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.

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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

A360337 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(3*n+1))^(n+1) for n >= 0.

Original entry on oeis.org

1, 1, 7, 124, 3446, 125706, 5540958, 282129207, 16148101259, 1020687876920, 70377734170699, 5246775452965364, 420104327765022458, 35937961751407922101, 3270668852260460283730, 315546031669853942486219, 32173855061751806476275665, 3457696770952845858846954590
Offset: 0

Views

Author

Paul D. Hanna, Feb 06 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 7*x^2 + 124*x^3 + 3446*x^4 + 125706*x^5 + 5540958*x^6 + 282129207*x^7 + 16148101259*x^8 + 1020687876920*x^9 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 8*x^2 + 146*x^3 + 4083*x^4 + 149077*x^5 + 6569555*x^6 + 334401750*x^7 + 19137707066*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(3*n+1))^(n+1) / (n+1),
as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
so that b(n) begin:
[1/1, 2/2, 24/3, 584/4, 20415/5, 894462/6, 45986885/7, 2675214000/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  7,  124,  3446,  125706,  5540958,  282129207, ...];
n=1: [1, 2, 15,  262,  7189,  260040, 11396948,  577954822, ...];
n=2: [1, 3, 24,  415, 11250,  403521, 17583859,  888063051, ...];
n=3: [1, 4, 34,  584, 15651,  556696, 24118370, 1213065672, ...];
n=4: [1, 5, 45,  770, 20415,  720141, 31017985, 1553601145, ...];
n=5: [1, 6, 57,  974, 25566,  894462, 38301069, 1910335764, ...];
n=6: [1, 7, 70, 1197, 31129, 1080296, 45986885, 2283964852, ...];
n=7: [1, 8, 84, 1440, 37130, 1278312, 54095632, 2675214000, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(3*n+1))^(n+1):
n=0: [1, 1,   1,    7,   124,    3446,   125706,    5540958, ...];
n=1: [1, 2,   9,   76,  1252,   32742,  1150522,   49515052, ...];
n=2: [1, 3,  24,  253,  4179,  103866,  3499510,  146421240, ...];
n=3: [1, 4,  46,  584, 10061,  242520,  7836278,  317454824, ...];
n=4: [1, 5,  75, 1115, 20415,  487566, 15193230,  594390940, ...];
n=5: [1, 6, 111, 1892, 37119,  894462, 27139545, 1025356992, ...];
n=6: [1, 7, 154, 2961, 62412, 1538698, 45986885, 1682957396, ...];
n=7: [1, 8, 204, 4368, 98894, 2519232, 75032832, 2675214000, ...]; ...
to see that the main diagonals of the tables are the same.

Links

Crossrefs

Cf. A360336, A360338, A360237, A360345, A302703.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(3*m+1))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(3*n+1))^(n+1) for n>=0.
(2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(3*n+1))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alpha, where d = 5.7189630165873859806..., alpha = 1.4141427006501..., c = 0.027880568114272... - Vaclav Kotesovec, Feb 06 2023
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