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 1-4 of 4 results.

A371709 Expansion of g.f. A(x) satisfying A( x*A(x)^2 + x*A(x)^3 ) = A(x)^3.

Original entry on oeis.org

1, 1, 1, 2, 6, 16, 39, 99, 271, 764, 2157, 6128, 17658, 51534, 151635, 448962, 1337493, 4008040, 12072594, 36524898, 110943633, 338218626, 1034509917, 3173811240, 9763898994, 30113782641, 93094164244, 288415278638, 895332445053, 2784580242557, 8675408291598, 27072326322939
Offset: 1

Views

Author

Paul D. Hanna, May 02 2024

Keywords

Comments

Compare to the following identities of the Catalan function C(x) = x + C(x)^2 (A000108):
(1) C(x)^2 = C( x*C(x)*(1 + C(x)) ),
(2) C(x)^4 = C( x*C(x)^3*(1 + C(x))*(1 + C(x)^2) ),
(3) C(x)^8 = C( x*C(x)^7*(1 + C(x))*(1 + C(x)^2)*(1 + C(x)^4) ),
(4) C(x)^(2^n) = C( x*C(x)^(2^n-1)*Product_{k=0..n-1} (1 + C(x)^(2^k)) ) for n > 0.
a(3^n) == 1 (mod 3) for n >= 0.
a(2*3^n) == 1 (mod 3) for n >= 0.
a(n) == 2 (mod 3) iff n is the sum of 2 distinct powers of 3 (A038464).

Examples

			G.f. A(x) = x + x^2 + x^3 + 2*x^4 + 6*x^5 + 16*x^6 + 39*x^7 + 99*x^8 + 271*x^9 + 764*x^10 + 2157*x^11 + 6128*x^12 + 17658*x^13 + 51534*x^14 + 151635*x^15 + 448962*x^16 + ...
where A( x*A(x)^2*(1 + A(x)) ) = A(x)^3.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 17*x^6 + 48*x^7 + 126*x^8 + 332*x^9 + 918*x^10 + 2616*x^11 + 7504*x^12 + ...
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 13*x^6 + 36*x^7 + 105*x^8 + 292*x^9 + 801*x^10 + 2256*x^11 + 6515*x^12 + 18981*x^13 + ...
A(x)^2 + A(x)^3 = x^2 + 3*x^3 + 6*x^4 + 12*x^5 + 30*x^6 + 84*x^7 + 231*x^8 + 624*x^9 + 1719*x^10 + 4872*x^11 + 14019*x^12 + 40599*x^13 + ...
Let B(x) be the series reversion of g.f. A(x), B(A(x)) = x, then
B(x) * (1+x)/(1+x^3) = x - 2*x^4 + 3*x^7 - 5*x^10 + 7*x^13 - 9*x^16 + 12*x^19 - 15*x^22 + 18*x^25 - 23*x^28 + ... + (-1)^n*A005704(n)*x^(3*n+1) + ...
where A005704 is the number of partitions of 3*n into powers of 3.
We can show that g.f. A(x) = A( x*A(x)^2*(1 + A(x)) )^(1/3) satisfies
(4) A(x) = x * Product_{n>=0} (1 + A(x)^(3^n))
by substituting x*A(x)^2*(1 + A(x)) for x in (4) to obtain
A(x)^3 = x * A(x)^2*(1 + A(x)) * Product_{n>=1} (1 + A(x)^(3^n))
which is equivalent to formula (4).
SPECIFIC VALUES.
A(3/10) = 0.526165645044542830201162330432965674027415264612114520...
A(1/4) = 0.353259384374080248921564026412797625837830114153200664...
A(1/5) = 0.255218141344695821239609680309162895225297482063273545...
A(t) = 1/2 and A(t*3/8) = 1/8 at t = (1/2)/Product_{n>=0} (1 + 1/2^(3^n)) = 0.295718718466711580562679377308518930409875701753934072...
A(t) = 1/3 and A(t*4/27) = 1/27 at t = (1/3)/Product_{n>=0} (1 + 1/3^(3^n)) = 0.241059181496179959557718992589733756750585121455883861...
A(t) = 1/4 and A(t*5/64) = 1/64 at t = (1/4)/Product_{n>=0} (1 + 1/4^(3^n)) = 0.196922325724019432212969925740117827612003158137366017...

