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 1-4 of 4 results.

A378582 G.f. Sum_{n=-oo..+oo} (x^n - x)^(n+1).

Original entry on oeis.org

2, -1, 1, -1, 4, -3, -1, -1, 13, -1, -7, -10, 10, -1, 22, -1, 10, -23, -25, -1, 43, -1, 50, -36, 14, -1, -19, -26, 16, -55, 78, -1, 201, -1, -129, -78, 20, -108, 211, -1, 22, -105, 9, -1, 349, -1, 274, -430, 26, -1, -421, -50, 568, -171, 441, -1, 769, -661, -238, -210, 32, -1, 1291, -1, 34, -591, -897, -1288, 1765, -1
Offset: 0

Views

Author

Paul D. Hanna, Jan 13 2025

Keywords

Comments

Compare g.f. to: Sum_{n=-oo..+oo} (x - x^(n+1))^n = 0.

Examples

			G.f.: A(x) = 2 - x + x^2 - x^3 + 4*x^4 - 3*x^5 - x^6 - x^7 + 13*x^8 - x^9 - 7*x^10 - 10*x^11 + 10*x^12 - x^13 + 22*x^14 - x^15 + 10*x^16 - 23*x^17 - 25*x^18 - x^19 + 43*x^20 - x^21 + 50*x^22 - 36*x^23 + 14*x^24 - x^25 + ...
RELATED SERIES.
F(x) = x + 4*x^2 + 16*x^3 + 255*x^4 + 4344*x^5 + 49104*x^6 + 543744*x^7 + 8203012*x^8 + 130252849*x^9 + 1857148424*x^10 + 26419178032*x^11 + 406394717168*x^12 + ... + A379764(n)*x^n + ...
where A(4*F(x)) = 2 - 4*x + 4*x^4 - 4*x^9 + 4*x^16 - 4*x^25 + 4*x^36 - 4*x^49 + 4*x^64 - 4*x^81 + 4*x^100 + ...
which equals 2*theta_4(x).
SPECIFIC VALUES.
A local minimum of A(x) is at x = z, A'(z) = 0,
  where z = 0.397529435491742842870725714009076671931564550115616181...
  and A(z) = 1.76925395689645126935316774753841505734121715127456863...
A(t) = 5 at t = 0.83799627848215104988844491211534885329390252098950...
  where 5 = Sum_{n=-oo..+oo} (t^n - t)^(n+1).
A(t) = 4 at t = 0.81576458148282505480367740238923856698108048584006...
A(t) = 3 at t = 0.77374138107025616474622853242292423166707448715221...
A(t) = 2 at t = 0.63388217567664405459819983625203116490495970722052...
  where 2 = Sum_{n=-oo..+oo} (t^n - t)^(n+1).
A(5/6) = 4.7450460128227344201120209017241324429858113440230...
A(4/5) = 3.5316007525899668447063164855200492187769453202119...
A(3/4) = 2.6807156006575496013515132361048335609447924890231...
A(2/3) = 2.1104745199027122018154499017357730364485251172261...
A(3/5) = 1.9196428858805646516374432938412152594386473418308...
A(1/2) = 1.7984818587175528236453909117209424070005516950258...
  where A(1/2) = Sum_{n=-oo..+oo} (1/2^n - 1/2)^(n+1).
A(2/5) = 1.7692678774574642220664568097843035516621774732422...
A(1/3) = 1.7777271141981820071010562139080023372566201778866...
  where A(1/3) = Sum_{n=-oo..+oo} (1/3^n - 1/3)^(n+1).
A(1/4) = 1.8094512892069907186358957280354026387421254606042...
A(1/5) = 1.8373950901417623688604868970319735279603621700924...
A(1/6) = 1.8591645951522733561268876665948564129719912163274...

Links

Crossrefs

Cf. A379764, A290003.

Programs

  • PARI
    {a(n) = my(A = sum(m=-n-1,n+1, (x^m - x +x*O(x^n))^(m+1))); polcoef(A,n)}
for(n=0,70, print1(a(n),", "))

Formula

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

A379195 G.f. A(x) satisfies x = Sum_{n=-oo..+oo} (A(x) - A(x)^n)^(n+1).

Original entry on oeis.org

1, 1, 1, 2, 5, 10, 21, 56, 148, 359, 906, 2450, 6571, 17338, 46777, 128681, 352859, 967315, 2679764, 7474260, 20860226, 58375826, 164197258, 463322792, 1309547562, 3710517258, 10543567357, 30021808808, 85628123727, 244694423127, 700553813377, 2008780153580, 5768264675938, 16587793685429, 47766704865133
Offset: 1

Views

Author

Paul D. Hanna, Jan 14 2025

Keywords

Comments

Compare to the identity 0 = Sum_{n=-oo..+oo} (x - x^(n+1))^n.

