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 1-3 of 3 results.

A302598 G.f.: Sum_{n>=0} (1 + (1+x)^n)^n / (2 + (1+x)^n)^(n+1).

Original entry on oeis.org

1, 1, 6, 58, 798, 14150, 307076, 7881756, 233536532, 7844786248, 294582696686, 12228351266210, 556017625969246, 27482790417322218, 1467194712330407238, 84134395928742550138, 5157545958316518485420, 336574587493456290969620, 23296320082405927961459550, 1704662916739625989249415610, 131480805016085834305348796128
Offset: 0

Views

Author

Paul D. Hanna, Apr 10 2018

Keywords

Comments

The following identity holds for |y| <= 1 and fixed real k > 0:
Sum_{n>=0} (k + y^n)^n/(1+k + y^n)^(n+1) = Sum_{n>=0} (y^n - 1)^n/(1+k - k*y^n)^(n+1).

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 58*x^3 + 798*x^4 + 14150*x^5 + 307076*x^6 + 7881756*x^7 + 233536532*x^8 + 7844786248*x^9 + 294582696686*x^10 + ...
such that
A(x) = 1/3  +  (1 + (1+x))/(2 + (1+x))^2  +  (1 + (1+x)^2)^2/(2 + (1+x)^2)^3  +  (1 + (1+x)^3)^3/(2 + (1+x)^3)^4  +  (1 + (1+x)^4)^4/(2 + (1+x)^4)^5  +  (1 + (1+x)^5)^5/(2 + (1+x)^5)^6  +  (1 + (1+x)^6)^6/(2 + (1+x)^6)^7  + ...
Also,
A(x) = 1  +  ((1+x) - 1)/(2 - (1+x))^2  +  ((1+x)^2 - 1)^2/(2 - (1+x)^2)^3  +  ((1+x)^3 - 1)^3/(2 - (1+x)^3)^4  +  ((1+x)^4 - 1)^4/(2 - (1+x)^4)^5  +  ((1+x)^5 - 1)^5/(2 - (1+x)^5)^6  +  ((1+x)^6 - 1)^6/(2 - (1+x)^6)^7 + ...
RELATED INFINITE SERIES.
(1) At x = -1/2: the following sums are equal
S1 = Sum_{n>=0} 2^n * (2^n + 1)^n/(2^(n+1) + 1)^(n+1)
S1 = Sum_{n>=0} (-2)^n * (2^n - 1)^n/(2^(n+1) - 1)^(n+1).
Explicitly,
S1 = 1/3 + 2*3/5^2 + 4*5^2/9^3 + 8*9^3/17^4 + 16*17^4/33^5 + 32*33^5/65^6 + 64*65^6/129^7 + 128*129^7/257^8 + 256*257^8/513^9 + 512*513^9/1025^10 + ...
S1 = 1 - 2*1/3^2 + 4*3^2/7^3 - 8*7^3/15^4 + 16*15^4/31^5 - 32*31^5/63^6 + 64*63^6/127^7 - 128*127^7/255^8 + 256*255^8/511^9 - 512*511^9/1023^10 + ...
where S1 = 0.84714730053329880291591114748812485885366310294051236295420...
(2) At x = -2/3: the following sums are equal
S2 = Sum_{n>=0} 3^n * (1 + 3^n)^n / (2*3^n + 1)^(n+1)
S2 = Sum_{n>=0} (-3)^n * (3^n - 1)^n / (2*3^n - 1)^(n+1).
Explicitly,
S2 = 1/3 + 3*4/7^2 + 9*10^2/19^3 + 27*28^3/55^4 + 81*82^4/163^5 + 243*244^5/487^6 + 729*730^6/1459^7 + 2187*2188^7/4375^8 + 6561*6562^8/13123^9 + 19683*19684^9/39367^10 + ...
S2 = 1 - 3*2^1/5^2 + 9*8^2/17^3 - 27*26^3/53^4 + 81*80^4/161^5 - 243*242^5/485^6 + 729*728^6/1457^7 - 2187*2186^7/4373^8 + 6561*6560^8/13121^9 - 19683*19682^9/39365^10 + ...
where S2 = 0.837457334418049175936255584889342515316005199043439291643371...
(3) At x = -1/3: the following sums are equal
S3 = Sum_{n>=0} 3^n * (2^n + 3^n)^n/(2*3^n + 2^n)^(n+1)
S3 = Sum_{n>=0} (-3)^n * (3^n - 2^n)^n/(2*3^n - 2^n)^(n+1).
Explicitly,
S3 = 1/3 + 3*5/8^2 + 9*13^2/22^3 + 27*35^3/62^4 + 81*97^4/178^5 + 243*275^5/518^6 + 729*793^6/1522^7 + 2187*2315^7/4502^8 + 6561*6817^8/13378^9 + 19683*20195^9/39878^10 + ...
S3 = 1 - 3*1/4^2 + 9*5^2/14^3 - 27*19^3/46^4 + 81*65^4/146^5 - 243*211^5/454^6 + 729*665^6/1394^7 - 2187*2059^7/4246^8 + 6561*6305^8/12866^9 - 19683*19171^9/38854^10 + ...
where S3 = 0.867357695200699139470956415922046910279987551651352471994920...