Links

Crossrefs

Cf. A371716, A370440, A370441, A370446, A264228, A198951.
Cf. A005704, A038464.

Programs

  • PARI
    /* Using series reversion of x/Product_{n>=0} (1 + x^(3^n)) */
{a(n) = my(A); A = serreverse( x/prod(k=0,ceil(log(n)/log(3)), (1 + x^(3^k) +x*O(x^n)) ) ); polcoeff(A,n)}
for(n=1,35, print1(a(n),", "))
  • PARI
    /* Using A(x)^3 = A( x*A(x)^2 + x*A(x)^3 ) */
{a(n) = my(A=[1],F); for(i=1,n, A = concat(A,0); F = x*Ser(A);
A[#A] = polcoeff( subst(F,x, x*F^2 + x*F^3 ) - F^3, #A+2) ); A[n]}
for(n=1,35, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = A( x*A(x)^2*(1 + A(x)) ).
(2) A(x)^9 = A( x*A(x)^8*(1 + A(x))*(1 + A(x)^3) ).
(3) A(x)^27 = A( x*A(x)^26*(1 + A(x))*(1 + A(x)^3)*(1 + A(x)^9) ).
(4) A(x) = x * Product_{n>=0} (1 + A(x)^(3^n)).
(5) A(x) = Series_Reversion( x / Product_{n>=0} (1 + x^(3^n)) ).
a(n) ~ c * d^n / n^(3/2), where d = 3.2753449994351908157330968510747739... and c = 0.1559869008021616116037651076359... - Vaclav Kotesovec, May 03 2024
The radius of convergence r of g.f. A(x) and A(r) satisfy 1 = Sum_{n>=0} 3^n * A(r)^(3^n) / (1 + A(r)^(3^n)) and r = A(r) / Product_{n>=0} (1 + A(r)^(3^n)), where r = 0.30531134893345362211... = 1/d (d is given above) and A(r) = 0.600582105427215700175254768411726892599... - Paul D. Hanna, May 03 2024

A372534 Expansion of g.f. A(x) satisfying A(x)^2 = A( x*A(x) + 3*x*A(x)^2 ).

Original entry on oeis.org

1, 3, 12, 63, 372, 2322, 15102, 101439, 698340, 4900914, 34931808, 252185238, 1840242546, 13551558336, 100579610790, 751610709279, 5650352546628, 42702935642082, 324256445598816, 2472613511240754, 18926918200655928, 145379893260849876, 1120198916414984148, 8656357557290045382
Offset: 1

Views

Author

Paul D. Hanna, May 29 2024

Keywords

Comments

Compare to the following identities of the Catalan function C(x) = x + C(x)^2 (A000108):
(1) C(x)^2 = C( x*C(x)*(1 + C(x)) ),
(2) C(x)^4 = C( x*C(x)^3*(1 + C(x))*(1 + C(x)^2) ),
(3) C(x)^8 = C( x*C(x)^7*(1 + C(x))*(1 + C(x)^2)*(1 + C(x)^4) ),
(4) C(x)^(2^n) = C( x*C(x)^(2^n-1)*Product_{k=0..n-1} (1 + C(x)^(2^k)) ) for n > 0.