Examples

			G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 5*x^5 + 10*x^6 + 21*x^7 + 56*x^8 + 148*x^9 + 359*x^10 + 906*x^11 + 2450*x^12 + 6571*x^13 + 17338*x^14 + ...
where x = Sum_{n=-oo..+oo} (A(x) - A(x)^n)^(n+1).
RELATED SERIES.
F(x) = Sum_{n=-oo..+oo} (x - x^n)^(n+1) = x - x^2 + x^3 - 2*x^4 + 3*x^5 - 3*x^6 + x^7 + x^8 + x^9 - 7*x^10 + 10*x^11 - 6*x^12 + x^13 + x^15 - 8*x^16 + 23*x^17 - 25*x^18 + x^19 + 17*x^20 + x^21 - 32*x^22 + 36*x^23 - 12*x^24 + x^25 + ... + A290003(n)*x^n + ...
where F(A(x)) = x.
SPECIFIC VALUES.
A(t) = 1/2 at t = 0.30725396830704316799197832656390411971168116373389...
  where t = Sum_{n=-oo..+oo} (1/2 - 1/2^n)^(n+1),
  also, t = Sum_{n=-oo..+oo} (2^(n-1) - 1)^(n+1) / 2^(n*(n+1)).
A(t) = 1/3 at t = 0.24338606674563424484910361835257533242309621632065...
  where t = Sum_{n=-oo..+oo} (1/3 - 1/3^n)^(n+1),
  also, t = Sum_{n=-oo..+oo} (3^(n-1) - 1)^(n+1) / 3^(n*(n+1)).
A(t) = 1/4 at t = 0.19758524006807690544490179709803177425355852401229...
  where t = Sum_{n=-oo..+oo} (1/4 - 1/4^n)^(n+1).
A(t) = 1/5 at t = 0.16558333624735433324843855679493132539350188690309...
  where t = Sum_{n=-oo..+oo} (1/5 - 1/5^n)^(n+1).
A(1/4) = 0.34697020435026836163926019675791627488695303305268...
  where 1/4 = Sum_{n=-oo..+oo} (A(1/4) - A(1/4)^n)^(n+1).
A(1/5) = 0.25400492231901630962271637839330240648984255624021...
A(1/6) = 0.20160813481244983396982286666489080077373441727643...
A(1/8) = 0.14327208862930858756346363646363969972815166338945...

Links

Crossrefs

Cf. A290003.

Programs

  • PARI
    N=40 \\ number of terms
{a(n) = my(R = sum(m=-N-1, N+1, (x - x^m +x^2*O(x^N))^(m+1) ), A=x);
A = serreverse(R); polcoef(A, n)}
for(n=1, N, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) x = Sum_{n=-oo..+oo} (A(x) - A(x)^n)^(n+1).
(2) x = Sum_{n=-oo..+oo} (-1)^n * A(x)^(n*(n-1)) / (1 - A(x)^(n+1))^(n-1).
From Paul D. Hanna, Jan 25 2025: (Start)
(3) x/A(x) = Sum_{n=-oo..+oo} (A(x) - A(x)^n)^n.
(4) x/A(x) = Sum_{n=-oo..+oo, n<>-1} (-1)^n * A(x)^(n^2) / (1 - A(x)^(n+1))^n.
(End)
a(n) ~ c * d^n / n^(3/2), where d = 3.00914051453408723176675508018... and c = 0.174541635630216521276160108... - Vaclav Kotesovec, Jan 22 2025

A294677 G.f.: 1 + Sum_{n=-oo..+oo, n<>0} (x - x^n)^n / (1 - (x - x^n)^n).

Original entry on oeis.org

1, -1, 2, -3, 7, -11, 17, -21, 35, -67, 125, -179, 246, -384, 715, -1199, 1871, -2850, 4593, -7589, 12811, -20366, 31545, -50483, 84597, -138964, 222534, -352910, 569680, -931694, 1523165, -2451150, 3924137, -6331780, 10329289, -16804843, 27109912, -43594466, 70485938, -114450985, 185713588, -300089184, 484106880, -782672321, 1269075821, -2056723036, 3325362211, -5371243069, 8688055226
Offset: 0

Views

Author

Paul D. Hanna, Nov 07 2017

Keywords

Comments

Compare g.f. to: Sum_{n=-oo..+oo} (x - x^(n+1))^n = 0.
Compare g.f. to: Sum_{n=-oo..+oo, n<>0} x^n/(1 - x^n) = 0, ignoring constant terms.
Limit a(n+1)/a(n) = -(sqrt(5) + 1)/2.

