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.

Previous Showing 11-15 of 15 results.

A120977 G.f. A(x) satisfies A(x) = 1 + x*A(x)^5 * A(x*A(x)^5)^5.

Original entry on oeis.org

1, 1, 10, 170, 3745, 96960, 2814752, 89221360, 3037327145, 109825686370, 4185287088735, 167139924222426, 6964610755602495, 301800832258018835, 13564159649547824735, 630916661388096564620, 30316241123672291911875
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2006

Keywords

Crossrefs

Cf. A120976; variants: A110447, A120971, A120973, A120975.

Programs

  • 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)^5))[ #G]); A=Vec(((Ser(G)-1)/x)^(1/5));A[n+1]}
  • PARI
    a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(5*n+k, j)/(5*n+k)*a(n-j, 5*j))); \\ Seiichi Manyama, Mar 01 2025

Formula

G.f. A(x) satisfies: A(x) = G(G(x)-1), A(G(x)-1) = G(A(x)-1), A(x) = G(x*A(x)^5) and A(x/G(x)^5) = G(x), where G(x) is the g.f. of A120976 and satisfies G(x/G(x)^5) = 1 + x.
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(5*n+k,j)/(5*n+k) * a(n-j,5*j). (End)

A140094 G.f. satisfies: A(x) = x/(1 - A(A(A(x)))).

Original entry on oeis.org

1, 1, 4, 25, 199, 1855, 19387, 221407, 2717782, 35455981, 487672243, 7029980797, 105732907498, 1653377947393, 26805765569863, 449568735630517, 7785116448484318, 138980739891821269, 2554369130466577138
Offset: 1

Views

Author

Paul D. Hanna, May 08 2008, May 20 2008

Keywords

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 25*x^4 + 199*x^5 + 1855*x^6 + 19387*x^7 +...
Iterations A_{n+1}(x) = A( A_{n}(x) ) are related as follows.
A_2(x) = 1 - Series_Reversion( A(x) ) / x;
A_3(x) = 1 - x / A(x);
A_4(x) = 1 - A(x) / A_2(x);
A_5(x) = 1 - A_2(x) / A_3(x);
A_6(x) = 1 - A_3(x) / A_4(x); ...
where the iterations of A(x) begin:
A_2(x) = x + 2*x^2 + 10*x^3 + 71*x^4 + 616*x^5 + 6119*x^6 + 67210*x^7 +...;
A_3(x) = x + 3*x^2 + 18*x^3 + 144*x^4 + 1365*x^5 + 14544*x^6 +...;
A_4(x) = x + 4*x^2 + 28*x^3 + 250*x^4 + 2584*x^5 + 29584*x^6 +...;
A_5(x) = x + 5*x^2 + 40*x^3 + 395*x^4 + 4435*x^5 + 54515*x^6 +...;
A_6(x) = x + 6*x^2 + 54*x^3 + 585*x^4 + 7104*x^5 + 93555*x^6 +...;
...
Iterations are also related by continued fractions:
A(x) = x/(1 - A_2(x)/(1 - A_4(x)/(1 - A_6(x)/(1 -...)))) ;
A_2(x) = A(x)/(1 - A_3(x)/(1 - A_5(x)/(1 - A_7(x)/(1 -...)))).

Links

Crossrefs

Cf. A140095, A088714.
Cf. A088717, A091713, A120971, A139702.

Programs

  • PARI
    {a(n)=local(A); if(n<0, 0, n++; A=x+O(x^2); for(i=2, n, A=x/(1-subst(A, x, subst(A, x, A)))); polcoeff(A, n))}

Formula

G.f. A(x) satisfies:
(1) A(x) = Series_Reversion(x - x*A(A(x))).
(2) A(x) = x + Sum_{n=1} d^(n-1)/dx^(n-1) x^n * A(A(x))^n / n!.
(3) A(x) = x*exp( Sum_{n=1} d^(n-1)/dx^(n-1) x^n * A(A(x))^n/x / n! ).
Define A_{n} such that A_{n+1}(x) = A( A_{n}(x) ) with A_0(x) = x,
then A_{n}(x) = A_{n-1}/[1 - A_{n+2}(x)] ;
thus A_{n}(x) = 1 - A_{n-3}(x) / A_{n-2}(x).
G.f. A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:
A = 1 + x*A*C;
B = A + x*B*D;
C = B + x*C*E;
D = C + x*D*F;
E = D + x*E*G; ...

Extensions

Name, formulas, and examples revised by Paul D. Hanna, Feb 03 2013

A140095 G.f. satisfies: A(x) = x/(1 - A(A(A(A(x))))).