Examples

			G.f.: A(x) = x + 3*x^2 + 12*x^3 + 63*x^4 + 372*x^5 + 2322*x^6 + 15102*x^7 + 101439*x^8 + 698340*x^9 + 4900914*x^10 + 34931808*x^11 + 252185238*x^12 + ...
where A(x)^2 = A( x*A(x) + 3*x*A(x)^2 ).
Also,
A(x) = x * (1 + 3*A(x)) * (1 + 3*A(x)^2) * (1 + 3*A(x)^4) * (1 + 3*A(x)^8) * (1 + 3*A(x)^16) * ... * (1 + 3*A(x)^(2^n)) * ...
RELATED SERIES.
A(x)^2 = x^2 + 6*x^3 + 33*x^4 + 198*x^5 + 1266*x^6 + 8388*x^7 + 57033*x^8 + 396090*x^9 + 2798718*x^10 + 20056824*x^11 + 145438146*x^12 + ...
x*A(x) + 3*x*A(x)^2 = x^2 + 6*x^3 + 30*x^4 + 162*x^5 + 966*x^6 + 6120*x^7 + 40266*x^8 + 272538*x^9 + 1886610*x^10 + 13297068*x^11 + ...
SPECIFIC VALUES.
A(1/9) = 0.20017482594200170883488591841314367600913783...
A(1/10) = 0.15939222988059047986391116283589184626082823...
A(1/11) = 0.13474373940944085584086064879196682498369755...
A(1/12) = 0.11741441277153705906655653078308588616286400...

Links

Crossrefs

Cf. A356782, A371716, A372530, A000120, A000108.

Programs

  • PARI
    {a(n) = my(A = serreverse( x/prod(k=0, #binary(n), 1 + 3*x^(2^k) +x*O(x^n)) ));
polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0); F=Ser(A);
A[#A] = polcoeff( subst(F, x, x*F*(1 + 3*F) ) - F^2, #A) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^2 = A( x*A(x) + 3*x*A(x)^2 ).
(2) A(x)^4 = A( x*A(x)^3*(1 + 3*A(x))*(1 + 3*A(x)^2) ).
(3) A(x)^8 = A( x*A(x)^7*(1 + 3*A(x))*(1 + 3*A(x)^2)*(1 + 3*A(x)^4) ).
(4) A(x)^(2^n) = A( x*A(x)^(2^n-1)*Product_{k=0..n-1} (1 + 3*A(x)^(2^k)) ) for n > 0.
(5) A(x) = x * Product_{n>=0} (1 + 3*A(x)^(2^n)).
(6) A(x) = x * Sum_{n>=0} 3^A000120(n) * A(x)^n, where A000120(n) = number of 1's in binary expansion of n.
(7) A(x) = Series_Reversion( x / Product_{n>=0} (1 + 3*x^(2^n)) ).
The radius of convergence r of g.f. A(x) and A(r) satisfy 1 = Sum_{n>=0} 3*2^n * A(r)^(2^n) / (1 + 3*A(r)^(2^n)) and r = A(r) / Product_{n>=0} (1 + 3*A(r)^(2^n)), where r = 0.121354219013536538658862726712953201279180864478537... and A(r) = 0.301069983372147236415588688159692129761365234627514...

A372526 Expansion of g.f. A(x) satisfying A( x*A(x)^2 + A(x)^4 ) = A(x)^3.

Original entry on oeis.org

1, 1, 2, 6, 20, 70, 256, 969, 3762, 14895, 59916, 244179, 1006026, 4183396, 17534888, 74007851, 314256048, 1341575769, 5754629794, 24789907450, 107202369386, 465209278326, 2025212712660, 8842042378050, 38707067608872, 169860383434800, 747096961093560, 3292855742992644
Offset: 1

Views

Author

Paul D. Hanna, May 04 2024

Keywords

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 70*x^6 + 256*x^7 + 969*x^8 + 3762*x^9 + 14895*x^10 + 59916*x^11 + 244179*x^12 + ...
where A( x*A(x)^2 + A(x)^4 ) = A(x)^3.
RELATED SERIES.
(1) A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 31*x^6 + 114*x^7 + 432*x^8 + 1676*x^9 + 6633*x^10 + 26676*x^11 + 108696*x^12 + ...
(2) x*A(x)^2 + A(x)^4 = x^3 + 3*x^4 + 9*x^5 + 30*x^6 + 108*x^7 + 405*x^8 + 1560*x^9 + 6138*x^10 + 24570*x^11 + 99738*x^12 + ...
(3) Let R(x) be the series reversion of A(x), R(A(x)) = x, then
R(x) = x - x^2 - x^4 - x^10 - x^28 - x^82 - x^244 - x^730 + ... + -x^(3^n+1) + ...
SPECIFIC VALUES.
A(1/5) = 0.2937167157779136500722875625899113632023...
A(1/6) = 0.2150539986528250703029216090552606059919...
A(1/7) = 0.1740789503092637057579787813575613522976...
A(1/8) = 0.1471095742959948638409574049543396207684...

