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.

A121687 G.f. satisfies A(x) = 1 + x*A(x) * A( x*A(x) )^2.

Original entry on oeis.org

1, 1, 3, 14, 83, 574, 4432, 37244, 335153, 3194510, 32001596, 335019839, 3649450270, 41227610316, 481724831132, 5809341783543, 72177761136925, 922539273876404, 12115001489115910, 163284755614174305
Offset: 0

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Author

Paul D. Hanna, Aug 15 2006, Aug 20 2008

Keywords

Examples

			G.f. A(x) = 1 + x + 3*x^2 + 14*x^3 + 83*x^4 + 574*x^5 + 4432*x^6 +...
A(x)^2 = 1 + 2*x + 7*x^2 + 34*x^3 + 203*x^4 + 1398*x^5 + 10706*x^6 +...
A(x*A(x)) = 1 + x + 4*x^2 + 23*x^3 + 160*x^4 + 1259*x^5 + 10833*x^6 +...
A(x*A(x))^2 = 1 + 2*x + 9*x^2 + 54*x^3 + 382*x^4 + 3022*x^5 + 25993*x^6 +...
A(x)*A(x*A(x))^2 = 1 + 3*x + 14*x^2 + 83*x^3 + 574*x^4 + 4432*x^5 +...
The logarithm of the g.f. is given by:
log(A(x)) = A(x)^2*x + {d/dx x*A(x)^4}*x^2/2! + {d^2/dx^2 x^2*A(x)^6}*x^3/3! + {d^3/dx^3 x^3*A(x)^8}*x^4/4! +...

Links

Crossrefs

Cf. A383563, variants: A030266, A182953, A182954.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=0,n,A=serreverse(x/(1+x*(A +x*O(x^n))^2))/x); polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1/(1-x*subst(A^2,x,x*A)));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+sum(i=1,n-1,a(i)*x^i+x*O(x^n)));
for(i=1,n,A=exp(sum(m=1,n,sum(k=0,n-m,binomial(m+k-1,k)*polcoeff(A^(2*m),k)*x^k)*x^m/m)+x*O(x^n)));polcoeff(A,n)} \\ Paul D. Hanna, Dec 15 2010
  • PARI
    {a(n, m=1)=if(n==0, 1, if(m==0, 0^n, sum(k=0, n, m*binomial(n+m, k)/(n+m)*a(n-k, 3*k))))} \\ Paul D. Hanna, Dec 15 2010

Formula

G.f. satisfies G(x) = x/(1 + x*A(x)^2) where G(x*A(x)) = x.
G.f. satisfies A(x) = 1/(1 - x*A(x*A(x))^2).
G.f. satisfies the following two equations (which are equivalent)
A(x) = exp( Sum_{m>=0} {d^m/dx^m x^m*A(x)^(2m+2)} * x^(m+1)/(m+1)! )
A(x) = exp( Sum_{m>=1} [Sum_{k>=0} C(m+k-1,k)*{[y^k] A(y)^(2m)}*x^k]*x^m/m). - Paul D. Hanna, Dec 15 2010
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(n+m,k)/(n+m) * a(n-k,2k). - Paul D. Hanna, Dec 15 2010

A147664 G.f. satisfies: A(x/A(x)) = 1 + x*A(x)^2.

Original entry on oeis.org

1, 1, 3, 15, 100, 801, 7296, 73174, 791751, 9116613, 110640310, 1405349658, 18585016509, 254855278565, 3612425924919, 52793266545585, 793851646358364, 12261570084250926, 194260753173421656, 3153098224666860712
Offset: 0