Original entry on oeis.org

1, 1, 5, 41, 437, 5513, 78477, 1225865, 20644021, 370334137, 7017055933, 139562915193, 2899946191077, 62722686552841, 1408033260333581, 32729098457253417, 786224322656857941, 19486950945070339801, 497649167866430159197, 13078602790892074110937
Offset: 1

Views

Author

Paul D. Hanna, May 08 2008, May 20 2008

Keywords

Examples

			G.f.: A(x) = x + x^2 + 5*x^3 + 41*x^4 + 437*x^5 + 5513*x^6 + 78477*x^7 +...
Iterations A_{n+1}(x) = A( A_{n}(x) ) are related as follows.
A_2(x) = 1 - Series_Reversion(A_2(x)) / Series_Reversion(A(x));
A_3(x) = 1 - Series_Reversion(A(x)) / x;
A_4(x) = 1 - x / A(x);
A_5(x) = 1 - A(x) / A_2(x);
A_6(x) = 1 - A_2(x) / A_3(x);
A_7(x) = 1 - A_3(x) / A_4(x);
A_8(x) = 1 - A_4(x) / A_5(x); ...
where the iterations of A(x) begin:
A_2(x) = x + 2*x^2 + 12*x^3 + 108*x^4 + 1220*x^5 + 16028*x^6 +...
A_3(x) = x + 3*x^2 + 21*x^3 + 207*x^4 + 2489*x^5 + 34259*x^6 +...
A_4(x) = x + 4*x^2 + 32*x^3 + 344*x^4 + 4408*x^5 + 63776*x^6 +...
A_5(x) = x + 5*x^2 + 45*x^3 + 525*x^4 + 7165*x^5 + 109125*x^6 +...
A_6(x) = x + 6*x^2 + 60*x^3 + 756*x^4 + 10972*x^5 + 175948*x^6 +...
A_7(x) = x + 7*x^2 + 77*x^3 + 1043*x^4 + 16065*x^5 + 271103*x^6 +...
A_8(x) = x + 8*x^2 + 96*x^3 + 1392*x^4 + 22704*x^5 + 402784*x^6 +...
...
Iterations are also related by continued fractions:
A(x) = x/(1 - A_3(x)/(1 - A_6(x)/(1 - A_9(x)/(1 -...)))) ;
A_2(x) = A(x)/(1 - A_4(x)/(1 - A_7(x)/(1 - A_10(x)/(1 -...)))) ;
A_3(x) = A_2(x)/(1 - A_5(x)/(1 - A_8(x)/(1 - A_11(x)/(1 -...)))) ;
A_4(x) = A_3(x)/(1 - A_6(x)/(1 - A_9(x)/(1 - A_12(x)/(1 -...)))) ; ...

Crossrefs

Cf. A140094, A088714, A088717, A091713, A120971, A139702.

Programs

  • PARI
    {a(n)=local(A); if(n<1, 0, n++; A=x+O(x^2); for(i=2, n,B=subst(A, x, A); A=x/(1-subst(B, x, B))); polcoeff(A, n))}

Formula

G.f. A(x) satisfies:
(1) A(x) = Series_Reversion(x - x*A(A(A(x)))).
(2) A(x) = x + Sum_{n=1} d^(n-1)/dx^(n-1) x^n * A(A(A(x)))^n / n!.
(3) A(x) = x*exp( Sum_{n=1} d^(n-1)/dx^(n-1) x^n * A(A(A(x)))^n/x / n! ).
Define A_{n} such that A_{n+1}(x) = A( A_{n}(x) ) with A_0(x) = x,
then A_{n}(x) = A_{n-1}/[1 - A_{n+3}(x)] ;
thus A_{n}(x) = 1 - A_{n-4}(x) / A_{n-3}(x).
G.f.: A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:
A = 1 + x*A*D;
B = A + x*B*E;
C = B + x*C*F;
D = C + x*D*G;
E = D + x*E*H; ...

A381029 G.f. A(x) satisfies A(x) = 1/(1 - x * A(x*A(x)^2)^2).

Original entry on oeis.org

1, 1, 3, 16, 113, 955, 9178, 97427, 1121705, 13836694, 181295019, 2507119320, 36416096984, 553461581406, 8774534872463, 144744539399484, 2479088917439527, 44004108702467428, 808171916050540308, 15335535608825061803, 300272362335527090277, 6059534345675248667550
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2025

Keywords

Crossrefs

Cf. A088714, A381615.
Cf. A120971, A143508, A381600.
Cf. A381572.

Programs

  • PARI
    a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-j+k, j)/(2*n-j+k)*a(n-j, 2*j)));

Formula

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-j+k,j)/(2*n-j+k) * a(n-j,2*j).

A381600 G.f. A(x) satisfies A(x) = 1/(1 - x * A(x)^2 * A(x*A(x)^2)^2).

Original entry on oeis.org

1, 1, 5, 39, 383, 4360, 55201, 758877, 11157081, 173623407, 2838995592, 48515016273, 862904739711, 15923514065053, 304089551295359, 5997295071211547, 121944040723497105, 2552667957311169834, 54944459391676448365, 1214747696691087352576, 27559533140410855702244
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2025

Keywords

Crossrefs

Column k=1 of A381592.
Cf. A120971.

Programs

  • PARI
    a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n+j+k, j)/(2*n+j+k)*a(n-j, 2*j)));

Formula

See A381592.
Previous Showing 11-15 of 15 results.

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