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 19 results. Next

A381566 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A087949.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 2, 0, 1, 4, 6, 6, 5, 0, 1, 5, 10, 13, 15, 16, 0, 1, 6, 15, 24, 33, 46, 59, 0, 1, 7, 21, 40, 63, 99, 164, 246, 0, 1, 8, 28, 62, 110, 188, 343, 662, 1131, 0, 1, 9, 36, 91, 180, 331, 638, 1344, 2961, 5655, 0
Offset: 0

Views

Author

Seiichi Manyama, Feb 28 2025

Keywords

Examples

			Square array begins:
  1,  1,   1,   1,   1,    1,    1, ...
  0,  1,   2,   3,   4,    5,    6, ...
  0,  1,   3,   6,  10,   15,   21, ...
  0,  2,   6,  13,  24,   40,   62, ...
  0,  5,  15,  33,  63,  110,  180, ...
  0, 16,  46,  99, 188,  331,  552, ...
  0, 59, 164, 343, 638, 1110, 1845, ...

Crossrefs

Columns k=0..1 give A000007, A087949.
Cf. A379598, A381567, A381569.

Programs

  • PARI
    a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(n-j+k, j)/(n-j+k)*a(n-j, j)));

Formula

See A087949.

A213591 G.f. A(x) satisfies A( x - A(x)^2 ) = x.

Original entry on oeis.org

1, 1, 4, 24, 178, 1512, 14152, 142705, 1528212, 17211564, 202460400, 2474708496, 31310415376, 408815254832, 5495451727376, 75907303147652, 1075685334980240, 15618612118252960, 232102241507321384, 3526880759915999016, 54755450619399484512, 867928449982022915984
Offset: 1

Views

Author

Paul D. Hanna, Jun 14 2012

Keywords

Comments

Unsigned version of A139702.
Self-convolution is A276370.
Row sums of triangle A277295.

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +...
where A(x) = x + A(A(x))^2:
A(A(x)) = x + 2*x^2 + 10*x^3 + 69*x^4 + 568*x^5 + 5250*x^6 + 52792*x^7 +...
A(A(x))^2 = x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +...
The g.f. satisfies the series:
A(x) = x + A(x)^2 + d/dx A(x)^4/2! + d^2/dx^2 A(x)^6/3! + d^3/dx^3 A(x)^8/4! +...
Logarithmic series:
log(A(x)/x) = A(x)^2/x + [d/dx A(x)^4/x]/2! + [d^2/dx^2 A(x)^6/x]/3! + [d^3/dx^3 A(x)^8/x]/4! +...
Also, A(x) = x*G(A(x)^2/x) where G(x) = x/A(x/G(x)^2) is the g.f. of A212411:
G(x) = 1 + x + 2*x^2 + 7*x^3 + 36*x^4 + 235*x^5 + 1792*x^6 + 15261*x^7 +...
Also, A(x)^2 = x*F(A(x)) where F(x) is the g.f. of A213628:
F(x) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 616*x^6 + 5072*x^7 + 46013*x^8 +...

Links

Crossrefs

Cf. A220379, A139702, A212411, A138740, A213628, A213639.
Cf. A088714, A088717, A091713, A120971, A140094, A140095.
Cf. A143426, A087949, A143435, A182969.
Cf. A275765, A276360, A276361, A276362, A276363, A276370.
Cf. A277295 (triangle).

Programs

  • Mathematica
    terms = 22; A[] = 0; Do[A[x] = x + A[A[x]]^2 + O[x]^(terms+1) // Normal, terms+1]; CoefficientList[A[x], x] // Rest (* Jean-François Alcover, Jan 09 2018 *)
  • PARI
    {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x - A^2+x*O(x^n))); polcoeff(A, n))}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, A^(2*m))/m!)+x*O(x^n)); polcoeff(A, n)}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, A^(2*m)/x)/m!)+x*O(x^n))); polcoeff(A, n)}
for(n=1,21,print1(a(n),", "))
  • PARI
    b(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(n+j+k, j)/(n+j+k)*b(n-j, 2*j)));
a(n) = b(n-1, 1); \\ Seiichi Manyama, Jun 05 2025

