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.

A357227 a(n) = coefficient of x^n, n >= 0, in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(n-1).

Original entry on oeis.org

1, 1, 5, 27, 156, 961, 6145, 40546, 273784, 1883468, 13153544, 93012247, 664640794, 4791939802, 34816034143, 254659426691, 1873698891024, 13858201221637, 102975937795619, 768385165594607, 5755185884844403, 43253819566052165, 326093530416255178, 2465456045342545908
Offset: 0

Views

Author

Paul D. Hanna, Oct 17 2022

Keywords

Comments

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

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 27*x^3 + 156*x^4 + 961*x^5 + 6145*x^6 + 40546*x^7 + 273784*x^8 + 1883468*x^9 + 13153544*x^10 + 93012247*x^11 + 664640794*x^12 + ...
where
1 = ... + x^(-3)/(2*A(x) - x^(-3))^4 + x^(-2)/(2*A(x) - x^(-2))^3 + x^(-1)/(2*A(x) - x^(-1))^2 + 1/(2*A(x) - 1) + x + x^2*(2*A(x) - x^2) + x^3*(2*A(x) - x^3)^2 + x^4*(2*A(x) - x^4)^3 + ... + x^n*(2*A(x) - x^n)^(n-1) + ...
SPECIFIC VALUES.
A(1/9) = 1.30108724398914093656591796643458817060949...
A(1/10) = 1.22176622612326449515553495048940456186175...

Links

Crossrefs

Cf. A358961, A358962, A358963, A358964, A358965, A358937.
Cf. A355868, A357232, A355865.
Cf. A363312, A363313, A363314, A363315.

Programs

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

Formula

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

A363312 Expansion of g.f. A(x) satisfying 1/2 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1), with a(0) = 3.

Original entry on oeis.org

3, 8, 68, 656, 6924, 77816, 912504, 11043616, 136909712, 1729812880, 22193496988, 288368706416, 3786876943856, 50180784019384, 670150485880336, 9010466250798080, 121871951481594296, 1657086342551799752, 22637216782139196588, 310547100988853539728
Offset: 0

Views

Author

Paul D. Hanna, May 28 2023

Keywords

Comments

a(n) == 0 (mod 2^2) for n > 0.

Examples

			G.f.: A(x) = 3 + 8*x + 68*x^2 + 656*x^3 + 6924*x^4 + 77816*x^5 + 912504*x^6 + 11043616*x^7 + 136909712*x^8 + 1729812880*x^9 + ...

Links

Crossrefs

Cf. A357227, A363141, A363313, A363314, A363315.

Programs

  • PARI
    {a(n) = my(A=[3]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(-2  + 2^2*sum(m=-#A, #A, x^m * (Ser(A) - x^m)^(m-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 may be defined by the following.
(1) 1/2 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1).
(2) 1/2 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / (1 - x^n*A(x))^(n+1).
(3) A(x)/2 = Sum_{n=-oo..+oo} x^(2*n) * (A(x) - x^n)^(n-1).
(4) A(x)/2 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - x^n*A(x))^(n+1).

A363314 Expansion of g.f. A(x) satisfying 1/4 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1), with a(0) = 5.

Original entry on oeis.org

5, 32, 496, 9024, 181296, 3882848, 86887712, 2007577472, 47530180736, 1147071160768, 28114384217104, 697913487791552, 17511114852998912, 443374443981736160, 11314170816869911232, 290688529521060711424, 7513202655833624201472, 195216134898681278515232
Offset: 0

Views

Author

Paul D. Hanna, May 28 2023

Keywords

Comments

a(n) == 0 (mod 4^2) for n > 0.

Examples

			G.f.: A(x) =  5 + 32*x + 496*x^2 + 9024*x^3 + 181296*x^4 + 3882848*x^5 + 86887712*x^6 + 2007577472*x^7 + 47530180736*x^8 + ...

Links

Crossrefs

Cf. A357227, A363141, A363312, A363313, A363315.

Programs

  • PARI
    {a(n) = my(A=[5]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(-4  + 4^2*sum(m=-#A, #A, x^m * (Ser(A) - x^m)^(m-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 may be defined by the following.
(1) 1/4 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1).
(2) 1/4 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / (1 - x^n*A(x))^(n+1).
(3) A(x)/4 = Sum_{n=-oo..+oo} x^(2*n) * (A(x) - x^n)^(n-1).
(4) A(x)/4 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - x^n*A(x))^(n+1).

A363315 Expansion of g.f. A(x) satisfying 1/5 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1), with a(0) = 6.

Original entry on oeis.org

6, 50, 950, 21350, 530700, 14067650, 389701050, 11147799700, 326779719500, 9764739197800, 296342706620800, 9108989853295550, 283002934668287000, 8872796279035164100, 280368062326854982450, 8919740526808334086550, 285476263708658548421000, 9185078302539674382641450
Offset: 0

Views

Author

Paul D. Hanna, May 28 2023

Keywords

Comments

a(n) == 0 (mod 5^2) for n > 0.

Examples

			G.f.: A(x) = 6 + 50*x + 950*x^2 + 21350*x^3 + 530700*x^4 + 14067650*x^5 + 389701050*x^6 + 11147799700*x^7 + 326779719500*x^8 + ...

Links

Crossrefs

Cf. A357227, A363141, A363312, A363313, A363314.

Programs

  • PARI
    {a(n) = my(A=[6]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(-5  + 5^2*sum(m=-#A, #A, x^m * (Ser(A) - x^m)^(m-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 may be defined by the following.
(1) 1/5 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1).
(2) 1/5 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / (1 - x^n*A(x))^(n+1).
(3) A(x)/5 = Sum_{n=-oo..+oo} x^(2*n) * (A(x) - x^n)^(n-1).
(4) A(x)/5 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - x^n*A(x))^(n+1).

A363141 Expansion of g.f. A(x) satisfying 1/x = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1), with a(0) = 1, a(1) = 1.

Original entry on oeis.org

1, 1, 0, 2, 3, 11, 23, 76, 188, 575, 1587, 4732, 13714, 40993, 121787, 367100, 1107371, 3367412, 10267404, 31468401, 96734992, 298488537, 923587457, 2866241029, 8916951360, 27808418089, 86910042122, 272180834822, 854004007736, 2684311988984, 8451232727631
Offset: 0

Views

Author

Paul D. Hanna, Jun 09 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^3 + 3*x^4 + 11*x^5 + 23*x^6 + 76*x^7 + 188*x^8 + 575*x^9 + 1587*x^10 + 4732*x^11 + 13714*x^12 + ...

Links

Crossrefs

Cf. A357227, A363312, A363313, A363314, A363315.

Programs

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

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) 1/x = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1).
(2) 1/x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / (1 - x^n*A(x))^(n+1).
(3) A(x)/x = Sum_{n=-oo..+oo} x^(2*n) * (A(x) - x^n)^(n-1).
(4) A(x)/x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - x^n*A(x))^(n+1).
a(n) ~ c * d^n / n^(3/2), where d = 3.3064656105288278... and c = 0.3845291573508... - Vaclav Kotesovec, Jun 09 2023
Showing 1-5 of 5 results.

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