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.

A304639 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1/(1-x)^n - A(x) )^n.

Original entry on oeis.org

1, 1, 2, 11, 117, 1735, 31853, 689043, 17079221, 476238926, 14742680162, 501584454703, 18605089712174, 747393133162471, 32332767332220442, 1498961537925543920, 74153115616699819304, 3899494667155151052688, 217246028175467702590241, 12783023090792392539557926, 792236994094236725330142276, 51585659784100723438219893047, 3520987513029712770759434038820
Offset: 0

Views

Author

Paul D. Hanna, May 16 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 117*x^4 + 1735*x^5 + 31853*x^6 + 689043*x^7 + 17079221*x^8 + 476238926*x^9 + 14742680162*x^10 + 501584454703*x^11 + ...
is such that
1 = 1 + (1/(1-x) - A(x)) + (1/(1-x)^2 - A(x))^2  + (1/(1-x)^3 - A(x))^3 + (1/(1-x)^4 - A(x))^4 + (1/(1-x)^5 - A(x))^5 + (1/(1-x)^6 - A(x))^6 + (1/(1-x)^7 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  (1-x)/((1-x) + A(x))^2  +  (1-x)^2/((1-x)^2 + A(x))^3  +  (1-x)^3/((1-x)^3  +  A(x))^4 + (1-x)^4/((1-x)^4 + A(x))^5  +  (1-x)^5/((1-x)^5 + A(x))^6  +  (1-x)^6/((1-x)^6 + A(x))^7 + ...
PARTICULAR VALUES.
Although the power series A(x) diverges at x = -1, it may be evaluated formally.
Let t = A(-1) = 0.5452189736359494312349502450349441069576127988881794567242641...
then t satisfies
(1) 1 = Sum_{n>=0} ( 1/2^n - t )^n.
(2) 1 = Sum_{n>=0} ( 1 - 2^n*t )^n / 2^(n^2).
(3) 1 = Sum_{n>=0} 2^n / ( 2^n + t )^(n+1).

Links

Crossrefs

Cf. A326262, A326263, A326264, A326265.
Cf. A303056.

Programs

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

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/(1-x)^n - A(x) )^n.
(2) 1 = Sum_{n>=0} ( 1 - (1-x)^n*A(x) )^n / (1-x)^(n^2).
(3) 1 = Sum_{n>=0} (1-x)^n / ( (1-x)^n + A(x) )^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.16108865386542881383... and c = 0.16107844724485... - Vaclav Kotesovec, Oct 14 2020

A321604 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(4*n) - A(x))^n.

Original entry on oeis.org

1, 4, 22, 564, 25157, 1499576, 109904860, 9480509576, 937113401201, 104240673195936, 12878161899791760, 1749261564410844864, 259132759251207789056, 41585481940418457992816, 7188476201158569394613976, 1331880173688346226092103696, 263358773243148578509342224153, 55363099822436514905885084770968, 12330972024423209530808891225876436, 2900976547500300324930009436969260936
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2018

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 22*x^2 + 564*x^3 + 25157*x^4 + 1499576*x^5 + 109904860*x^6 + 9480509576*x^7 + 937113401201*x^8 + 104240673195936*x^9 + ...
such that
1 = 1  +  ((1+x)^4 - A(x))  +  ((1+x)^8 - A(x))^2  +  ((1+x)^12 - A(x))^3  +  ((1+x)^16 - A(x))^4  +  ((1+x)^20 - A(x))^5  +  ((1+x)^24 - A(x))^6  +  ((1+x)^28 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  (1+x)^4/(1 + (1+x)^4*A(x))^2  +  (1+x)^16/(1 + (1+x)^8*A(x))^3  +  (1+x)^36/(1 + (1+x)^12*A(x))^4  +  (1+x)^64/(1 + (1+x)^16*A(x))^5  +  (1+x)^100/(1 + (1+x)^20*A(x))^6  + ...
RELATED SERIES.
The logarithmic derivative of the g.f. begins
A'(x)/A(x) = 4 + 28*x + 1492*x^2 + 91788*x^3 + 6981484*x^4 + 621939700*x^5 + 63151305340*x^6 + 7181135905380*x^7 + 903210250234696*x^8 + ...
the coefficients of which are all divisible by 4:
(1/4) * A'(x)/A(x) = 1 + 7*x + 373*x^2 + 22947*x^3 + 1745371*x^4 + 155484925*x^5 + 15787826335*x^6 + 1795283976345*x^7 + 225802562558674*x^8 + ...

Links

Crossrefs

Cf. A303056, A321602, A321603, A321605.
Cf. A326264.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(4*m) - Ser(A))^m ) )[#A] );H=A; A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ((1+x)^(4*n) - A(x))^n.
(2) 1 = Sum_{n>=0} (1+x)^(4*n^2) / (1 + (1+x)^(4*n)*A(x))^(n+1).

A326262 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1/(1-x)^(2*n) - A(x) )^n.