Formula

G.f. satisfies:
(1) A(x) = x + A(A(x))^2.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n)/x / n! ).
(4) A(x) = x*G(A(x)^2/x) where G(x) = 1 + x*G(1-1/G(x))^2 is the g.f. of A212411.
(5) A(x)^2 = x*F(A(x)) where F(x) = 1 - (x^2/F(x))/F(x^2/F(x)) is the g.f. of A213628.
(6) x = A(A( x-x^2 - A(x)^2 )). - Paul D. Hanna, Jul 01 2012
(7) A(x) is the unique solution to variable A in the infinite system of simultaneous equations starting with:
A = x + B^2;
B = A + C^2;
C = B + D^2;
D = C + E^2; ...
where B = A(A(x)), C = A(A(A(x))), D = A(A(A(A(x)))), etc.
...
a(n) = Sum_{k=0..n-1} A277295(n,k).
From Seiichi Manyama, Jun 05 2025: (Start)
Let b(n,k) = [x^n] (A(x)/x)^k.
b(n,0) = 0^n; b(n,k) = k * Sum_{j=0..n} binomial(n+j+k,j)/(n+j+k) * b(n-j,2*j).
a(n) = b(n-1,1). (End)

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

A120971 G.f. A(x) satisfies A(x) = 1 + x*A(x)^2 * A( x*A(x)^2 )^2.

Original entry on oeis.org

1, 1, 4, 26, 218, 2151, 23854, 289555, 3783568, 52624689, 772928988, 11918181144, 192074926618, 3224153299106, 56213565222834, 1015694652332437, 18982833869517376, 366384235565593176, 7292660345274942402
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2006

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 26*x^3 + 218*x^4 + 2151*x^5 + 23854*x^6 +...
From _Paul D. Hanna_, Apr 16 2007: G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + x*B^2;
B = A*(1 + x*C^2);
C = B*(1 + x*D^2);
D = C*(1 + x*E^2);
E = D*(1 + x*F^2); ...
The above series begin:
B(x) = 1 + 2*x + 11*x^2 + 87*x^3 + 841*x^4 + 9288*x^5 + 113166*x^6 +...
C(x) = 1 + 3*x + 21*x^2 + 198*x^3 + 2204*x^4 + 27431*x^5 + 371102*x^6 +...
D(x) = 1 + 4*x + 34*x^2 + 374*x^3 + 4747*x^4 + 66350*x^5 + 996943*x^6 +...
E(x) = 1 + 5*x + 50*x^2 + 630*x^3 + 9015*x^4 + 140510*x^5 + 2334895*x^6 +...
F(x) = 1 + 6*x + 69*x^2 + 981*x^3 + 15658*x^4 + 270016*x^5 + 4933294*x^6 +...

Crossrefs

Cf. A120970; variants: A110447, A120973, A120975, A120977.
Cf. A002449, A087949, A088714, A088717, A091713.

Programs

  • Mathematica
    m = 19; A[] = 0; Do[A[x] = 1 + x A[x]^2 A[x A[x]^2]^2 + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 07 2019 *)
  • PARI
    {a(n)=local(A,G=[1,1]);for(i=1,n,G=concat(G,0); G[ #G]=-Vec(subst(Ser(G),x,x/Ser(G)^2))[ #G]); A=Vec(((Ser(G)-1)/x)^(1/2));A[n+1]}
  • PARI
    a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n+k, j)/(2*n+k)*a(n-j, 2*j))); \\ Seiichi Manyama, Mar 01 2025

Formula

G.f. A(x) satisfies:
(1) A(x) = G(G(x)-1),
(2) A(G(x)-1) = G(A(x)-1),
(3) A(x) = G(x*A(x)^2),
(4) A(x/G(x)^2) = G(x),
where G(x) is the g.f. of A120970 and satisfies G(x/G(x)^2) = 1 + x.
G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n+1)^2) for n>0 with F(x,0)=1. - Paul D. Hanna, Apr 16 2007
Let B(x) = Sum_{n>=0} a(n)*x^(2*n+1), then B( x/(1+B(x)^2) ) = x. - Paul D. Hanna, Oct 30 2013
From Seiichi Manyama, Mar 01 2025: (Start)
Let a(n,k) = [x^n] A(x)^k.
a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(2*n+k,j)/(2*n+k) * a(n-j,2*j). (End)

A091713 G.f. satisfies A(x) = x + x*A(A(A(x))).

