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 1-6 of 6 results.

A355865 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * (x^n - (-1)^n*2*A(x))^(2*n+1).

Original entry on oeis.org

1, 3, 25, 254, 2844, 34031, 426498, 5526399, 73433377, 995167783, 13701794657, 191122323160, 2695092314319, 38357425655599, 550268824751092, 7948720164361366, 115517358604881329, 1687796954715824015, 24777722054035138573, 365305177280838473896
Offset: 0

Views

Author

Paul D. Hanna, Aug 04 2022

Keywords

Comments

Compare to the identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n which holds for all y.

Examples

			G.f.: A(x) = 1 + 3*x + 25*x^2 + 254*x^3 + 2844*x^4 + 34031*x^5 + 426498*x^6 + 5526399*x^7 + 73433377*x^8 + 995167783*x^9 + 13701794657*x^10 + ...

Links

Crossrefs

Cf. A355866.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, x^m * (x^m - (-1)^m*2*Ser(A))^(2*m+1) ), #A-1)/2);A[n+1]}
for(n=0,20,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 0 = Sum_{n=-oo..+oo} (-x)^n * ((-x)^n - 2*A(x))^(2*n+1).
(2) 0 = Sum_{n=-oo..+oo} x^(2*n*(n-1)) / (1 - 2*A(x)*(-x)^n)^(2*n-1).

A366229 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (x^(3*n+1) - A(x))^n.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 55, 138, 349, 904, 2377, 6323, 16993, 46036, 125625, 344973, 952565, 2643257, 7366942, 20613366, 57884187, 163071852, 460769168, 1305466309, 3707928596, 10555941648, 30115379589, 86087330322, 246541672062, 707274898726, 2032285666846
Offset: 0

Views

Author

Paul D. Hanna, Oct 29 2023

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(3*n+2))^n, which holds formally for all y.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 23*x^5 + 55*x^6 + 138*x^7 + 349*x^8 + 904*x^9 + 2377*x^10 + 6323*x^11 + 16993*x^12 + ...

Links

Crossrefs

Cf. A358937, A355866.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A, #A, x^n * (x^(3*n+1) - Ser(A))^n ), #A) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 1 = Sum_{n=-oo..+oo} x^n * (x^(3*n+1) - A(x))^n.
(2) 1 = Sum_{n=-oo..+oo} x^(n*(3*n-2)) / (1 - x^(3*n-1)*A(x))^n.
(3) 0 = Sum_{n=-oo..+oo} (-1)^n * x^n * (x^(3*n+2) - A(x))^n (trivial).
(4) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n*(n-1)) / (1 - x^(3*n-1)*A(x))^n (trivial).

A380676 G.f. A(x) satisfies 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^n)^(3*n+1).

Original entry on oeis.org

1, 2, 9, 76, 591, 5127, 46919, 444617, 4333010, 43132310, 436715297, 4483520704, 46564078707, 488335074439, 5164287656762, 55010054836724, 589682412920880, 6356441723399838, 68858811108713642, 749250723117079260, 8185098919015604558, 89739660783143322586, 987110817010576637569
Offset: 0

Views

Author

Paul D. Hanna, Feb 02 2025

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 76*x^3 + 591*x^4 + 5127*x^5 + 46919*x^6 + 444617*x^7 + 4333010*x^8 + 43132310*x^9 + 436715297*x^10 + ...
SPECIFIC VALUES.
A(t) = 3/2 at t = 0.084454810721317538501174440773777047952092460562060...
  where 2 = Sum_{n=-oo..+oo} (-t)^n * (3/2 + t^n)^(3*n+1).
A(t) = 4/3 at t = 0.077952215522932621280995556726745992779521168178442...
A(t) = 5/4 at t = 0.069865542488187377549700484712724108090103217291400...
A(t) = 6/5 at t = 0.062525019563729453209334340397151869258204650105887...
A(1/12) = 1.4451475449531942766582635648883506035661276873944...
  where 2 = Sum_{n=-oo..+oo} (-1/12)^n * (A(1/12) + (1/12)^n)^(3*n+1).
A(1/13) = 1.3197666375699291221191258833369709715040515804644...
A(1/14) = 1.2629677124586701325494126247872966004241466655536...
A(1/15) = 1.2263276036037963341062042248250428743844880153971...
A(1/16) = 1.1998529038743458677434930677034050910039899372219...

Links

Crossrefs

Cf. A380067, A355866, A380677.

Programs

  • PARI
    {a(n) = my(V=[1]); for(i=1, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef(2 - sum(n=-#V, #V, (-1)^n * x^n * (A + x^n)^(3*n+1) ), #V-1) ); H=A; V[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^n)^(3*n+1).
(2) 2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(3*n-2)) / (1 + x^n*A(x))^(3*n-1).

A380677 G.f. A(x) satisfies 1 = Sum_{n=-oo..+oo} x^(2*n) * (x^n - A(x))^(3*n+1).

