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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A301435 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 3*x*A(x) )^n / 2^(n+1).

Original entry on oeis.org

1, 5, 85, 2413, 92501, 4394663, 246960721, 15952488893, 1161947365721, 94123508334877, 8390631582459161, 816285612080072183, 86069411025655759073, 9778818588385117669485, 1191176369495005591666205, 154886342347657508336231809, 21414816209632043592416524165, 3137473307880710686085483679771, 485584927860050612832028930482597, 79169341280742628145184619086229089
Offset: 0

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Author

Paul D. Hanna, Mar 24 2018

Keywords

Examples

			G.f.: A(x) = 1 + 5*x + 85*x^2 + 2413*x^3 + 92501*x^4 + 4394663*x^5 + 246960721*x^6 + 15952488893*x^7 + 1161947365721*x^8 + 94123508334877*x^9 + ...
such that
1 = 1/2  +  ((1+x) - 3*x*A(x))/2^2  +  ((1+x)^2 - 3*x*A(x))^2/2^3  +  ((1+x)^3 - 3*x*A(x))^3/2^4  +  ((1+x)^4 - 3*x*A(x))^4/2^5  +  ((1+x)^5 - 3*x*A(x))^5/2^6  + ...
Also,
1 = 1/(2 + 3*x*A(x)) + (1+x)/(2 + 3*x*A(x)*(1+x))^2 + (1+x)^4/(2 + 3*x*A(x)*(1+x)^2)^3 + (1+x)^9/(2 + 3*x*A(x)*(1+x)^3)^4 + (1+x)^16/(2 + 3*x*A(x)*(1+x)^4)^5 + ...
RELATED SERIES.
Let R(k,x) = Sum_{n>=0} binomial(n+k,k) * (1+x)^(n*(n+k)) / 2^(n+k+1)
then
1 = R(0,x) - 3*x*A(x)*R(1,x) + 3^2*x^2*A(x)^2*R(2,x) - 3^3*x^3*A(x)^3*R(3,x) + 3^4*x^4*A(x)^4*R(4,x) - 3^5*x^5*A(x)^5*R(5,x) + ...
The table of coefficients in R(k,x) begins:
k=0: [1, 3, 36, 744, 21606, 807912, 36948912, 1997801520, ...];
k=1: [1, 10, 197, 5600, 206880, 9387864, 504836996, 31376330400, ...];
k=2: [1, 21, 621, 23447, 1078980, 58590504, 3667676768, ...];
k=3: [1, 36, 1494, 72516, 4075569, 261336096, 18861815280, ...];
k=4: [1, 55, 3050, 185190, 12492745, 934629539, 77091424200, ...];
k=5: [1, 78, 5571, 413764, 33004131, 2850142590, 266518090901, ...];
k=6: [1, 105, 9387, 837165, 77946645, 7696470411, 810015165897, ...];
k=7: [1, 136, 14876, 1568632, 168591350, 18874524760, 2221139481932, ...]; ...

Links

Crossrefs

Cf. A303288, A301436, A323314, A323315, A323316, A323317, A323318, A323319, A323320, A323321.

Programs

  • Mathematica
    nmax = 19;
nmax2 = 300 (* = empirical sum terms *);
sol = {a[0] -> 1};
A[x_] = Sum[a[n] x^n, {n, 0, nmax}];
Do[A[x] = A[x] /. sol; s = 1-Sum[((1+x)^n - 3x A[x] + O[x]^(k+1))^n / 2^(n+1), {n, 0, nmax2}] /. sol; c = SeriesCoefficient[s, {x, 0, k}]; sol = sol ~Join~ Solve[c == 0][[1]] /. HoldPattern[a[n_] -> an_] :> (a[n] -> Round[an]), {k, 2, nmax+1}];
a /@ Range[0, nmax] /. sol (* Jean-François Alcover, Nov 05 2019 *)

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( (1+x)^n - 3*x*A(x) )^n / 2^(n+1).
(2) 1 = Sum_{n>=0} (1+x)^(n^2) / (2 + 3*x*A(x)*(1+x)^n)^(n+1). - Paul D. Hanna, Jan 10 2019
(3) 1 = Sum_{k>=0} (-3*x)^k * A(x)^k * Sum_{n>=0} C(n+k,k) * (1+x)^(n*(n+k)) / 2^(n+k+1).
(4) 1 = Sum_{n>=0} Sum_{k=0..n} C(n,k) * (1+x)^(n*(n-k)) / 2^(n+1) * (-3*x)^k * A(x)^k.