Links

Crossrefs

Cf. A371713, A371716.

Programs

  • PARI
    /* Using series reversion of x - x*Sum_{n>=0} x^(3^n) */
{a(n) = my(A); A = serreverse( x - x*sum(k=0, ceil(log(n)/log(3)), x^(3^k) +x*O(x^n)) ); polcoeff(A, n)}
for(n=1, 35, print1(a(n), ", "))
  • PARI
    /* Using A(x)^3 = A( x*A(x)^2 + A(x)^4 ) */
{a(n) = my(A=[1], F); for(i=1, n, A = concat(A, 0); F = x*Ser(A);
A[#A] = polcoeff( subst(F, x, x*F^2 + F^4 ) - F^3, #A+2) ); A[n]}
for(n=1, 35, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = A( x*A(x)^2 + A(x)^4 ).
(2) A(x)^9 = A( x*A(x)^8 + A(x)^10 + A(x)^12 ).
(3) A(x)^27 = A( x*A(x)^26 + A(x)^28 + A(x)^30 + A(x)^36 ).
(4) A(x)^(3^n) = A( x*A(x)^(3^n-1) + Sum_{k=0..n-1} A(x)^(3^n+3^k) ) for n > 0.
(5) A(x) = x + Sum_{n>=0} A(x)^(3^n+1).
(6) A(x) = Series_Reversion(x - x*Sum_{n>=0} x^(3^n) ).
The radius of convergence r and A(r) satisfy: 1 = Sum_{n>=0} (3^n+1) * A(r)^(3^n) and r = A(r) - Sum_{n>=0} A(r)^(3^n+1), where r = 0.214732801800375010254079407876131682823903064701286670006... and A(r) = 0.384967312289976324530970877165834568783164468488676531438...

A376230 Expansion of g.f. A(x) satisfying A(x)^2 = A( x*A(x) + x*A(x)^2 + x*A(x)^3 ).

Original entry on oeis.org

1, 1, 3, 8, 28, 98, 370, 1423, 5643, 22753, 93299, 387324, 1625768, 6886156, 29399430, 126377000, 546527682, 2376094442, 10379414436, 45532904886, 200511864604, 886055084460, 3927826810396, 17462128520246, 77838085223570, 347813389031746, 1557683052973482, 6990670698115144
Offset: 1

Views

Author

Paul D. Hanna, Sep 23 2024

Keywords

Comments

Compare to C(x)^2 = C( x*C(x) + x*C(x)^2 ) where C(x) = x + C(x)^2 is the g.f. of the Catalan numbers (A000108).
Compare to D(x)^2 = D( x*D(x) + 2*x*D(x)^2 + x*D(x)^3 ) where D(x) = x*F(x)^2 and F(x) = 1 + F(x)^3 is the g.f. of A001764.
It appears that, for n >= 1, a(n) is odd iff n = 4*A000695(k) + {0,1,2,3} for some k >= 0, where A000695 is the Moser-de Bruijn sequence (sums of distinct powers of 4).