Original entry on oeis.org

1, 2, 8, 36, 198, 1128, 6837, 42690, 273960, 1792650, 11922735, 80342746, 547403208, 3764568202, 26097746670, 182183863242, 1279566641040, 9035527984360, 64109825254786, 456834687004440, 3267926616628182, 23458797921291994, 168936073477132102, 1220121029135864026, 8835737467337361482
Offset: 0

Views

Author

Paul D. Hanna, Feb 02 2025

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 36*x^3 + 198*x^4 + 1128*x^5 + 6837*x^6 + 42690*x^7 + 273960*x^8 + 1792650*x^9 + 11922735*x^10 + ...
SPECIFIC VALUES.
A(t) = 7/4 at t = 0.12654949614445186746403892264694555335923498557738...
  where 1 = Sum_{n=-oo..+oo} t^(2*n) * (t^n - 7/4)^(3*n+1).
A(t) = 5/3 at t = 0.12374694612565134762563311753154796236873902596812...
A(t) = 3/2 at t = 0.11392195456863186572686610752037791827642247932473...
A(t) = 4/3 at t = 0.09535917714046949923896929084305426642940930464927...
A(t) = 5/4 at t = 0.08098320583796566321668508295130093344916245020730...
A(1/8) = 1.69987163237671043867918157348979527169465395859405...
  where 1 = Sum_{n=-oo..+oo} (1/8)^(2*n) * ((1/8)^n - A(1/8))^(3*n+1).
A(1/9) = 1.46724009425513930419976858432180568713155056224164...
A(1/10) = 1.3665270076239843695076027726524469708778850053524...
A(1/11) = 1.3048130783240200786482939740924774873262324649207...
A(1/12) = 1.2620494023042372384830602119971826992309809007730...
A(1/14) = 1.2057100150678855865365454675611764497376238367914...
A(1/16) = 1.1698113057379453133949062841882391284824341375308...

Links

Crossrefs

Cf. A380067, A355866, A380676.

Programs

  • PARI
    {a(n) = my(V=[1]); for(i=1, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef(-1 + sum(n=-#V, #V, x^(2*n) * (x^n - A)^(3*n+1) ), #V-1) ); H=A; V[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 1 = Sum_{n=-oo..+oo} x^(2*n) * (x^n - A(x))^(3*n+1).
(2) 1 = Sum_{n=-oo..+oo} x^(3*n*(n-1)) / (1 - x^n*A(x))^(3*n-1).

A361766 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * (1 - x^n/A(-x))^(n+2).

Original entry on oeis.org

1, 1, 2, 5, 12, 27, 57, 123, 280, 666, 1614, 3955, 9733, 23949, 58967, 145844, 363137, 910339, 2295192, 5811070, 14754567, 37542078, 95715596, 244567665, 626388406, 1608131393, 4137707994, 10667045757, 27546269363, 71241831762, 184508259405, 478501423792
Offset: 0

Views

Author

Paul D. Hanna, Mar 26 2023

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds for all y as a formal power series in x.
Related identity: 0 = Sum_{n=-oo..+oo} x^(3*n) * (y - x^n)^(n+2), which holds for all y as a formal power series in x. - Paul D. Hanna, Jan 19 2025

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 27*x^5 + 57*x^6 + 123*x^7 + 280*x^8 + 666*x^9 + 1614*x^10 + 3955*x^11 + 9733*x^12 + ...

Links

Crossrefs

Cf. A358952, A355866.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = -polcoeff( sum(m=-#A,#A, (-x)^m * (1 - (-x)^m/Ser(A))^(m+2) ), #A-3));A[n+1]}
for(n=0,35,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 0 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) - (-x)^n)^(n+2) / A(x)^n.
(2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)) * A(x)^n / (1 - (-x)^n*A(x))^(n-2).
From Paul D. Hanna, Jan 19 2025: (Start)
(3) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (1 - (-x)^n)^n * A(x)^n.
(4) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)) / ((1 - (-x)^n)^n * A(x)^n).
(End)
a(n) ~ c * d^n / n^(3/2), where d = 2.71312501383... and c = 3.43853109... - Vaclav Kotesovec, Mar 31 2023

A385909 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * (x^(2*n) - A(x))^(3*n+1).

Original entry on oeis.org

1, 1, 3, 9, 31, 122, 493, 2086, 9106, 40764, 186206, 865068, 4076020, 19437711, 93655043, 455293416, 2230636436, 11003483165, 54607084364, 272453502850, 1365876088389, 6876896373019, 34757806185051, 176291771193079, 897001780346928, 4577362669389502, 23420275560794225, 120123996076924029
Offset: 0

Views

Author

Paul D. Hanna, Jul 29 2025

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(2*n+1))^n, which holds formally for all y.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 31*x^4 + 122*x^5 + 493*x^6 + 2086*x^7 + 9106*x^8 + 40764*x^9 + 186206*x^10 + 865068*x^11 + 4076020*x^12 + ...

Links

Crossrefs

Cf. A355866.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A, #A, x^n*(x^(2*n) - Ser(A))^(3*n+1) ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 0 = Sum_{n=-oo..+oo} x^n * (x^n - A(x))^(3*n+1).
(2) 0 = Sum_{n=-oo..+oo} x^(3*n*(2*n-1)) / (1 - A(x)*x^(2*n))^(3*n-1).
a(n) ~ c * d^n / n^(3/2), where d = 5.437310827623... and c = 0.230314472... - Vaclav Kotesovec, Aug 04 2025
Showing 1-6 of 6 results.

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