Links

Crossrefs

Cf. A302700, A122400, A302614, A302615.
Cf. A317662, A317663, A317664.

Programs

  • PARI
    {a(n) = my(A=1,o=x*O(x^n)); A = sum(m=0,n,((1+x +o)^m - 1)^m / (2 - (1+x +o)^m)^(m+1)); polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))

Formula

G.f.: Sum_{n>=0} ((1+x)^n - 1)^n / (2 - (1+x)^n)^(n+1).
G.f.: Sum_{n>=0} ((1+x)^n + 1)^n / (2 + (1+x)^n)^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = A317904 = 3.9561842030261697545408021818783008332999988095... and c = 0.274656497660429769528095546948772676444158... - Vaclav Kotesovec, Aug 09 2018

A317662 G.f.: Sum_{n>=0} ( (1+x)^n - 1 )^n * 2^n / (3 - 2*(1+x)^n)^(n+1).

Original entry on oeis.org

1, 2, 24, 448, 11820, 401392, 16668960, 818355488, 46367354632, 2977828665832, 213763450387456, 16961461169786752, 1474091484740240064, 139256465915227044352, 14208358055857371300864, 1557104405499802200814464, 182416569911688799401148816, 22749429746475540390909166048, 3009102958766163591152586574464, 420767787785916464100556297780608
Offset: 0

Views

Author

Paul D. Hanna, Aug 03 2018

Keywords

Comments

The following identities hold for |y| <= 1 and fixed real k > 0:
(C1) Sum_{n>=0} (y^n + k)^n/(1+k + y^n)^(n+1) = Sum_{n>=0} (y^n - 1)^n/(1+k - k*y^n)^(n+1).
(C2) Sum_{n>=0} (y^n + 1)^n*k^n/(1+k + k*y^n)^(n+1) = Sum_{n>=0} (y^n - 1)^n*k^n/(1+k - k*y^n)^(n+1).
This sequence is an example of (C2) when y = 1+x and k = 2.