Original entry on oeis.org

1, 1, 3, 15, 99, 781, 7001, 69253, 742071, 8506775, 103411463, 1324477033, 17785238513, 249432247233, 3642471258187, 55246757713367, 868523130653947, 14127076257342933, 237386074703124457, 4115341407421082869, 73516094755096807279, 1351801707136238290351
Offset: 1

Views

Author

Paul D. Hanna, Jan 31 2004, Jun 04 2008

Keywords

Comments

Conjecture: all terms are odd. - Paul D. Hanna, Dec 01 2024

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 15*x^4 + 99*x^5 + 781*x^6 +...
From _Paul D. Hanna_, Apr 16 2007:
G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + xC;
B = A*(1 + xD);
C = B*(1 + xE);
D = C*(1 + xF);
E = D*(1 + xG) ; ...
The solution to the variables in the system of equations are
A=A(x), B=A(A(x)), C=A(A(A(x))), D=A(A(A(A(x)))), etc., where:
A(x) = x + x^2 + 3*x^3 + 15*x^4 + 99*x^5 + 781*x^6 + 7001*x^7 +...
A(A(x)) = x + 2*x^2 + 8*x^3 + 46*x^4 + 330*x^5 + 2756*x^6 +...
A(A(A(x))) = x + 3*x^2 + 15*x^3 + 99*x^4 + 781*x^5 + 7001*x^6 +...
A(A(A(A(x)))) = x + 4*x^2 + 24*x^3 + 180*x^4 + 1564*x^5 +...
A(A(A(A(A(x))))) = x + 5*x^2 + 35*x^3 + 295*x^4 + 2815*x^5 +...
ALTERNATE GENERATING METHOD.
The g.f. A(x) equals the sum of products of even iterations of A(x):
A(x) = x + x*A_2(x) + x*A_2(x)*A_4(x) + x*A_2(x)*A_4(x)*A_6(x) + x*A_2(x)*A_4(x)*A_6(x)*A_8(x) +...+ Product_{k=0..n} A_{2*k}(x) +...
where A_n(x) = A_{n-1}(A(x)) is the n-th iteration of A(x) with A_0(x)=x.

Links

Crossrefs

Cf. A002449, A030266, A087949, A088714, A088717, A120971.
Cf. A141117.

Programs

  • Mathematica
    Nest[x + x (# /. x -> # /. x -> #) &, O[x], 30][[3]] (* Vladimir Reshetnikov, Aug 08 2019 *)
  • PARI
    {a(n)=local(A);A=x+x^2;for(i=3,n, A=x+x*subst(A,x,subst(A,x,A))+x*O(x^n)); polcoeff(A,n,x)}
  • PARI
    /* Define the n-th iteration of F: */
{ITERATE(F,n,p)=local(G=x);for(i=1,n,G=subst(F,x,G+x*O(x^p)));G}
/* A(x) equals the sum of products of even iterations of A(x): */
{a(n)=local(A=x);for(i=1,n,A=sum(m=0,n-1,prod(k=0,m,ITERATE(A,2*k,n)+x*O(x^n))));polcoeff(A,n)}

Formula

G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n+2)) for n>0 with F(x,0)=1. - Paul D. Hanna, Apr 16 2007
G.f.: A(x) = G(x)/[1 + G(G(x))] where G(x) = A(A(x)) = g.f. of A141117.
G.f.: A(x) = Series_Reversion[ x/(1 + A(A(x))) ].
G.f. satisfies: A(x) = Sum_{n>=0} Product_{k=0..n} A_{2*k}(x), where A_n(x) denotes the n-th iteration of A(x) with A_0(x)=x. - Paul D. Hanna, Jul 21 2011

A139702 G.f. satisfies: x = A( x + A(x)^2 ).

Original entry on oeis.org

1, -1, 4, -24, 178, -1512, 14152, -142705, 1528212, -17211564, 202460400, -2474708496, 31310415376, -408815254832, 5495451727376, -75907303147652, 1075685334980240, -15618612118252960, 232102241507321384, -3526880759915999016
Offset: 1

Views

Author

Paul D. Hanna, Apr 30 2008, May 20 2008

Keywords

Comments

Signed version of A213591.

