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 1-5 of 5 results.

A144682 E.g.f. satisfies: A(x/A(x)^2) = exp(x).

Original entry on oeis.org

1, 1, 5, 73, 2073, 92481, 5775133, 471058953, 47961475313, 5904863932609, 858738633997941, 144899744540718729, 27970301202134146441, 6106540658691499524993, 1493749158085983126737165
Offset: 0

Views

Author

Paul D. Hanna, Sep 19 2008

Keywords

Examples

			E.g.f. A(x) = 1 + x + 5*x^2/2! + 73*x^3/3! + 2073*x^4/4! +...
ILLUSTRATE FORMULA a(n+1) = [x^n/n!] exp(x)*A(x)^(2*n+2) as follows.
Form a table of coefficients of x^k/k! in exp(x)*A(x)^(2*n) for n>=1, k>=0:
exp(x)*A(x)^2: [(1), 3, 17, 219, 5665, 239283, 14432433, ...];
exp(x)*A(x)^4: [1,(5), 41, 605, 15633, 638325, 37250233, ...];
exp(x)*A(x)^6: [1, 7,(73), 1207, 31825, 1274407, 72322201, ...];
exp(x)*A(x)^8: [1, 9, 113,(2073), 56545, 2249769, 124959057, ...];
exp(x)*A(x)^10:[1, 11, 161, 3251,(92481), 3695451, 202282081, ...];
exp(x)*A(x)^12:[1, 13, 217, 4789, 142705,(5775133), 313637833, ...];
exp(x)*A(x)^14:[1, 15, 281, 6735, 210673, 8688975,(471058953), ...]; ...
then the terms along the main diagonal form this sequence shift left.

Crossrefs

Cf. A144681, A144683, A144684.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(n=0,n,A=exp(serreverse(x/A^2)));n!*polcoeff(A,n)}
{a(n)=local(A=1+x+sum(k=2, n-1, a(k)*x^k/k!)+x*O(x^n)); if(n==0, 1, (n-1)!*polcoeff(exp(x+x*O(x^n))*A^(2*n), n-1))}

Formula

E.g.f. satisfies: A(x) = exp( x*A(log A(x))^2 ).
E.g.f. satisfies: a(n+1) = [x^n/n!] exp(x)*A(x)^(2*n+2).
E.g.f. A(x) = G(x/2)^2 where G(x/G(x)) = exp(x) and G(x) is the g.f. of A144681.

A144683 E.g.f. satisfies: A(x/A(x)^3) = exp(x).

Original entry on oeis.org

1, 1, 7, 154, 6625, 446776, 42088609, 5171653432, 792466370161, 146737621112464, 32079442602647521, 8134165156479090136, 2358873194743497162889, 773523517692799444058632, 284154419348084944647780289
Offset: 0

Views

Author

Paul D. Hanna, Sep 19 2008

Keywords

Examples

			E.g.f. A(x) = 1 + x + 7*x^2/2! + 154*x^3/3! + 6625*x^4/4! +...
ILLUSTRATE FORMULA a(n+1) = [x^n/n!] exp(x)*A(x)^(3*n+3) as follows.
Form a table of coefficients of x^k/k! in exp(x)*A(x)^(3*n) for n>=1, k>=0:
exp(x)*A(x)^3: [(1), 4, 34, 685, 27256, 1747159, 159049000, ...];
exp(x)*A(x)^6: [1, (7), 85, 1909, 75193, 4654417, 410053357, ...];
exp(x)*A(x)^9: [1, 10,(154), 3835, 153208, 9284725, 795356632, ...];
exp(x)*A(x)^12:[1, 13, 241, (6625), 272641, 16385713, 1373165425, ...];
exp(x)*A(x)^15:[1, 16, 346, 10441,(446776), 26918851, 2221660936, ...];
exp(x)*A(x)^18:[1, 19, 469, 15445, 690841, (42088609), 3443635405, ...];
exp(x)*A(x)^21:[1, 22, 610, 21799, 1022008, 63371617,(5171653432), ...]; ...
then the terms along the main diagonal form this sequence shift left.

Crossrefs

Cf. A144681, A144682, A144684.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(n=0,n,A=exp(serreverse(x/A^3)));n!*polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x+sum(k=2, n-1, a(k)*x^k/k!)+x*O(x^n)); if(n==0, 1, (n-1)!*polcoeff(exp(x+x*O(x^n))*A^(3*n), n-1))}

Formula

E.g.f. satisfies: A(x) = exp( x*A(log A(x))^3 ).
E.g.f. satisfies: a(n+1) = [x^n/n!] exp(x)*A(x)^(3*n+3).
E.g.f. A(x) = G(3x)^(1/3) where G(x/G(x)) = exp(x) and G(x) is the g.f. of A144681.

A367390 Expansion of e.g.f. A(x) satisfying A(x)^2 = exp(x) * A(x*A(x)) with A(0) = 0.

