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.

A317904 Decimal expansion of a constant related to the asymptotics of A302598.

Original entry on oeis.org

3, 9, 5, 6, 1, 8, 4, 2, 0, 3, 0, 2, 6, 1, 6, 9, 7, 5, 4, 5, 4, 0, 8, 0, 2, 1, 8, 1, 8, 7, 8, 3, 0, 0, 8, 3, 3, 2, 9, 9, 9, 9, 8, 8, 0, 9, 5, 2, 5, 5, 5, 4, 0, 9, 8, 1, 6, 4, 9, 6, 2, 1, 1, 3, 0, 9, 3, 2, 8, 8, 5, 1, 3, 1, 4, 6, 2, 5, 2, 1, 2, 3, 0, 3, 1, 1, 5, 8, 9, 4, 8, 8, 4, 1, 6, 4, 0, 9, 6, 0, 7, 4, 7, 6, 5
Offset: 1

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Author

Vaclav Kotesovec, Aug 10 2018

Keywords

Examples

			3.9561842030261697545408021818783008332999988095...

Crossrefs

Cf. A302598, A302700, A317350, A317351, A317355, A317356, A322737, A323311, A323313.

A323311 E.g.f. A(x) satisfies: 1 = Sum_{n>=0} (exp(n*x) - 1)^n/(A(x) + 1 - exp(n*x))^(n+1).

Original entry on oeis.org

1, 1, 11, 283, 14855, 1310011, 172520351, 31513669363, 7595793146855, 2330879613371851, 886383762411615791, 408963256168949033443, 225040270250903527024055, 145601653678200482159541691, 109437844707983885536850408831, 94572173789825201408460630621523, 93118733370917669491764504635160455, 103644400582305503214140030821130959531, 129490690058782610512772741408027302955471, 180464581077334737195826400036356606725361603
Offset: 0

Views

Author

Paul D. Hanna, Feb 02 2019

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 11*x^2/2! + 283*x^3/3! + 14855*x^4/4! + 1310011*x^5/5! + 172520351*x^6/6! + 31513669363*x^7/7! + 7595793146855*x^8/8! + 2330879613371851*x^9/9! +  + 886383762411615791*x^10/10! + ...
such that
1 = 1/A(x)  +  (exp(x) - 1)/(A(x) + 1 - exp(x))^2  +  (exp(2*x) - 1)^2/(A(x) + 1 - exp(2*x))^3  +  (exp(3*x) - 1)^3/(A(x) + 1 - exp(3*x))^4  +  (exp(4*x) - 1)^4/(A(x) + 1 - exp(4*x))^5  +  (exp(5*x) - 1)^5/(A(x) + 1 - exp(5*x))^6 + ...
also,
1 = 1/(A(x) + 2)  +  (exp(x) + 1)/(A(x) + 1 + exp(x))^2  +  (exp(2*x) + 1)^2/(A(x) + 1 + exp(2*x))^3  +  (exp(3*x) + 1)^3/(A(x) + 1 + exp(3*x))^4  +  (exp(4*x) + 1)^4/(A(x) + 1 + exp(4*x))^5  +  (exp(5*x) + 1)^5/(A(x) + 1 + exp(5*x))^6 + ...
RELATED SERIES.
log(A(x)) = x + 10*x^2/2! + 252*x^3/3! + 13486*x^4/4! + 1213260*x^5/5! + 162204670*x^6/6! + 29956649772*x^7/7! + 7279075598686*x^8/8! + 2247264600871500*x^9/9! + ...

Links

Crossrefs

Cf. A323313.

Programs

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

Formula

E.g.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} (exp(n*x) - 1)^n/(A(x) + 1 - exp(n*x))^(n+1).
(2) 1 = Sum_{n>=0} (exp(n*x) + 1)^n/(A(x) + 1 + exp(n*x))^(n+1).
a(n) ~ c * A317904^n * n^(2*n + 1/2) / exp(2*n), where c = 1.5545244013... - Vaclav Kotesovec, Aug 11 2021

A323573 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^n - 1)^n/(A(x) + (1+x)^n - 1)^(n+1).

Original entry on oeis.org

1, 1, 1, 6, 41, 381, 4387, 59961, 946119, 16901219, 336924252, 7412401676, 178372705409, 4660680881897, 131410732869312, 3977081948965664, 128600945014475040, 4424941538152614645, 161433547224627797940, 6224586371820817112652, 252934418382142622780667, 10803348636926511625239387, 483881915960470248201012949
Offset: 0

Views

Author

Paul D. Hanna, Feb 07 2019

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 6*x^3 + 41*x^4 + 381*x^5 + 4387*x^6 + 59961*x^7 + 946119*x^8 + 16901219*x^9 + 336924252*x^10 + 7412401676*x^11 + 178372705409*x^12 + ...
such that
1 = 1/A(x) +  x/(A(x) + x)^2 + ((1+x)^2 - 1)^2/(A(x) + (1+x)^2 - 1)^3 + ((1+x)^3 - 1)^3/(A(x) + (1+x)^3 - 1)^4 + ((1+x)^4 - 1)^4/(A(x) + (1+x)^4 - 1)^5 + ((1+x)^5 - 1)^5/(A(x) + (1+x)^5 - 1)^6 + ((1+x)^2 - 1)^6/(A(x) + (1+x)^6 - 1)^7 + ...

Links

Crossrefs

Cf. A323313.

