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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A375434 Expansion of g.f. A(x) satisfying A(x) = (1 + x*A(x)) * (1 + 3*x*A(x)^2).

Original entry on oeis.org

1, 4, 31, 301, 3274, 38158, 465919, 5883040, 76189177, 1006440238, 13508178448, 183689450959, 2525336086630, 35041483528522, 490125130328455, 6902993856515389, 97814486474787898, 1393470813699724726, 19946461692566594413, 286742046721454817358, 4138001844031453456120
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2024

Keywords

Comments

In general, if G(x) = (1 + p*x*G(x)) * (1 + q*x*G(x)^2) for fixed p and q, then
(C.1) G(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * p^(n-k) * q^k * G(x)^k ).
(C.2) G(x) = (1/x) * Series_Reversion( x/(1 + p*x) - q*x^2 ).
(C.3) x = (sqrt((p - q*y)^2 + 4*p*q*y^2) - (p + q*y))/(2*p*q*y^2), where y = G(x).

Examples

			G.f. A(x) = 1 + 4*x + 31*x^2 + 301*x^3 + 3274*x^4 + 38158*x^5 + 465919*x^6 + 5883040*x^7 + 76189177*x^8 + 1006440238*x^9 + 13508178448*x^10 + ...
where A(x) = (1 + x*A(x)) * (1 + 3*x*A(x)^2).
RELATED SERIES.
Let B(x) = A(x/B(x)) and B(x*A(x)) = A(x), then
B(x) = 1 + 4*x + 15*x^2 + 57*x^3 + 216*x^4 + 819*x^5 + 3105*x^6 + 11772*x^7 + ... + A125145(n)*x^n + ...
where B(x) = (1 + x)/(1 - 3*x - 3*x^2).

Links

Crossrefs

Cf. A125145, A215661, A375435, A375436, A375437.
Cf. A007863, A216314, A364374.

Programs

  • PARI
    {a(n) = my(A=1+x); for(i=1, n, A=(1 + x*A)*(1 + 3*x*(A+x*O(x^n))^2)); polcoef(A, n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n) = polcoef( (1/x)*serreverse( x*(1-3*x-3*x^2)/(1+x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2 * 3^j * A^j)*x^m/m))); polcoef(A, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum{n>=0} a(n)*x^n satisfies:
(1) A(x) = (1 + x*A(x)) * (1 + 3*x*A(x)^2).
(2) A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * 3^k * A(x)^k ).
(3) A(x) = (1/x) * Series_Reversion( x*(1 - 3*x - 3*x^2)/(1 + x) ).
(4) A(x) = Sum_{n>=0} A125145(n) * x^n * A(x)^n, where g.f. of A125145 = (1 + x)/(1 - 3*x - 3*x^2).
(5) x = (sqrt(21*A(x)^2 - 6*A(x) + 1) - (1 + 3*A(x)))/(6*A(x)^2).
a(n) = Sum_{k=0..n} 3^k * binomial(n+k+1,k) * binomial(n+k+1,n-k) / (n+k+1). - Seiichi Manyama, Sep 08 2024
a(n) ~ ((36 + (48266 - 714*sqrt(17))^(1/3) + (48266 + 714*sqrt(17))^(1/3))/7)^n / (sqrt(6*Pi*((20517 - 4861*sqrt(17))^(1/3) + (20517 + 4861*sqrt(17))^(1/3) - 42)) * n^(3/2)). - Vaclav Kotesovec, Sep 14 2024

A128235 Triangle read by rows: T(n,k) is the number of sequences of length n on the alphabet {0,1,2,3}, containing k subsequences 00 (0<=k<=n-1).

Original entry on oeis.org

1, 4, 15, 1, 57, 6, 1, 216, 33, 6, 1, 819, 162, 36, 6, 1, 3105, 756, 189, 39, 6, 1, 11772, 3402, 945, 216, 42, 6, 1, 44631, 14931, 4536, 1143, 243, 45, 6, 1, 169209, 64314, 21168, 5778, 1350, 270, 48, 6, 1, 641520, 273051, 96633, 28323, 7128, 1566, 297, 51
Offset: 0

Views

Author

Emeric Deutsch, Feb 27 2007

Keywords

Comments

Row n has n terms (n>=1). T(n,0) = A125145(n). Sum(k*T(n,k), k=0..n-1) = (n-1)*4^(n-2) = A002697(n-1).

Examples

			T(4,2) = 6 because we have 0001, 0002, 0003, 1000, 2000 and 3000.
Triangle starts:
1;
4;
15,    1;
57,    6,  1;
216,  33,  6, 1;
819, 162, 36, 6, 1;

Links

Crossrefs

Cf. A125145, A002697.

Programs

  • Maple
    G:=(1+z-t*z)/(1-3*z-3*z^2-t*z+3*t*z^2): Gser:=simplify(series(G,z=0,14)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: 1; for n from 1 to 11 do seq(coeff(P[n],t,j),j=0..n-1) od; # yields sequence in triangular form

Formula

G.f.: (1+z-tz)/(1-3z-3z^2-tz+3tz^2).

A351529 The number of quaternary strings of length n containing 00.

Original entry on oeis.org

0, 0, 1, 7, 40, 205, 991, 4612, 20905, 92935, 407056, 1762117, 7556095, 32148940, 135892321, 571232647, 2389810360, 9956870845, 41335010911, 171055514452, 705891052825, 2905717608775, 11934337612576, 48918212175157, 200149835407615, 817572886925980
Offset: 0

Views

Author

R. J. Mathar, Feb 13 2022

Keywords

Links

Crossrefs

Cf. A008466 (2-ary), A186244 (3-ary), A351530 (5-ary), A125145 (not containing 00).

Programs

  • Mathematica
    LinearRecurrence[{7,-9,-12},{0,0,1},30] (* Harvey P. Dale, Feb 27 2023 *)

Formula

G.f.: x^2 / ( (4*x-1)*(3*x^2+3*x-1) ).
a(n) = 4^n - A125145(n).
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