = x + 3*x*A(x).
RELATED SERIES.
A(x - x*A(x)) = x + 3*x^2 + 12*x^3 + 72*x^4 + 588*x^5 + 5832*x^6 + 65832*x^7 + 816240*x^8 +...
which equals x + 3*x*A(x).
Series_Reversion( x - x*A(x) ) = x + x^2 + 6*x^3 + 49*x^4 + 486*x^5 + 5486*x^6 + 68020*x^7 + 906853*x^8 +...
which equals (1/4)*A(x) + 3*x/4.
A( (A(x) + 3*x)/4 ) = x + 5*x^2 + 38*x^3 + 369*x^4 + 4158*x^5 + 51870*x^6 + 698036*x^7 + 9974297*x^8 + 149755186*x^9 + 2345335606*x^10 +...
which equals (A(x) - x) / (A(x) + 3*x).

Crossrefs

Cf. A291820, A291813, A291814, A291815, A291816, A291817, A291818.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = 4*serreverse( x - x*A +x*O(x^n) ) - 3*x ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = 4*Series_Reversion( x - x*A(x) ) - 3*x.
(2) A( (A(x) + 3*x)/4 ) = (A(x) - x) / (A(x) + 3*x).
a(n) = Sum_{k=0..n-1} A291820(n, k) * 4^(n-k-1).
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