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 1-3 of 3 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

Views

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

A145345 G.f. satisfies: A(x/A(x)) = 1 + x*A(x).

Original entry on oeis.org

1, 1, 2, 7, 34, 203, 1398, 10706, 89120, 794347, 7502170, 74511150, 773864654, 8368430208, 93905460014, 1090519614152, 13077315637592, 161643281777801, 2056306418177832, 26887064722265250, 360939404438509866
Offset: 0

Views

Author

Paul D. Hanna, Nov 05 2008

Keywords

Comments

From Paul D. Hanna, Nov 15 2008: (Start)
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. (End)

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 34*x^4 + 203*x^5 + 1398*x^6 + ...
A(x/A(x)) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 34*x^5 + 203*x^6 + 1398*x^7 + ...
A(x) = 1 + x*G(x)^2 where
G(x) = 1 + x + 3*x^2 + 14*x^3 + 83*x^4 + 574*x^5 + 4432*x^6 + ...
is the g.f. of A121687.
ALTERNATE GENERATING METHOD.
This sequence forms column zero in the following array.
Let A denote this sequence.
Start in row zero with this sequence, A, 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, 1, 2, 7, 34, 203, 1398, 10706, 89120, 794347, 7502170, ...];
[1, 2, 5, 18, 86, 502, 3387, 25496, 209242, 1843134, 17235671, ...];
[2, 7, 27, 128, 727, 4763, 34912, 280006, 2418537, 22240055, ...];
[7, 34, 169, 958, 6173, 44364, 349152, 2965098, 26864357, ...];
[34, 203, 1195, 7707, 54792, 425216, 3560600, 31842929, ...];
[203, 1398, 9308, 66310, 510689, 4231188, 37425922, ...];
[1398, 10706, 78414, 605401, 4987185, 43742924, 406387957, ...];
[10706, 89120, 705227, 5824356, 50853813, 469182452, ...];
[89120, 794347, 6707823, 58712463, 539651646, 5211277285, ...];
[794347, 7502170, 67008980, 617340184, 5942316416, 59827126712, ...]; ...

Links

Crossrefs

Cf. A383563, A121687, A145350, A145349, A147664, A302703.

Programs

  • PARI
    {a(n)=local(F=1+x);for(i=0,n,G=serreverse(x/(F+x*O(x^n)))/x;F=1+x*subst(F,x,x*G)^2);polcoeff(F,n)}
  • PARI
    {a(n)=local(F=1+x);for(i=0,n,G=serreverse(x/(F+x*O(x^n)))/x;F=1+x*G^2);polcoeff(F,n)} \\ Paul D. Hanna, Nov 08 2008
  • PARI
    /* This sequence is generated when k=1, m=1: A(x/A(x)^k) = 1 + x*A(x)^m */ {a(n,k=1,m=1)=local(A=sum(i=0,n-1,a(i,k,m)*x^i));if(n==0,1,polcoeff((m+k)/(m+k*n)*A^(m+k*n),n-1))} \\ Paul D. Hanna, Nov 15 2008
  • PARI
    /* Prints terms 0..30 */
{A=[1];
for(m=1,30,
  B=A;
  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
  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(m+1)))[m+1] - Vec(Ser(A))[m+1])/(m+1)); A[n+1]}
for(n=0, 30, print1(2^n*a(n), ", ")) \\ Vaclav Kotesovec, Jan 31 2023

Formula

G.f. satisfies: A(x) = 1 + x*G(x)^2 where G(x) = g.f. of A121687.
G.f. satisfies: A(x) = G(x/A(x)) where G(x) = A(x*G(x)) = g.f. of A121687. - Paul D. Hanna, Nov 08 2008
a(n) = [x^(n-1)] (2/(n+1))*A(x)^(n+1) for n>=1 with a(0)=1; i.e., a(n) equals 2/(n+1) times the coefficient of x^(n-1) in A(x)^(n+1) for n>=1. - Paul D. Hanna, Nov 15 2008

A384265 G.f. A(x) satisfies A( x/A(x)^2 + x^2 ) = 1 + x*A(x)^2.