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 8*x^4 + 28*x^5 + 98*x^6 + 370*x^7 + 1423*x^8 + 5643*x^9 + 22753*x^10 + 93299*x^11 + 387324*x^12 + ...
where A(x)^2 = A( x*A(x) + x*A(x)^2 + x*A(x)^3 ).
RELATED SERIES.
Let B(x) be the g.f. of Stern's diatomic series (A002487):
B(x) = x + x^2 + 2*x^3 + x^4 + 3*x^5 + 2*x^6 + 3*x^7 + x^8 + 4*x^9 + 3*x^10 + 5*x^11 + 2*x^12 + 5*x^13 + 3*x^14 + 4*x^15 + x^16 + ...
then A(x)^2 = x * B( A(x) ) and A( x^2/B(x) ) = x.
Other related series begin as follows.
A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 22*x^5 + 81*x^6 + 300*x^7 + 1168*x^8 + 4622*x^9 + 18704*x^10 + 76738*x^11 + 319054*x^12 + ...
A(x)^3 = x^3 + 3*x^4 + 12*x^5 + 43*x^6 + 168*x^7 + 657*x^8 + 2649*x^9 + 10803*x^10 + 44733*x^11 + 187130*x^12 + ...
SPECIFIC VALUES.
A(t) = 2/5 at t = 0.21059927171685797509442589615351221149308548505349232763302...
A(t) = 1/3 at t = 0.20284350479378267499481171039846064829566135845385597426623...
A(t) = 1/4 at t = 0.17791461653470954766421766118399907657065676113208935616640...
A(t) = 1/5 at t = 0.15460046705113920070261079885331294477343970681952336915318...
A(1/5) = 0.31995483821441278259163540295892975411660207078522958569307...
A(1/6) = 0.22418805161328879302723377308422267982113532037722470920139...
A(1/7) = 0.17887744157158359437462802053243127220002079340556427992475...
A(1/8) = 0.15001315387877904231502214457445835910409883718703588438530...
A(1/10) = 0.11423875178408085947774841103426888472717517658954942399943...

Links

Crossrefs

Cf. A002487, A371709, A371716.

Programs

  • PARI
    /* Series_Reversion(x / Product_{n>=0} (1 + x^(2^n) + x^(2^(n+1))) ) */
{a(n) = my(A = serreverse(x / prod(k=0, #binary(n), (1 + x^(2^k) + x^(2^(k+1))) +x*O(x^n))));  polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    /* A(x)^2 = A( x*A(x) + x*A(x)^2 + x*A(x)^3 ) */
{a(n) = my(A=[1], F); for(i=1, n, A=concat(A, 0); F=x*Ser(A);
A[#A] = -polcoeff( F^2 - subst(F, x, x*F + x*F^2 + x*F^3), #A+1) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^2 = A( x*A(x) + x*A(x)^2 + x*A(x)^3 ).
(2) A(x)^4 = A( x*A(x)^3 * (1 + A(x) + A(x)^2) * (1 + A(x)^2 + A(x)^4) ).
(3) A(x)^8 = A( x*A(x)^7 * (1 + A(x) + A(x)^2) * (1 + A(x)^2 + A(x)^4) * (1 + A(x)^4 + A(x)^8) ).
(4) A(x)^(2^n) = A( x*A(x)^(2^n-1) * Product_{k=0..n-1} (1 + A(x)^(2^k) + A(x)^(2^(k+1))) ) for n > 0.
(5) A(x) = x * Product_{n>=0} (1 + A(x)^(2^n) + A(x)^(2^(n+1))).
(6) A(x) = Series_Reversion(x / Product_{n>=0} (1 + x^(2^n) + x^(2^(n+1))) ).
(7) A( x^2/B(x) ) = x where B(x) is the g.f. of Stern's diatomic series (A002487).
(8) A(x)^2 = x * B( A(x) ) where B(x) is the g.f. of Stern's diatomic series (A002487).
(9) A(x)^2 = x * Sum_{n>=0} A(x)^n * Sum_{k=0..n-1} (binomial(k, n-k-1) mod 2).
The radius of convergence r of g.f. A(x) and A(r) satisfy 1 = Sum_{n>=0} (2^n*A(r)^(2^n) + 2^(n+1)*A(r)^(2^(n+1))) / (1 + A(r)^(2^n) + A(r)^(2^(n+1))) and r = A(r) / Product_{n>=0} (1 + A(r)^(2^n) + A(r)^(2^(n+1))), where r = 0.210913447825795710516245118118286786032842961511008744076383999... and A(r) = 0.416616392852528289560364740676108057746635416495044590336240813...
Showing 1-4 of 4 results.

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