Examples

			G.f.: A(x) = x - x^2 + 4*x^3 - 24*x^4 + 178*x^5 - 1512*x^6 +-...
A(x)^2 = x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-...
where A(x + A(x)^2) = x.
Let G(x) = Series_Reversion( A(x) ) = x + A(x)^2, then:
G(x) = x + x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 -+... and
G(G(x)) = x + 2*x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 -+...
so that G(x) = G(G(x)) - x^2 = g.f. of A138740.
Logarithmic series:
log(A(x)/x) = -A(x)^2/x + [d/dx A(x)^4/x]/2! - [d^2/dx^2 A(x)^6/x]/3! + [d^3/dx^3 A(x)^8/x]/4! -+...

Links

Crossrefs

Cf. A138740, A212411, A213591.
Cf. A088714, A088717, A091713, A120971, A140094, A140095.
Cf. A143426, A087949, A143435, A182969.

Programs

  • Mathematica
    nmax = 20; sol = {a[1] -> 1}; nmin = Length[sol]+1;
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[x - A[x + A[x]^2] + O[x]^(n+1), x][[nmin;;]] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, nmin, nmax}];
a /@ Range[nmax] /. sol (* Jean-François Alcover, Nov 06 2019 *)
  • PARI
    {a(n)=local(A=x); if(n<1, 0, for(i=1,n, A=serreverse(x + (A+x*O(x^n))^2)); polcoeff(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, Dec 18 2010] */
{a(n)=local(A=x-x^2+x*O(x^n));for(i=1,n,
A=x*exp(sum(m=0,n,(-1)^(m+1)*Dx(m,A^(2*m+2)/x)/(m+1)!)+x*O(x^n)));polcoeff(A,n)}

Formula

Let G(x) = Series_Reversion( A(x) ) = x + A(x)^2, then G(x) = G(G(x)) - x^2 = g.f. of A138740.
G.f. satisfies: A(x) = x*G(-A(x)^2/x) where G(x) = 1 + x*G(1-1/G(x))^2 is the g.f. of A212411.
G.f.: A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:
A = 1 - x*B^2;
B = A - x*C^2;
C = B - x*D^2;
D = C - x*E^2;
E = D - x*F^2; ...
G.f. satisfies: A(x) = x*exp( Sum_{n>=0} (-1)^(n+1)*[d^n/dx^n A(x)^(2n+2)/x]/(n+1)! ). [Paul D. Hanna, Dec 18 2010]

A088717 G.f. satisfies: A(x) = 1 + x*A(x)^2*A(x*A(x)^2).

Original entry on oeis.org

1, 1, 3, 14, 84, 596, 4785, 42349, 406287, 4176971, 45640572, 526788153, 6392402793, 81247489335, 1078331283648, 14907041720241, 214187010762831, 3192620516380376, 49287883925072010, 786925082232918304, 12976244331714379149, 220728563512663520510
Offset: 0

Views

Author

Paul D. Hanna, Oct 12 2003 and Mar 10 2007

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 14*x^3 + 84*x^4 + 596*x^5 + 4785*x^6 +...
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*(1 + x*B*C);
C = B*(1 + x*C*D);
D = C*(1 + x*D*E);
E = D*(1 + x*E*F); ...
where B(x) = A(x)*A(x*A(x)^2), C(x) = A(x)*B(x*A(x)^2),  D(x) = A(x)*C(x*A(x)^2), ...
Expansions of a few of the functions described above begin:
B(x) = 1 + 2*x + 9*x^2 + 55*x^3 + 402*x^4 + 3328*x^5 + 30312*x^6 +...
C(x) = 1 + 3*x + 18*x^2 + 138*x^3 + 1218*x^4 + 11856*x^5 + 124467*x^6 +...
D(x) = 1 + 4*x + 30*x^2 + 278*x^3 + 2901*x^4 + 32846*x^5 + 395913*x^6 +...
ALTERNATE GENERATING METHOD.
Suppose functions A=A(x), B=B(x), C=C(x), etc., satisfy:
A = 1 + x*A^2*B,
B = 1 + x*(A*B)^2*C,
C = 1 + x*(A*B*C)^2*D,
D = 1 + x*(A*B*C*D)^2*E, etc.,
then B(x) = A(x*A(x)^2), C(x) = B(x*A(x)^2), D(x) = C(x*A(x)^2), etc.,
where A(x) = 1 + x*A(x)^2*A(x*A(x)^2) is the g.f. of this sequence.
Expansions of a few of the functions described above begin:
B(x) = 1 + x + 5*x^2 + 33*x^3 + 256*x^4 + 2223*x^5 + 21058*x^6 +...
C(x) = 1 + x + 7*x^2 + 60*x^3 + 578*x^4 + 6045*x^5 + 67421*x^6 +...
D(x) = 1 + x + 9*x^2 + 95*x^3 + 1098*x^4 + 13526*x^5 + 175176*x^6 +...