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Author

Paul D. Hanna, Nov 09 2008

Keywords

Comments

More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then
A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);
thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 100*x^4 + 801*x^5 + 7296*x^6 +...
A(x)^2 = 1 + 2*x + 7*x^2 + 36*x^3 + 239*x^4 + 1892*x^5 + 17019*x^6 +...
A(x/A(x)) = 1 + x + 2*x^2 + 7*x^3 + 36*x^4 + 239*x^5 + 1892*x^6 +...
A(x) = 1 + x*G(x)^3 where G(x) = A(x*G(x)) is the g.f. of A182953:
G(x) = 1 + x + 4*x^2 + 25*x^3 + 197*x^4 + 1797*x^5 + 18178*x^6 +...
To illustrate the formula a(n) = [x^(n-1)] 3*A(x)^(n+2)/(n+2),
form a table of coefficients in A(x)^(n+2) as follows:
A^3: [(1), 3, 12, 64, 426, 3345, 29766, 291999, ...];
A^4: [1, (4), 18, 100, 671, 5244, 46248, 449264, ...];
A^5: [1, 5, (25), 145, 985, 7686, 67305, 648085, ...];
A^6: [1, 6, 33, (200), 1380, 10782, 93922, 897402, ...];
A^7: [1, 7, 42, 266, (1869), 14658, 127246, 1207753, ...];
A^8: [1, 8, 52, 344, 2466, (19456), 168604, 1591496, ...];
A^9: [1, 9, 63, 435, 3186, 25335, (219522), 2063052, ...]; ...
in which the main diagonal forms the initial terms of this sequence:
[3/3*(1), 3/4*(4), 3/5*(25), 3/6*(200), 3/7*(1869), 3/8*(19456), ...].
ALTERNATE GENERATING METHOD.
This sequence forms column zero in the follow array.
Let A denote this sequence, and A^2 the self-convolution square of A.
Start in row zero with A^2, after prepending an initial '1', then repeat: drop the initial term and perform convolution with A and the remaining terms in a given row to obtain the next row:
[1, 1, 2, 7, 36, 239, 1892, 17019, 168746, 1807656, 20634852, 248560373, ...];
[1, 3, 12, 64, 426, 3345, 29766, 291999, 3097746, 35059659, 419160576,...];
[3, 15, 85, 571, 4443, 38952, 376090, 3930156, 43875903, 518191486, ...];
[15, 100, 701, 5494, 47883, 457451, 4724372, 52138050, 609215321, ...];
[100, 801, 6495, 56980, 542331, 5558082, 60793521, 704009142, ...];
[801, 7296, 65878, 630811, 6448878, 70124397, 806356465, 9749112990, ...];
[7296, 73174, 718577, 7386763, 80183478, 917721557, 11031656810, ...];
[73174, 791751, 8324862, 90823582, 1038161379, 12431200320, 155525155360, ...]; ...

Links

Crossrefs

Cf. A182953, A145348, A145350, A120972, A145345, A145349.

Programs

  • PARI
    {a(n)=local(F=1+x,G);for(i=0,n,G=serreverse(x/(F+x*O(x^n))^1)/x;F=1+x*G^3);polcoeff(F,n)}
  • PARI
    /* This sequence is generated when k=1, m=2: A(x/A(x)^k) = 1 + x*A(x)^m */
{a(n, k=1, m=2)=local(A=sum(i=0, n-1, a(i, k, m)*x^i)+x*O(x^n)); if(n==0, 1, polcoeff((m+k)/(m+k*n)*A^(m+k*n), n-1))}
for(n=0,20,print1(a(n),", "))
  • PARI
    /* Prints terms 0..30 */
{A=[1];
for(m=1,30,
  B=Vec(Ser(A)^2);
  for(i=1,m-1, C=Vec(Ser(A)*Ser(B)); B=vector(#C-1,n,C[n+1]) );
  A=concat(A,0); A[#A]=B[1];
);
A} \\ Paul D. Hanna, Jan 10 2016

Formula

G.f.: A(x) = 1 + x*G(x)^3 where G(x) = A(x*G(x)) and A(x) = G(x/A(x)) is the g.f. of A182953.
a(n) = [x^(n-1)] 3*A(x)^(n+2)/(n+2) for n>=1 with a(0)=1; i.e., a(n) equals the coefficient of x^(n-1) in 3*A(x)^(n+2)/(n+2) for n>=1 (see comment).

A182954 G.f. satisfies: A(x) = 1 + x*A(x) * A( x*A(x) )^4.