Examples

			G.f.: A(x) = 1 - x + 2*x^2 - 3*x^3 + 7*x^4 - 11*x^5 + 17*x^6 - 21*x^7 + 35*x^8 - 67*x^9 + 125*x^10 - 179*x^11 + 246*x^12 - 384*x^13 + 715*x^14 - 1199*x^15 + 1871*x^16 - 2850*x^17 + 4593*x^18 - 7589*x^19 + 12811*x^20 - 20366*x^21 + 31545*x^22 - 50483*x^23 + 84597*x^24 - 138964*x^25 +...

Links

Crossrefs

Cf. A290003.

Programs

  • PARI
    {a(n) = my(A); A = sum(m=-n-1,n+1, if(m==0,1, (x-x^m)^m/(1 - (x-x^m +x*O(x^n))^m ))); polcoeff(A,n)}
for(n=0,40,print1(a(n),", "))

Formula

a(n) ~ (-1)^n * (1 + sqrt(5))^(n+1) / 2^(n+2). - Vaclav Kotesovec, Nov 08 2017

A359463 Coefficient a(n) of x^n in power series A(x), n >= 0, such that A(x) = Sum_{n=-oo..+oo} (-x*A(x))^n * (1 - (-x*A(x))^(n-1))^n.

Original entry on oeis.org

1, 1, 2, 6, 20, 69, 245, 896, 3362, 12869, 50024, 196896, 783205, 3143713, 12717532, 51798089, 212233756, 874193355, 3617797596, 15035379576, 62724649455, 262579756558, 1102680011825, 4643936681122, 19609621413193, 83005706694022, 352145760387515, 1497067760933244
Offset: 0

Views

Author

Paul D. Hanna, Jan 17 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 20*x^4 + 69*x^5 + 245*x^6 + 896*x^7 + 3362*x^8 + 12869*x^9 + 50024*x^10 + 196896*x^11 + 783205*x^12 + ...
SPECIFIC VALUES.
A(x) = 2 at x = 0.22210374835192555734961892166866769267669905135315...
A(1/5) = 1.45174689360673617561694352881716190508117725206270...
A(1/6) = 1.28852385900727494844427701605174847197781970881818...

Links

Crossrefs

Cf. A290003.

Programs

  • PARI
    {a(n) = my(A=1);
A = (-1/x)*serreverse(-x/sum(m=-n-1,n+1, (x - x^m +x*O(x^(n+1)))^m )); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n) = my(A=1); for(i=1,n,
A = sum(m=-n,n, (-x*A)^m * (1 - (-x*A)^(m-1) +x*O(x^n))^m)); polcoeff(A,n)}
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) A(x) = Sum_{n=-oo..+oo} (-x*A(x))^n * (1 - (-x*A(x))^(n-1))^n.
(2) A(x) = Sum_{n=-oo..+oo} (-x*A(x))^n * (1 - (-x*A(x))^(n-1))^(n+1).
(3) A(x) = Sum_{n=-oo..+oo} (x*A(x))^(2*n+1) * (1 - (-x*A(x))^n)^n.
(4) A(x) = Sum_{n=-oo..+oo} (x*A(x))^(n^2) / (1 - (-x*A(x))^(n+1))^(n+1).
(5) x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * (x*A(x))^(n*(n-1)) / (1 - (-x*A(x))^(n+1))^(n-1).
(6) -1/x = Sum_{n=-oo..+oo} (-x*A(x))^n * (1 - (-x*A(x))^n)^(n+1).
(7) -1/x = Sum_{n=-oo..+oo} (-x*A(x))^n * (1 - (-x*A(x))^n)^(n+2).
(8) 1/x = Sum_{n=-oo..+oo} (x*A(x))^(2*n) * (1 - (-x*A(x))^n)^n.
(9) 0 = Sum_{n=-oo..+oo} (-x*A(x))^n * (1 - (-x*A(x))^n)^n.
(10) 0 = Sum_{n=-oo..+oo} (x*A(x))^(2*n) * (1 - (-x*A(x))^n)^(n+1).
(11) A(-x/G(x)) = G(x) where G(x) = Sum_{n=-oo..+oo} (x - x^n)^n is the g.f. of A290003.
(12) A(x) = (-1/x) * Series_Reversion( -x / Sum_{n=-oo..+oo} (x - x^n)^n ).
a(n) ~ c * d^n / n^(3/2), where d = 4.4911010651615255101195452998052055698... and c = 0.53507007927413038001531299966030791... - Vaclav Kotesovec, Mar 14 2023
Showing 1-4 of 4 results.

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