Original entry on oeis.org

1, 1, 3, 16, 119, 1087, 11408, 132468, 1663047, 22234598, 313303201, 4618133168, 70815362628, 1124901511837, 18450127411436, 311636597558992, 5409374008212747, 96326565666389514, 1757212245656330130, 32798907320986196010, 625759879697614204041, 12192353855092076824051, 242419156191210763668352
Offset: 0

Views

Author

Paul D. Hanna, May 30 2025

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 119*x^4 + 1087*x^5 + 11408*x^6 + 132468*x^7 + 1663047*x^8 + 22234598*x^9 + 313303201*x^10 + ...
 where A( x/A(x)^2 + x^2 ) = 1 + x*A(x)^2.
RELATED SERIES.
The g.f. of A145347 begins
B(x) = 1 + x + 4*x^2 + 26*x^3 + 220*x^4 + 2203*x^5 + 24836*x^6 + 306104*x^7 + 4047988*x^8 + ... + A145347(n)*x^n + ...
 where B(x/B(x)) = 1 + x*B(x)^3
 also, B(x) = A( x*B(x) ).
The g.f. of A182954 begins
C(x) = 1 + x + 5*x^2 + 39*x^3 + 381*x^4 + 4284*x^5 + 53163*x^6 + 710810*x^7 + 10085621*x^8 + ... + A182954(n)*x^n + ...
 where C(x) = 1 + x*C(x) * C( x*C(x) )^4
 also, C(x) = A( x*C(x)^2 ).
C(x)^4 = 1 + 4*x + 26*x^2 + 220*x^3 + 2203*x^4 + 24836*x^5 + 306104*x^6 + ...
 where B(x) = 1 + x*C(x)^4.
RELATED TABLE.
The table of coefficients of x^k in A(x)^n begins
  n = 1: [ 1,  1,   3,   16,   119,   1087, ...];
  n = 2: [ 1, (2),  7,   38,   279,   2508, ...];
  n = 3: [ 1,  3, (12),  67,   489,   4338, ...];
  n = 4: [(1), 4,  18, (104),  759,   6664, ...];
  n = 5: [ 1,  5,  25,  150, (1100),  9586, ...];
  n = 6: [ 1, (6), 33,  206,  1524, (13218),...];
  n = 7: [ 1,  7,  42,  273,  2044,  17689, ...];
  n = 8: [ 1,  8, (52), 352,  2674,  23144, ...];
  n = 9: [ 1,  9,  63,  444,  3429,  29745, ...];
  n =10: [ 1, 10,  75, (550), 4325,  37672, ...];
  n =11: [ 1, 11,  88,  671,  5379,  47124, ...];
  n =12: [ 1, 12, 102,  808, (6609), 58320, ...];
  ...
in which the terms in parenthesis illustrate the property
(2.b) [x^n] A(x)^(n+1) = 2 * [x^(n-1)] A(x)^(2*n+2) for n >= 1.
Further, other terms along diagonals obey
(5.a) A145345(n) = [x^n] B(x) = [x^n] A(x)^(n+1)/(n+1), and
(5.b) A121687(n) = [x^n] C(x) = [x^n] A(x)^(2*n+1)/(2*n+1) for n >= 0.

Links

Crossrefs

Cf. A383563, A145347 (B(x)), A182954 (C(x)).

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoef(x - serreverse(x/Ser(A)) + serreverse(x/Ser(A)^2)^2, #A) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n along with B(x) = g.f. of A145347 and C(x) = g.f. of A182954 satisfy the following formulas.
(1.a) A( x/A(x)^2 + x^2 ) = 1 + x*A(x)^2.
(1.b) A(x) = A(x)^2 - x*A( sqrt(x - x/A(x)) )^4.
(2.a) x = Series_Reversion( x/A(x) ) - Series_Reversion( x/A(x)^2 )^2.
(2.b) [x^n] A(x)^(n+1) = 2 * [x^(n-1)] A(x)^(2*n+2) for n >= 1.
(2.c) B(x) = 1 + x*C(x)^4 where B(x) = A( x*B(x) ) and C(x) = A( x*C(x)^2 ).
(3.a) A(x) = B( x/A(x) ) where B(x/B(x)) = 1 + x*B(x)^3.
(3.b) A(x) = C( x/A(x)^2 ) where C(x) = 1 + x*C(x) * C( x*C(x) )^4.
(4.a) A(x) = A(x)^2 - x*C( x/A(x) )^4.
(4.b) C(x/A(x)) = A( sqrt(x - x/A(x)) ) = A( x/A(x) * C(x/A(x))^2 ).
(4.c) B(x/A(x)^2) = 1 + x*A(x)^2 = A( x/A(x)^2 * B(x/A(x)^2) ).
(5.a) A145347(n) = [x^n] B(x) = [x^n] A(x)^(n+1)/(n+1) for n >= 0.
(5.b) A182954(n) = [x^n] C(x) = [x^n] A(x)^(2*n+1)/(2*n+1) for n >= 0.
(5.c) A145347(n) = [x^(n-1)] C(x)^4 = [x^(n-1)] A(x)^(2*n+2)*2/(n+1) for n >= 1.
Showing 1-3 of 3 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.