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 1-2 of 2 results.

A355868 G.f. A(x) satisfies: 1 = Sum_{n=-oo..+oo} (x^n - 2*x*A(x))^n.

Original entry on oeis.org

1, 2, 3, 3, 5, 39, 206, 697, 1656, 3208, 8727, 41667, 192142, 688944, 1965643, 5117374, 15888133, 63924038, 263759291, 955198539, 3017571957, 9101208987, 30075674452, 113177783141, 437460265979, 1583161667787, 5299622270275, 17294182815347, 59169678008804
Offset: 0

Views

Author

Paul D. Hanna, Aug 09 2022

Keywords

Comments

Related identity: Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + y)^n = 0 for all y.
Related identity: Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + y*x^n)^n = 0 for all y.

Examples

			G.f.: A(x) = 1 + 2*x + 3*x^2 + 3*x^3 + 5*x^4 + 39*x^5 + 206*x^6 + 697*x^7 + 1656*x^8 + 3208*x^9 + 8727*x^10 + 41667*x^11 + 192142*x^12 + ...
where
1 = ... + (x^(-3) - 2*x*A(x))^(-3) + (x^(-2) - 2*x*A(x))^(-2) + (x^(-1) - 2*x*A(x))^(-1) + 1 + (x - 2*x*A(x)) + (x^2 - 2*x*A(x))^2 + (x^3 - 2*x*A(x))^3 + ... + (x^n - 2*x*A(x))^n + ...
and
1 = ... + x^(-5)/(x^(-3) + 2*A(x))^3 + x^(-3)/(x^(-2) + 2*A(x))^2 + x^(-1)/(x^(-1) + 2*A(x)) + x + x^3*(x + 2*A(x)) + x^5*(x^2 + 2*A(x))^2 + x^7*(x^3 + 2*A(x))^3 + ... + x^(2*n+1)*(x^n + 2*A(x))^n + ...
also,
1 = ... + x^9*(1 - 2*A(x)/x^2)^3 + x^4*(1 - 2*A(x)/x)^2 + x*(1 - 2*A(x)) + 1 + x/(1 - 2*A(x)*x^2) + x^4/(1 - 2*A(x)*x^3)^2 + x^9/(1 - 2*A(x)*x^4)^3 + ... + x^(n^2)/(1 - 2*A(x)*x^(n+1))^n + ...
further,
1 = ... + x^9*(1 + 2*A(x)/x^2)^2 + x^4*(1 + 2*A(x)/x) + x + 1/(1 + 2*A(x)*x) + x/(1 + 2*A(x)*x^2)^2 + x^4/(1 - 2*A(x)*x^3)^3 + x^9/(1 - 2*A(x)*x^4)^4 + ... + x^(n^2)/(1 + 2*A(x)*x^(n+1))^(n+1) + ...
SPECIFIC VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.04720243920412572796492634515550526365563452970121157309...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 2*A)^n = 1.
(V.2) Let A = A(exp(-2*Pi)) = 0.001874436990256710694689538031391789940066981740061145959...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 2*A)^n = 1.
(V.3) Let A = A(-exp(-Pi)) = -0.03971121915244100584186154683625533541823516978831008865...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 2*A)^n = 1.
(V.4) Let A = A(-exp(-2*Pi)) = -0.001860487547859226152163099117755736250804492732905479139...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 2*A)^n = 1.

Links

Crossrefs

Cf. A355867, A359671, A359673.
Cf. A370041, A370030, A370031, A370033, A370034, A370035, A370036, A370037, A370038, A370039, A370043.

Programs

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

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n=-oo..+oo} (x^n - 2*x*A(x))^n.
(2) 1 = Sum_{n=-oo..+oo} x^(2*n+1) * (x^n + 2*A(x))^n.
(3) 0 = Sum_{n=-oo..+oo} (-1)^n * (x^n - 2*x*A(x))^(n-1).
(4) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (x^n + 2*x*A(x))^(n+1).
(5) 1 = Sum_{n=-oo..+oo} x^(n^2) / (1 - 2*A(x)*x^(n+1))^n.
(6) 1 = Sum_{n=-oo..+oo} x^(n^2) / (1 + 2*A(x)*x^(n+1))^(n+1).
(7) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 2*A(x)*x^n)^n.
a(n) ~ c * d^n / n^(3/2), where d = 3.70839... and c = 1.176... - Vaclav Kotesovec, Feb 18 2024

A378829 G.f. A(x) satisfies 1 = Sum_{n=-oo..+oo} (A(x)^n - 2*x)^n.