A303653 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 3*(1+x)^n - A(x) )^n / 3^(n+1).

Original entry on oeis.org

1, 15, 291, 20868, 2501535, 406641390, 82021892979, 19576367780568, 5370958558206975, 1661471768423203359, 571522497313691705223, 216322544080204799422227, 89344723486622904627485286, 39989870323587920736747152457, 19285197574525200774860259575856, 9970552400727667627167081347333058, 5502200681071110455003310691040648913
Offset: 0

Views

Author

Paul D. Hanna, Apr 28 2018

Keywords

Examples

			G.f.: A(x) = 1 + 15*x + 291*x^2 + 20868*x^3 + 2501535*x^4 + 406641390*x^5 + 82021892979*x^6 + 19576367780568*x^7 + 5370958558206975*x^8 + ...
such that
1 = 1/3  +  (3*(1+x) - A(x))/3^2  +  (3*(1+x)^2 - A(x))^2/3^3  +  (3*(1+x)^3 - A(x))^3/3^4  +  (3*(1+x)^4 - A(x))^4/3^5  +  (3*(1+x)^5 - A(x))^5/3^6 + ...
Also,
1 = 1/(3 + A(x))  +  3*(1+x)/(3 + (1+x)*A(x))^2  +  3^2*(1+x)^4/(3 + (1+x)^2*A(x))^3  +  3^3*(1+x)^9/(3 + (1+x)^3*A(x))^4  +  3^4*(1+x)^16/(3 + (1+x)^4*A(x))^5  +  3^5*(1+x)^25/(3 + (1+x)^5*A(x))^6  + ...

Links

Crossrefs

Cf. A301436.

Formula

G.f.: 1 = Sum_{n>=0} 3^n * (1+x)^(n^2) / (3 + (1+x)^n * A(x))^(n+1).

A305134 E.g.f. A(x) satisfies: 1 = Sum_{n>=0} ( 2*exp(n*x) - A(x) )^n / 2^(n+1).

Original entry on oeis.org

1, 6, 106, 9798, 2042986, 721198086, 378754904746, 274462194065478, 261211828432706026, 315282684090141417606, 470124979835875652863786, 848422945353825106452994758, 1822526603267557240862350671466, 4596139606368556055825161023870726, 13448584326250762088160567798167642026, 45199506338787031550197525974862852621638
Offset: 0

Views

Author

Paul D. Hanna, May 29 2018

Keywords

Examples

			E.g.f.: A(x) = 1 + 6*x + 106*x^2/2! + 9798*x^3/3! + 2042986*x^4/4! + 721198086*x^5/5! + 378754904746*x^6/6! + 274462194065478*x^7/7! + 261211828432706026*x^8/8! + 315282684090141417606*x^9/9! + 470124979835875652863786*x^10/10! + ...
such that
1 = 1/2  +  (2*exp(x) - A(x))/2^2  +  (2*exp(2*x) - A(x))^2/2^3  +  (2*exp(3*x) - A(x))^3/2^4  +  (2*exp(4*x) - A(x))^4/2^5  +  (2*exp(5*x) - A(x))^5/2^6 + ...
Also,
1 = 1/(2 + A(x))  +  2*exp(x)/(2 + exp(x)*A(x))^2  +  2^2*exp(4*x)/(2 + exp(2*x)*A(x))^3  +  2^3*exp(9*x)/(2 + exp(3*x)*A(x))^4  +  2^4*exp(16*x)/(2 + exp(4*x)*A(x))^5  +  2^5*exp(25*x)/(2 + exp(5*x)*A(x))^6  + ...
RELATED SERIES.
log(A(x)) = 6*x + 70*x^2/2! + 8322*x^3/3! + 1812142*x^4/4! + 657412530*x^5/5! + 351254035150*x^6/6! + 257586196964082*x^7/7! + 247297892785673422*x^8/8! + 300478711708843324530*x^9/9! + 450397140484880214948430*x^10/10! + ...

Links

Crossrefs

Cf. A304640, A301436.

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
(1) 1 = Sum_{n>=0} ( 2*exp(n*x) - A(x) )^n / 2^(n+1).
(2) 1 = Sum_{n>=0} 2^n * exp(n^2*x) / (2 + exp(n*x) * A(x))^(n+1).
Showing 1-3 of 3 results.

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