Original entry on oeis.org

1, 2, 7, 80, 1742, 51842, 1902589, 82219592, 4071164749, 226803165574, 14029472009781, 953926536359084, 70723894649169937, 5679305945331227594, 491179287055641264989, 45527108214667404725616, 4503148842172835722939285, 473502491643614888369261116, 52748299277043902326373361722, 6206479798643382507763241117360, 769187266152748793100664986340382, 100156538984193022704291755068539370
Offset: 0

Views

Author

Paul D. Hanna, Jun 20 2019

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 80*x^3 + 1742*x^4 + 51842*x^5 + 1902589*x^6 + 82219592*x^7 + 4071164749*x^8 + 226803165574*x^9 + 14029472009781*x^10 + ...
such that
1 = 1  +  (1/(1-x)^2 - A(x))  +  (1/(1-x)^4 - A(x))^2  +  (1/(1-x)^6 - A(x))^3  +  (1/(1-x)^8 - A(x))^4  +  (1/(1-x)^10 - A(x))^5  +  (1/(1-x)^12 - A(x))^6  +  (1/(1-x)^14 - A(x))^7  + ...
Also,
1 = 1/(1 + A(x))  +  (1-x)^2/((1-x)^2 + A(x))^2  +  (1-x)^4/((1-x)^4 + A(x))^3  +  (1-x)^6/((1-x)^6  +  A(x))^4 + (1-x)^8/((1-x)^8 + A(x))^5  +  (1-x)^10/((1-x)^10 + A(x))^6  +  (1-x)^12/((1-x)^12 + A(x))^7 + ...

Crossrefs

Cf. A304639, A326263, A326264, A326265.
Cf. A321602.

Programs

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

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/(1-x)^(2*n) - A(x) )^n.
(2) 1 = Sum_{n>=0} ( 1 - (1-x)^(2*n)*A(x) )^n / (1-x)^(2*n^2).
(3) 1 = Sum_{n>=0} (1-x)^(2*n) / ( (1-x)^(2*n) + A(x) )^(n+1).

A326263 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1/(1-x)^(3*n) - A(x) )^n.

Original entry on oeis.org

1, 3, 15, 262, 8616, 384873, 21181421, 1372455324, 101895990777, 8511828635054, 789539638329648, 80506096148928303, 8951189588697000825, 1078020157296224938479, 139830500253903232730304, 19438947952499889395212003, 2883820412306778479104733811, 454810046719340404484233328331, 75993667094400965507408118716882, 13411571696501962452150617362998648, 2493074269436929464139674369969509811
Offset: 0

Views

Author

Paul D. Hanna, Jun 20 2019

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 15*x^2 + 262*x^3 + 8616*x^4 + 384873*x^5 + 21181421*x^6 + 1372455324*x^7 + 101895990777*x^8 + 8511828635054*x^9 + 789539638329648*x^10 + ...
such that
1 = 1  +  (1/(1-x)^3 - A(x))  +  (1/(1-x)^6 - A(x))^2  +  (1/(1-x)^9 - A(x))^3  +  (1/(1-x)^12 - A(x))^4  +  (1/(1-x)^15 - A(x))^5  +  (1/(1-x)^18 - A(x))^6  +  (1/(1-x)^21 - A(x))^7  + ...
Also,
1 = 1/(1 + A(x))  +  (1-x)^3/((1-x)^3 + A(x))^2  +  (1-x)^6/((1-x)^6 + A(x))^3  +  (1-x)^9/((1-x)^9  +  A(x))^4 + (1-x)^12/((1-x)^12 + A(x))^5  +  (1-x)^15/((1-x)^15 + A(x))^6  +  (1-x)^18/((1-x)^18 + A(x))^7 + ...

Crossrefs

Cf. A304639, A326262, A326264, A326265.
Cf. A321603.

Programs

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

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/(1-x)^(3*n) - A(x) )^n.
(2) 1 = Sum_{n>=0} ( 1 - (1-x)^(3*n)*A(x) )^n / (1-x)^(3*n^2).
(3) 1 = Sum_{n>=0} (1-x)^(3*n) / ( (1-x)^(3*n) + A(x) )^(n+1).

A326265 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1/(1-x)^(5*n) - A(x) )^n.

Original entry on oeis.org

1, 5, 40, 1185, 65270, 4861126, 445776670, 48124064710, 5952881626790, 828544320379330, 128058593506875627, 21758230559633783765, 4031357498037096661170, 809070343591564791211705, 174888309616496370413590235, 40517215307075701804767255261, 10017278630199891781122121185615, 2632883558256463087445119555912870, 733167697272377998186394054589647855, 215641985221691590110546294934099963285
Offset: 0

Views

Author

Paul D. Hanna, Jun 20 2019

Keywords

Examples

			G.f.: A(x) = 1 + 5*x + 40*x^2 + 1185*x^3 + 65270*x^4 + 4861126*x^5 + 445776670*x^6 + 48124064710*x^7 + 5952881626790*x^8 + 828544320379330*x^9 + 128058593506875627*x^10 + ...
such that
1 = 1  +  (1/(1-x)^5 - A(x))  +  (1/(1-x)^10 - A(x))^2  +  (1/(1-x)^15 - A(x))^3  +  (1/(1-x)^20 - A(x))^4  +  (1/(1-x)^25 - A(x))^5  +  (1/(1-x)^30 - A(x))^6  +  (1/(1-x)^35 - A(x))^7  + ...
Also,
1 = 1/(1 + A(x))  +  (1-x)^5/((1-x)^5 + A(x))^2  +  (1-x)^10/((1-x)^10 + A(x))^3  +  (1-x)^15/((1-x)^15  +  A(x))^4 + (1-x)^20/((1-x)^20 + A(x))^5  +  (1-x)^25/((1-x)^25 + A(x))^6  +  (1-x)^30/((1-x)^30 + A(x))^7 + ...

Crossrefs

Cf. A304639, A326262, A326263, A326264.
Cf. A321605.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1-x)^(-5*m) - Ser(A))^m ) )[#A] ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/(1-x)^(5*n) - A(x) )^n.
(2) 1 = Sum_{n>=0} ( 1 - (1-x)^(5*n)*A(x) )^n / (1-x)^(5*n^2).
(3) 1 = Sum_{n>=0} (1-x)^(5*n) / ( (1-x)^(5*n) + A(x) )^(n+1).
Showing 1-5 of 5 results.

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