Original entry on oeis.org

1, -2, 5, -13, 30, -74, 202, -616, 2126, -7828, 29366, -110398, 414214, -1556848, 5892713, -22524354, 86954484, -338421674, 1324660464, -5204326208, 20498580511, -80907096678, 320002290542, -1268500509496, 5040195484362, -20073242195580, 80120884387322, -320442284717582, 1283939790460139
Offset: 1

Views

Author

Paul D. Hanna, Dec 13 2024

Keywords

Comments

A signed version of A359673.

Examples

			G.f.: A(x) = x - 2*x^2 + 5*x^3 - 13*x^4 + 30*x^5 - 74*x^6 + 202*x^7 - 616*x^8 + 2126*x^9 - 7828*x^10 + 29366*x^11 - 110398*x^12 + ...
where 1 = Sum_{n=-oo..+oo} (A(x)^n - 2*x)^n.
SPECIFIC VALUES.
A(t) = 1/6 at t = 0.24134833288352420167420358490093379236139061653959...
  where 1 = Sum_{n=-oo..+oo} (1/6^n - 2*t)^n.
A(t) = 1/7 at t = 0.19473287649699543474178954182484954936895675300220...
  where 1 = Sum_{n=-oo..+oo} (1/7^n - 2*t)^n.
A(t) = 1/8 at t = 0.16330047299490635761734791354706359079698287572429...
  where 1 = Sum_{n=-oo..+oo} (1/8^n - 2*t)^n.
A(t) = exp(-Pi) at t = 0.04720243920412572796492634515550526365563452970121157309...
  where 1 = Sum_{n=-oo..+oo} (exp(-n*Pi) - 2*t)^n,
  also, 1 = Sum_{n=-oo..+oo} exp(-n^2*Pi) / (1 - 2*t*exp(-n*Pi))^n;
  compare to Sum_{n=-oo..+oo} exp(-n^2*Pi) = Pi^(1/4)/gamma(3/4).
A(t) = exp(-2*Pi) at t = 0.001874436990256710694689538031391789940066981740061145959...
  where 1 = Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 2*t)^n,
  also, 1 = Sum_{n=-oo..+oo} exp(-2*n^2*Pi) / (1 - 2*t*exp(-2*n*Pi))^n;
  compare to Sum_{n=-oo..+oo} exp(-2*n^2*Pi) = Pi^(1/4)/gamma(3/4) * sqrt(2+sqrt(2))/2.
A(1/5) = 0.14570268760195709902234365534810153966906514204980...
  where 1 = Sum_{n=-oo..+oo} (A(1/5)^n - 2/5)^n.
A(1/6) = 0.12698642862956730423090954809810167590805619510041...
  where 1 = Sum_{n=-oo..+oo} (A(1/6)^n - 1/3)^n.
A(1/7) = 0.11253270334433369822784652362071431711460474251926...
A(1/8) = 0.10104551587569245791494155789285565556961920656039...
  where 1 = Sum_{n=-oo..+oo} (A(1/8)^n - 1/4)^n.

Links

Crossrefs

Cf. A355868, A359673.

Programs

  • PARI
    {a(n) = my(V=[0,1],A); for(i=1, n, V=concat(V, 0); A=Ser(V);
V[#V] = polcoef( -sum(m=-#V, #V, (A^m - 2*x)^m ), #V-1)/2); V[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) 1 = Sum_{n=-oo..+oo} (A(x)^n - 2*x)^n.
(2) 1 = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 2*x)^(n-1).
(3) 0 = Sum_{n=-oo..+oo} (-1)^n * (A(x)^n - 2*x)^(n-1).
(4) 0 = Sum_{n=-oo..+oo} (-1)^n * A(x)^(2*n) * (A(x)^n + 2*x)^(n+1).
(5) 1 = Sum_{n=-oo..+oo} A(x)^(n^2) / (1 - 2*x*A(x)^n)^n.
(6) 1 = Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + 2*x*A(x)^n)^(n+1).
(7) 0 = Sum_{n=-oo..+oo} (-1)^n * A(x)^(n*(n+1)) / (1 + 2*x*A(x)^n)^(n+1).
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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