Programs

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

Formula

a(n) ~ c * n^n / (exp(n) * log(2)^(2*n)), where c = 0.51205951699411... - Vaclav Kotesovec, Aug 11 2021

A323574 G.f. A(x) satisfies: 1 = Sum_{n>=0} 2^n * ((1+x)^n - 1)^n / (A(x) + 2 - 2*(1+x)^n)^(n+1).

Original entry on oeis.org

1, 2, 20, 320, 7996, 269272, 11293792, 563253696, 32433009160, 2113347523336, 153579286783456, 12309659862402976, 1078628781953636960, 102578628758305245024, 10523148808846566898816, 1158407291029244188955264, 136214299772837816557703120, 17040721610970237566148646464, 2260018461602151565512432884608, 316748455363386162460484685488512
Offset: 0

Views

Author

Paul D. Hanna, Feb 06 2019

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 20*x^2 + 320*x^3 + 7996*x^4 + 269272*x^5 + 11293792*x^6 + 563253696*x^7 + 32433009160*x^8 + 2113347523336*x^9 + 153579286783456*x^10 + ...
such that
1 = 1/A(x) + 2*((1+x) - 1)/(A(x) + 2 - 2*(1+x))^2  +  2^2*((1+x)^2 - 1)^2/(A(x) + 2 - 2*(1+x)^2)^3  +  2^3*((1+x)^3 - 1)^3/(A(x) + 2 - 2*(1+x)^3)^4  +  2^4*((1+x)^4 - 1)^4/(A(x) + 2 - 2*(1+x)^4)^5  +  2^5*((1+x)^5 - 1)^5/(A(x) + 2 - 2*(1+x)^5)^6 + ...
also,
1 = 1/(A(x) + 4)  +  2*(1 + (1+x))/(A(x) + 2 + 2*(1+x))^2  +  2^2*(1 + (1+x)^2)^2/(A(x) + 2 + 2*(1+x)^2)^3  +  2^3*(1 + (1+x)^3)^3/(A(x) + 2 + 2*(1+x)^3)^4  +  2^4*(1 + (1+x)^4)^4/(A(x) + 2 + 2*(1+x)^4)^5  +  2^5*(1 + (1+x)^5)^5/(A(x) + 2 + 2*(1+x)^5)^6 + ...

Links

Crossrefs

Cf. A323313.

Programs

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

Formula

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

A323572 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^n + i)^n / (A(x) + 1 + i*(1+x)^n)^(n+1), where i^2 = -1.

Original entry on oeis.org

1, 5, 45, 1142, 47253, 2664573, 187170069, 15598588065, 1497110942013, 162226788530207, 19566798092698042, 2598785222401424468, 376850999493886187699, 59248452153964672923677, 10039900576546291696149404, 1824412367286993070795917580, 353943959915575446954764374094, 73024199735586268826145811783169, 15966496692824534985042866376857576, 3688160965656359052252569464435170928, 897528733209823570848685886402050648933
Offset: 0

Views

Author

Paul D. Hanna, Feb 10 2019

Keywords

Comments

It is remarkable that the generating function results in a power series in x with only real coefficients.

Examples

			G.f.: A(x) = 1 + 5*x + 45*x^2 + 1142*x^3 + 47253*x^4 + 2664573*x^5 + 187170069*x^6 + 15598588065*x^7 + 1497110942013*x^8 + 162226788530207*x^9 + ...
such that
1 = 1/(A(x) + 1+i) + ((1+x) + i)/(A(x) + 1 + i*(1+x))^2  +  ((1+x)^2 + i)^2/(A(x) + 1 + i*(1+x)^2)^3  +  ((1+x)^3 + i)^3/(A(x) + 1 + i*(1+x)^3)^4  +  ((1+x)^4 + i)^4/(A(x) + 1 + i*(1+x)^4)^5  +  ((1+x)^5 + i)^5/(A(x) + 1 + i*(1+x)^5)^6 + ...
also,
1 = 1/(A(x) + 1-i) + ((1+x) - i)/(A(x) + 1 - i*(1+x))^2  +  ((1+x)^2 - i)^2/(A(x) + 1 - i*(1+x)^2)^3  +  ((1+x)^3 - i)^3/(A(x) + 1 - i*(1+x)^3)^4  +  ((1+x)^4 - i)^4/(A(x) + 1 - i*(1+x)^4)^5  +  ((1+x)^5 - i)^5/(A(x) + 1 - i*(1+x)^5)^6 + ...

Crossrefs

Cf. A323313.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = round( Vec( sum(m=0, #A*20+300, ((1+x+x*O(x^n))^m + I)^m / (Ser(A) + 1 + I*(1+x+x*O(x^n))^m)^(m+1)*1. ) )[#A]) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = round( Vec( sum(m=0, #A*20+300, ((1+x+x*O(x^n))^m - I)^m / (Ser(A) + 1 - I*(1+x+x*O(x^n))^m)^(m+1)*1. ) )[#A]) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ((1+x)^n + i)^n / (A(x) + 1 + i*(1+x)^n)^(n+1).
(2) 1 = Sum_{n>=0} ((1+x)^n - i)^n / (A(x) + 1 - i*(1+x)^n)^(n+1).
(3) 1 = Sum_{n>=0} ((1+x)^n + i)^n * (A(x) + 1 - i*(1+x)^n)^(n+1) / ((A(x) + 1)^2 + (1+x)^(2*n))^(n+1).
(4) 1 = Sum_{n>=0} ((1+x)^n - i)^n * (A(x) + 1 + i*(1+x)^n)^(n+1) / ((A(x) + 1)^2 + (1+x)^(2*n))^(n+1).
Showing 1-5 of 5 results.

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