Crossrefs

Cf. A215505, A215506, A215507.
Cf. A182224, A002449, A030266, A087949, A088714, A091713, A120971.

Programs

  • Mathematica
    m = 22; A[] = 0; Do[A[x] = 1 + x A[x]^2 A[x A[x]^2] + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 07 2019 *)
  • PARI
    {a(n)=local(A=1+x);for(i=0,n,A=1+x*A^2*subst(A,x,x*A^2+x*O(x^n)));polcoeff(A,n)}
  • PARI
    /* a(n) = [x^n] (1+x*A(x))^(2*n+1)/(2*n+1): */
{a(n)=local(A=1+x); for(i=0, n, A=sum(m=0,n,polcoeff((1+x*A+x*O(x^m))^(2*m+1)/(2*m+1),m)*x^m)+x*O(x^n));polcoeff(A,n)}
  • PARI
    {a(n, m=1)=if(n==0, 1, if(m==0, 0^n, sum(k=0, n, m*binomial(2*n+m, k)/(2*n+m)*a(n-k, k))))}

Formula

a(n) = coefficient of x^n in (1+x*A(x))^(2*n+1)/(2*n+1) where A(x) = Sum_{n=0} a(n)*x^n.
Recurrence:
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(2n+m,k)/(2n+m) * a(n-k,k). [Paul D. Hanna, Dec 16 2010]
G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n)*F(x,n+1)) for n>0 with F(x,0)=1. - Paul D. Hanna, Apr 16 2007

A143501 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^3).

Original entry on oeis.org

1, 1, 1, 4, 16, 92, 616, 4729, 40776, 388057, 4028230, 45207583, 544680014, 7004865885, 95694153485, 1382946630490, 21067128029388, 337224872043659, 5656357906530796, 99168643108816180, 1813250965008114981, 34506927801196386939, 682210688659107549765, 13988252589131863518730
Offset: 0

Views

Author

Paul D. Hanna, Aug 20 2008

Keywords

Examples

			G.f. A(x) = 1 + x + x^2 + 4*x^3 + 16*x^4 + 92*x^5 + 616*x^6 + 4729*x^7 +...
A(x)^3 = 1 + 3*x + 6*x^2 + 19*x^3 + 78*x^4 + 411*x^5 + 2617*x^6 +...
A(x*A(x)^3) = 1 + x + 4*x^2 + 16*x^3 + 92*x^4 + 616*x^5 + 4729*x^6 +...
If G(x*A(x)^3) = x then
G(x) = x - 3*x^2 + 12*x^3 - 64*x^4 + 372*x^5 - 2385*x^6 + 15675*x^7 -+...
A(G(x)) = 1 + A(x)*G(x) = (x/G(x))^(1/3) where
A(x)*G(x) = x - 2*x^2 + 10*x^3 - 51*x^4 + 324*x^5 - 1985*x^6 + 13938*x^7 -...

Crossrefs

Cf. A087949, A143500.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,x*A^3));polcoeff(A,n)}
  • PARI
    a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-3*j+k, j)/(3*n-3*j+k)*a(n-j, j))); \\ Seiichi Manyama, Jun 04 2025

Formula

G.f. satisfies: G(x) = x/[1 + A(x)*G(x)]^3 = x/A(G(x))^3 where G(x*A(x)^3) = x.
From Seiichi Manyama, Jun 04 2025: (Start)
Let a(n,k) = [x^n] A(x)^k.
a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(3*n-3*j+k,j)/(3*n-3*j+k) * a(n-j,j). (End)

A143426 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x))^2.