Original entry on oeis.org

1, 1, 5, 39, 381, 4284, 53163, 710810, 10085621, 150326044, 2336828792, 37687170215, 628069684439, 10782885724300, 190248852445782, 3442896376032300, 63804661588968521, 1209314277690837796
Offset: 0

Views

Author

Paul D. Hanna, Dec 15 2010

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 39*x^3 + 381*x^4 + 4284*x^5 + ...
Related expansions:
A(x*A(x)) = 1 + x + 6*x^2 + 54*x^3 + 592*x^4 + 7331*x^5 + 98870*x^6 + ...
A(x*A(x))^4 = 1 + 4*x + 30*x^2 + 292*x^3 + 3305*x^4 + 41420*x^5 + ...
The g.f. satisfies:
log(A(x)) = A(x)^4*x + {d/dx x*A(x)^8}*x^2/2! + {d^2/dx^2 x^2*A(x)^12}*x^3/3! + {d^3/dx^3 x^3*A(x)^16}*x^4/4! + ...

Links

Crossrefs

Cf. A384265, A030266, A121687, A182953, A182955.

Programs

  • PARI
    {a(n) = my(A = 1 + sum(i=1,n-1,a(i)*x^i+x*O(x^n)));
for(i=1,n, A = exp( sum(m=1,n, sum(k=0,n-m, binomial(m+k-1,k)*polcoef(A^(4*m),k)*x^k) * x^m/m ) + x*O(x^n))); polcoef(A,n)}
  • PARI
    {a(n, m=1) = if(n==0, 1, if(m==0, 0^n, sum(k=0, n, m*binomial(n+m, k)/(n+m)*a(n-k, 4*k))))}

Formula

G.f. A(x) satisfies:
* A(x) = exp( Sum_{m>=0} {d^m/dx^m x^m*A(x)^(4m+4)} * x^(m+1)/(m+1)! );
* A(x) = exp( Sum_{m>=1} [Sum_{k>=0} C(m+k-1,k)*{[y^k] A(y)^(4m)}*x^k]*x^m/m);
which are equivalent.
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(n+m,k)/(n+m) * a(n-k,4k).

A182955 G.f. satisfies: A(x) = 1 + x*A(x) * A( x*A(x) )^5.

Original entry on oeis.org

1, 1, 6, 56, 651, 8671, 126997, 1997798, 33260799, 580270730, 10534337521, 197986746949, 3837397114948, 76473239154148, 1563252546786254, 32716989219013821, 699959257347957763, 15288884723649589585
Offset: 0

Views

Author

Paul D. Hanna, Dec 15 2010

Keywords

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 56*x^3 + 651*x^4 + 8671*x^5 +...
Related expansions:
A(x*A(x)) = 1 + x + 7*x^2 + 74*x^3 + 953*x^4 + 13846*x^5 +...
A(x*A(x))^5 = 1 + 5*x + 45*x^2 + 520*x^3 + 6950*x^4 + 102481*x^5 +...
The g.f. satisfies:
log(A(x)) = A(x)^5*x + {d/dx x*A(x)^10}*x^2/2! + {d^2/dx^2 x^2*A(x)^15}*x^3/3! + {d^3/dx^3 x^3*A(x)^20}*x^4/4! +...

Crossrefs

Cf. A030266, A121687, A182953, A182954.

Programs

  • PARI
    {a(n)=local(A=1+sum(i=1,n-1,a(i)*x^i+x*O(x^n)));
for(i=1,n,A=exp(sum(m=1,n,sum(k=0,n-m,binomial(m+k-1,k)*polcoeff(A^(5*m),k)*x^k)*x^m/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(n+m, k)/(n+m)*a(n-k, 5*k))))}

Formula

G.f. A(x) satisfies:
* A(x) = exp( Sum_{m>=0} {d^m/dx^m x^m*A(x)^(5m+5)} * x^(m+1)/(m+1)! );
* A(x) = exp( Sum_{m>=1} [Sum_{k>=0} C(m+k-1,k)*{[y^k] A(y)^(5m)}*x^k]*x^m/m);
which are equivalent.
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(n+m,k)/(n+m) * a(n-k,5k).
Showing 1-4 of 4 results.

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