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).

A358961 a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(2*n+1))^(n-1).

Original entry on oeis.org

1, 3, 7, 33, 163, 858, 4708, 26662, 154699, 914885, 5494719, 33423598, 205493244, 1274928510, 7972042450, 50188844583, 317861388939, 2023777490895, 12945901676736, 83163975425669, 536279878717858, 3470134399230086, 22525040920670333, 146633283078321531
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2022

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 + 3*x + 7*x^2 + 33*x^3 + 163*x^4 + 858*x^5 + 4708*x^6 + 26662*x^7 + 154699*x^8 + 914885*x^9 + 5494719*x^10 + ...
where A = A(x) satisfies the doubly infinite sum
1 = ... + x^(-2)*(A - x^(-3))^(-3) + x^(-1)*(A - x^(-1))^(-2) + (A - x)^(-1) + x*(A - x^3)^0 + x^2*(A - x^5) + x^3*(A - x^7)^2 + x^4*(A - x^9)^3 + ... + x^n * (A - x^(2*n+1))^(n-1) + ...
also,
A(x) = ... + x^24/(1 - x^(-7)*A)^(-2) - x^12/(1 - x^(-5)*A)^(-1) + x^4 - 1/(1 - x^(-1)*A) + 1/(1 - x*A)^2 - x^4/(1 - x^3*A)^3 + x^12/(1 - x^5*A)^4 - x^24/(1 - x^7*A)^5 + ... + (-1)^(n+1)*x^(2*n*(n-1))/(1 - x^(2*n-1)*A(x))^(n+1) + ...

Links

Crossrefs

Cf. A357227, A358962, A358963, A358964, A358965, A358937.

Programs

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

A358962 a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(3*n+2))^(n-1).

Original entry on oeis.org

1, 2, 8, 30, 146, 748, 4002, 22114, 125220, 722850, 4238148, 25169064, 151084168, 915235106, 5587985801, 34351213384, 212436911849, 1320744403708, 8250065775120, 51752790871466, 325887027304769, 2059216160242430, 13052805881695018, 82976612756731258
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2022

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 + 2*x + 8*x^2 + 30*x^3 + 146*x^4 + 748*x^5 + 4002*x^6 + 22114*x^7 + 125220*x^8 + 722850*x^9 + 4238148*x^10 + ...
where A = A(x) satisfies the doubly infinite sum
1 = ... + x^(-2)*(A - x^(-4))^(-3) + x^(-1)*(A - x^(-1))^(-2) + (A - x^2)^(-1) + x*(A - x^5)^0 + x^2*(A - x^8) + x^3*(A - x^11)^2 + x^4*(A - x^14)^3 + ... + x^n * (A - x^(3*n+2))^(n-1) + ...
also,
A(x) = ... + x^36/(1 - x^(-11)*A)^(-2) - x^18/(1 - x^(-8)*A)^(-1) + x^6 - 1/(1 - x^(-2)*A) + 1/(1 - x*A)^2 - x^6/(1 - x^4*A)^3 + x^18/(1 - x^7*A)^4 - x^36/(1 - x^10*A)^5 + ... + (-1)^(n+1)*x^(3*n*(n-1))/(1 - x^(3*n-2)*A)^(n+1) + ...

Links

Crossrefs

Cf. A358961, A358963, A358964, A358965.

Programs

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

A358963 a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(4*n+3))^(n-1).

Original entry on oeis.org

1, 2, 7, 31, 143, 731, 3896, 21444, 120967, 695699, 4063879, 24045306, 143808836, 867972228, 5280039896, 32339575813, 199266229047, 1234340158837, 7682216027973, 48014943810066, 301247658649431, 1896587278353158, 11978138505184044, 75867527248248561
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(4*n+3))^n, which holds formally for all y.

Examples

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 31*x^3 + 143*x^4 + 731*x^5 + 3896*x^6 + 21444*x^7 + 120967*x^8 + 695699*x^9 + 4063879*x^10 + ...
where A = A(x) satisfies the doubly infinite sum
1 = ... + x^(-2)*(A - x^(-5))^(-3) + x^(-1)*(A - x^(-1))^(-2) + (A - x^3)^(-1) + x*(A - x^7)^0 + x^2*(A - x^11) + x^3*(A - x^15)^2 + x^4*(A - x^19)^3 + ... + x^n * (A - x^(4*n+3))^(n-1) + ...
also,
A(x) = ... + x^48/(1 - x^(-15)*A)^(-2) - x^24/(1 - x^(-11)*A)^(-1) + x^8 - 1/(1 - x^(-3)*A) + 1/(1 - x*A)^2 - x^8/(1 - x^5*A)^3 + x^24/(1 - x^9*A)^4 - x^48/(1 - x^13*A)^5 + ... + (-1)^(n+1)*x^(4*n*(n-1))/(1 - x^(4*n-3)*A)^(n+1) + ...

Links

Crossrefs

Cf. A358961. A358962, A358964, A358965.

Programs

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

A358964 a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(5*n+4))^(n-1).

Original entry on oeis.org

1, 2, 7, 30, 144, 728, 3879, 21338, 120301, 691482, 4037020, 23873308, 142702222, 860823760, 5233702949, 32038319854, 197302553658, 1221511228130, 7598234842024, 47464203317986, 297630203452010, 1872792573164662, 11821420702394153, 74834134991237178
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(5*n+4))^n, which holds formally for all y.

Examples

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 30*x^3 + 144*x^4 + 728*x^5 + 3879*x^6 + 21338*x^7 + 120301*x^8 + 691482*x^9 + 4037020*x^10 + ...
where A = A(x) satisfies the doubly infinite sum
1 = ... + x^(-2)*(A - x^(-6))^(-3) + x^(-1)*(A - x^(-1))^(-2) + (A - x^4)^(-1) + x*(A - x^9)^0 + x^2*(A - x^14) + x^3*(A - x^19)^2 + x^4*(A - x^24)^3 + ... + x^n * (A - x^(5*n+4))^(n-1) + ...
also,
A(x) = ... + x^60/(1 - x^(-19)*A)^(-2) - x^30/(1 - x^(-14)*A)^(-1) + x^10 - 1/(1 - x^(-4)*A) + 1/(1 - x*A)^2 - x^10/(1 - x^6*A)^3 + x^30/(1 - x^11*A)^4 - x^60/(1 - x^16*A)^5 + ... + (-1)^(n+1)*x^(5*n*(n-1))/(1 - x^(5*n-4)*A)^(n+1) + ...

Links

Crossrefs

Cf. A358961, A358962, A358963, A358965.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( sum(n=-#A,#A, x^n * (Ser(A) - x^(5*n+4))^(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:
(1) 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(5*n+4))^(n-1).
(2) x^4 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(5*n^2) / (1 - x^(5*n-4)*A(x))^(n+1).
(3) A(x) = Sum_{n=-oo..+oo} x^(6*n+4)* (A(x) - x^(5*n+4))^(n-1).
(4) A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(5*n*(n-1)) / (1 - x^(5*n-4)*A(x))^(n+1).
(5) 0 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(5*n*(n-1)) / (1 - x^(5*n-4)*A(x))^n.
Showing 1-5 of 5 results.

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