Original entry on oeis.org

1, 1, 2, 7, 32, 175, 1086, 7429, 54994, 435120, 3647686, 32192596, 297654824, 2872372828, 28841766844, 300592170551, 3244942353856, 36219458512421, 417365572999944, 4958429472475171, 60659660219655616, 763325035692109389, 9870492111677035538
Offset: 0

Views

Author

Paul D. Hanna, Aug 14 2008

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 32*x^4 + 175*x^5 + 1086*x^6 +...
A(x*A(x)) = 1 + x + 3*x^2 + 13*x^3 + 70*x^4 + 434*x^5 + 2986*x^6 +...
A(x*A(x))^2 = 1 + 2*x + 7*x^2 + 32*x^3 + 175*x^4 + 1086*x^5 +...
Logarithmic series:
log(A(x)) = x + [d/dx x^3*A(x)^4]*A(x)^(-4)/2! + [d^2/dx^2 x^5*A(x)^6]*A(x)^(-6)/3! + [d^3/dx^3 x^7*A(x)^8]*A(x)^(-8)/4! +...

Links

Crossrefs

Cf. A139702, A087949, A143435, A182969.
Cf. A143500.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A^2,x,x*A));polcoeff(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, Dec 18 2010] */
{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,
A=exp(sum(m=0,n,Dx(m,x^(2*m+1)*A^(2*m+2))*A^(-2*m-2)/(m+1)!)+x*O(x^n)));polcoeff(A,n)}
  • PARI
    a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(n-j+k, j)/(n-j+k)*a(n-j, 2*j))); \\ Seiichi Manyama, Jun 05 2025

Formula

G.f. satisfies: x - G(x) = G(x)^2*A(x)^2 where G(x*A(x)) = x.
G.f. satisfies: A(x) = exp( Sum_{n>=0} [d^n/dx^n x^(2n+1)*A(x)^(2n+2)]*A(x)^(-2n-2)/(n+1)! ). [Paul D. Hanna, Dec 18 2010]
From Seiichi Manyama, Jun 05 2025: (Start)
Let a(n,k) = [x^n] A(x)^k.
a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(n-j+k,j)/(n-j+k) * a(n-j,2*j). (End)

A143435 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x))^3.

Original entry on oeis.org

1, 1, 3, 15, 97, 738, 6297, 58630, 585543, 6200916, 69071103, 804470751, 9753459717, 122670681073, 1596129692136, 21437840848440, 296680980737270, 4224090724829151, 61794432127467450, 927795254532531834, 14282871462981487854, 225247807261125989496, 3636185180695164503129
Offset: 0

Views

Author

Paul D. Hanna, Aug 14 2008

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 97*x^4 + 738*x^5 + 6297*x^6 +...
A(x*A(x)) = 1 + x + 4*x^2 + 24*x^3 + 178*x^4 + 1511*x^5 + 14130*x^6 +...
A(x*A(x))^3 = 1 + 3*x + 15*x^2 + 97*x^3 + 738*x^4 + 6297*x^5 +...
Logarithmic series:
log(A(x)) = x*A(x) + [d/dx x^3*A(x)^6]*A(x)^(-4)/2! + [d^2/dx^2 x^5*A(x)^9]*A(x)^(-6)/3! + [d^3/dx^3 x^7*A(x)^12]*A(x)^(-8)/4! +...

Crossrefs

Cf. A143426, A143436, A143437.
Cf. A139702, A087949, A182969.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A^3,x,x*A));polcoeff(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, Dec 18 2010] */
{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,
A=exp(sum(m=0,n,Dx(m,x^(2*m+1)*A^(3*m+3))*A^(-2*m-2)/(m+1)!)+x*O(x^n)));polcoeff(A,n)}
  • PARI
    a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(n-j+k, j)/(n-j+k)*a(n-j, 3*j))); \\ Seiichi Manyama, Jun 05 2025

Formula

G.f. satisfies: x - G(x) = G(x)^2*A(x)^3 where G(x*A) = x.
G.f. satisfies: A(x) = exp( Sum_{n>=0} [d^n/dx^n x^(2n+1)*A(x)^(3*n+3)]*A(x)^(-2n-2)/(n+1)! ). [Paul D. Hanna, Dec 18 2010]
From Seiichi Manyama, Jun 05 2025: (Start)
Let a(n,k) = [x^n] A(x)^k.
a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(n-j+k,j)/(n-j+k) * a(n-j,3*j). (End)
Showing 1-10 of 19 results. Next

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