Original entry on oeis.org

1, 2, 9, 52, 545, 6366, 98707, 1700840, 35405505, 817958170, 21500633891, 618661892652, 19636408658737, 675144805723766, 25147073628948195, 1004734122294047056, 42965745214637476097, 1955039747566085781426, 94404335950307686644163, 4818562790963397438214100
Offset: 1

Views

Author

Paul D. Hanna, Jan 08 2024

Keywords

Comments

Note that if F(x)^2 = exp(x) * F(x*F(x)) with F(0) = 1, then F(x) is the e.g.f. of A367391.

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 545*x^5/5! + 6366*x^6/6! + 98707*x^7/7! + 1700840*x^8/8! + 35405505*x^9/9! + 817958170*x^10/10! + ...
where A(x)^2 = exp(x) * A(x*A(x)) as can be seen from the following expansions
A(x)^2 = 2*x^2/2! + 12*x^3/3! + 96*x^4/4! + 880*x^5/5! + 11280*x^6/6! + 167664*x^7/7! + 3030944*x^8/8! + ...
A(x*A(x)) = 2*x^2/2! + 6*x^3/3! + 60*x^4/4! + 500*x^5/5! + 7230*x^6/6! + 104202*x^7/7! + 1962296*x^8/8! + ...
Let B(x) = x*A(x), then log( A(x)/x ) equals the sum of all iterations of B(x)
log( A(x)/x ) = x + B(x) + B(B(x)) + B(B(B(x))) + B(B(B(B(x)))) + ...
which is equivalent to
log( A(x)/x ) = x + x*A(x) + x*A(x)*A(x*A(x)) + x*A(x)*A(x*A(x)) * A( x*A(x)*A(x*A(x)) ) + ...
RELATED SERIES.
A(x)/x = 1 + x + 3*x^2/2! + 13*x^3/3! + 109*x^4/4! + 1061*x^5/5! + 14101*x^6/6! + 212605*x^7/7! + 3933945*x^8/8! + 81795817*x^9/9! + ...
log( A(x)/x ) = x + 2*x^2/2! + 6*x^3/3! + 60*x^4/4! + 500*x^5/5! + 6870*x^6/6! + 96642*x^7/7! + 1824536*x^8/8! + 36995688*x^9/9! + ...
Successive iterations of B(x) = x*A(x) begin
B(x) = 2*x^2/2! + 6*x^3/3! + 36*x^4/4! + 260*x^5/5! + 3270*x^6/6! + 44562*x^7/7! + 789656*x^8/8! + ...
B(B(x)) = 24*x^4/4! + 240*x^5/5! + 3600*x^6/6! + 52080*x^7/7! + 994560*x^8/8! + ...
B(B(B(x))) = 40320*x^8/8! + 1451520*x^9/9! + 50803200*x^10/10! + ...
B(B(B(B(x)))) = 20922789888000*x^16/16! + 2845499424768000*x^17/17! + ...
etc.
where A(x) = x * exp(x + B(x) + B(B(x)) + B(B(B(x))) + B(B(B(B(x)))) + ...).

Links

Crossrefs

Cf. A367391, A144681, A367386, A367387.
Cf. A369090.

Programs

  • PARI
    {a(n) = my(A=x, V=[0,1]); for(i=1,n, V = concat(V,0); A = Ser(V);
V[#V] = polcoeff( subst(A,x,x*A) - exp(-x +x*O(x^(#V)))*A^2, #V) ); n!*V[n+1]}
for(n=1,40, print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! and B(x) = x*A(x) satisfies the following formulas.
(1) A(x)^2 = exp(x) * A(x*A(x)).
Let B^n(x) denote the n-th iteration of B(x) = x*A(x), where B^(n+1)(x) = B( B^n(x) ) with B^0(x) = x, then
(2) log( A(x)/x ) = Sum_{n>=0} B^n(x).
(3) B^n(x) = x*A(x)^(2^n - 1) / exp( Sum_{k=0..n-2} (2^(n-k-1) - 1) * B^k(x) ) for n > 1.
(3.a) B^2(x) = x*A(x)^3 / exp(x).
(3.b) B^3(x) = x*A(x)^7 / exp(3*x + B(x)).
(3.c) B^4(x) = x*A(x)^15 / exp(7*x + 3*B(x) + B^2(x)).
(3.d) B^5(x) = x*A(x)^31 / exp(15*x + 7*B(x) + 3*B^2(x) + B^3(x)).
(4) A( B^n(x) ) = A(x)^(2^n) / exp( Sum_{k=0..n-1} 2^(n-k-1) * B^k(x) ) for n > 0.
(4.a) A(B(x)) = A(x)^2 / exp(x).
(4.b) A(B^2(x)) = A(x)^4 / exp(2*x + B(x)).
(4.c) A(B^3(x)) = A(x)^8 / exp(4*x + 2*B(x) + B^2(x)).
(4.d) A(B^4(x)) = A(x)^16 / exp(8*x + 4*B(x) + 2*B^2(x) + B^3(x)).

A367385 Expansion of e.g.f. A(x) satisfying A(x/A(x)) = exp(x*A(x)).

Original entry on oeis.org

1, 1, 5, 61, 1329, 43841, 1987153, 116322249, 8430315169, 733890562273, 75025552012641, 8851196086238969, 1188516164483406289, 179619377095898214801, 30271231938826215582001, 5645050489627807288153321, 1157185379272549414363693377, 259281400277115714365664526529
Offset: 0

Views

Author

Paul D. Hanna, Dec 22 2023

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 61*x^3/3! + 1329*x^4/4! + 43841*x^5/5! + 1987153*x^6/6! + 116322249*x^7/7! + 8430315169*x^8/8! + 733890562273*x^9/9! + ...
where A(x/A(x)) = exp(x*A(x)) and
exp(x*A(x)) = 1 + x + 3*x^2/2! + 22*x^3/3! + 329*x^4/4! + 8396*x^5/5! + 318577*x^6/6! + 16388086*x^7/7! + 1075939601*x^8/8! + 86549687704*x^9/9! + ...
Also,
A(x) = exp(x*B(x)^2) where B(x) = A(x*B(x)) begins
B(x) = 1 + x + 7*x^2/2! + 112*x^3/3! + 2989*x^4/4! + 115136*x^5/5! + 5899159*x^6/6! + 381657928*x^7/7! + 30082660633*x^8/8! + 2814548348224*x^9/9! + ...
B(x)^2 = 1 + 2*x + 16*x^2/2! + 266*x^3/3! + 7168*x^4/4! + 275842*x^5/5! + 14058520*x^6/6! + 903187826*x^7/7! + 70653972896*x^8/8! + 6560662418306*x^9/9! + ...
Further,
A(x/C(x)^2) = exp(x) where C(x) = A(x/C(x)) begins
C(x) = 1 + x + 3*x^2/2! + 28*x^3/3! + 533*x^4/4! + 16096*x^5/5! + 680827*x^6/6! + 37544368*x^7/7! + 2577391273*x^8/8! + 213306280480*x^9/9! + ...
C(x)^2 = 1 + 2*x + 8*x^2/2! + 74*x^3/3! + 1344*x^4/4! + 39202*x^5/5! + 1618456*x^6/6! + 87693090*x^7/7! + 5940234656*x^8/8! + 486479747906*x^9/9! + ...

Links

Crossrefs

Cf. A144681.

Programs

  • PARI
    {a(n) = my(A=1+x); for(i=0,n, A = exp( (1/x)*serreverse( x/(A + x*O(x^n)) )^2 )); n!*polcoeff(A,n)}
for(n=0,20, print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x/A(x)) = exp(x*A(x)).
(2) A(x) = exp(x*B(x)^2) where B(x) = A(x*B(x)) = (1/x)*Series_Reversion(x/A(x)).
(3) A(x/C(x)^2) = exp(x) where C(x) = A(x/C(x)) = x / Series_Reversion(x*A(x)).

A144684 E.g.f. satisfies: A(x/A(x)^4) = exp(x).

Original entry on oeis.org

1, 1, 9, 265, 15281, 1379441, 173762425, 28528990393, 5838405067745, 1443332192692321, 421171717332106601, 142520112730201819625, 55149333235223148407953, 24128837930726025305020369
Offset: 0

Views

Author

Paul D. Hanna, Sep 19 2008

Keywords

Examples

			E.g.f. A(x) = 1 + x + 9*x^2/2! + 265*x^3/3! + 15281*x^4/4! +...
ILLUSTRATE FORMULA a(n+1) = [x^n/n!] exp(x)*A(x)^(4*n+4) as follows.
Form a table of coefficients of x^k/k! in exp(x)*A(x)^(4*n) for n>=1, k>=0:
exp(x)*A(x)^4: [(1), 5, 57, 1565, 84017, 7220725, 879058921, ...];
exp(x)*A(x)^8: [1, (9), 145, 4377, 231713, 19222569, 2265114033, ...];
exp(x)*A(x)^12:[1, 13,(265), 8821, 472273, 38328733, 4391383897, ...];
exp(x)*A(x)^16:[1, 17, 417,(15281), 841025, 67630417, 7578669793, ...];
exp(x)*A(x)^20:[1, 21, 601, 24141, (1379441), 111109701, 12258211401, ...];
exp(x)*A(x)^24:[1, 25, 817, 35785, 2135137, (173762425), 18997491601, ...];
exp(x)*A(x)^28:[1, 29, 1065, 50597, 3161873, 261721069,(28528990393), ...]; ...
then the terms along the main diagonal form this sequence shift left.

Crossrefs

Cf. A144681, A144682, A144683.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(n=0,n,A=exp(serreverse(x/A^4)));n!*polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x+sum(k=2, n-1, a(k)*x^k/k!)+x*O(x^n)); if(n==0, 1, (n-1)!*polcoeff(exp(x+x*O(x^n))*A^(4*n), n-1))}

Formula

E.g.f. satisfies: A(x) = exp( x*A(log A(x))^4 ).
E.g.f. satisfies: a(n+1) = [x^n/n!] exp(x)*A(x)^(4*n+4).
E.g.f. A(x) = G(4x)^(1/4) where G(x/G(x)) = exp(x) and G(x) is the g.f. of A144681.
E.g.f. A(x) = G(2x)^(1/2) where G(x/G(x)^2) = exp(x) and G(x) is the g.f. of A144682.
Showing 1-5 of 5 results.

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