Examples

			G.f.: A(x) = 1 + 2*x + 24*x^2 + 448*x^3 + 11820*x^4 + 401392*x^5 + 16668960*x^6 + 818355488*x^7 + 46367354632*x^8 + ...
such that
A(x) = 1  +  ((1+x) - 1)*2/(3 - 2*(1+x))^2  +  ((1+x)^2 - 1)^2*2^2/(3 - 2*(1+x)^2)^3  +  ((1+x)^3 - 1)^3*2^3/(3 - 2*(1+x)^3)^4  +  ((1+x)^4 - 1)^4*2^4/(3 - 2*(1+x)^4)^5  +  ((1+x)^5 - 1)^5*2^5/(3 - 2*(1+x)^5)^6  +  ((1+x)^6 - 1)^6*2^6/(3 - 2*(1+x)^6)^7 + ...
Also,
A(x) = 1/5  +  ((1+x) + 1)*2/(3 + 2*(1+x))^2  +  ((1+x)^2 + 1)^2*2^2/(3 + 2*(1+x)^2)^3  +  ((1+x)^3 + 1)^3*2^3/(3 + 2*(1+x)^3)^4  +  ((1+x)^4 + 1)^4*2^4/(3 + 2*(1+x)^4)^5  +  ((1+x)^5 + 1)^5*2^5/(3 + 2*(1+x)^5)^6  +  ((1+x)^6 + 1)^6*2^6/(3 + 2*(1+x)^6)^7 + ...
EXAMPLE OF SUMS.
Evaluating the g.f. formally at x = -1/2, we obtain the sums
S1 = Sum_{n>=0} (1 - 2^n)^n * 4^n / (3*2^n - 2)^(n+1),
S2 = Sum_{n>=0} (1 + 2^n)^n * 4^n / (3*2^n + 2)^(n+1),
explicitly,
S1 = 1 - 4/4^2 + 3^2*4^2/10^3 - 7^3*4^3/22^4 + 15^4*4^4/46^5 - 31^5*4^5/94^6 + 63^6*4^6/190^7 - 127^7*4^7/382^8 + 255^8*4^8/766^9 - 511^9*4^9/1534^10 + 1023^10*4^10/3070^11 + ...
S2 = 1/5 + 3*4/8^2 + 5^2*4^2/14^3 + 9^3*4^3/26^4 + 17^4*4^4/50^5 + 33^5*4^5/98^6 + 65^6*4^6/194^7 + 129^7*4^7/386^8 + 257^8*4^8/770^9 + 513^9*4^9/1538^10 + 1025^10*4^10/3074^11 + ...
where S1 = S2 = 0.8378452129227094466992700455568437913726753230322...

Links

Crossrefs

Cf. A302598, A317663, A317664, A302614, A317350.

Programs

  • PARI
    {a(n) = my(A=1); A = sum(m=0, n, ( (1+x)^m - 1 +x*O(x^n) )^m * 2^m / (3 - 2*(1+x)^m +x*O(x^n) )^(m+1) ); ;polcoeff(A,n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} ( (1+x)^n - 1 )^n * 2^n / (3 - 2*(1+x)^n)^(n+1).
(2) A(x) = Sum_{n>=0} ( (1+x)^n + 1 )^n * 2^n / (3 + 2*(1+x)^n)^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = 7.5592435681748721825440151469382350654183499600538671407998439255608144356... and c = 0.30852178850187571906358489049387403704035769403106379389644818349... - Vaclav Kotesovec, Aug 09 2018

A317663 G.f.: Sum_{n>=0} ( (1+x)^n - 1 )^n * 3^n / (4 - 3*(1+x)^n)^(n+1).

Original entry on oeis.org

1, 3, 54, 1494, 58248, 2921346, 179119836, 12981530772, 1085678924472, 102911898062376, 10903265835968178, 1276820000444309958, 163765996498409795046, 22831772648676453195534, 3437850237146915605162722, 555999871854064840852040190, 96123855383854022518187481072, 17690662829477220311541393099324, 3453143617992367493730150308612370
Offset: 0

Views

Author

Paul D. Hanna, Aug 03 2018

Keywords

Comments

The following identities hold for |y| <= 1 and fixed real k > 0:
(C1) Sum_{n>=0} (y^n + k)^n/(1+k + y^n)^(n+1) = Sum_{n>=0} (y^n - 1)^n/(1+k - k*y^n)^(n+1).
(C2) Sum_{n>=0} (y^n + 1)^n*k^n/(1+k + k*y^n)^(n+1) = Sum_{n>=0} (y^n - 1)^n*k^n/(1+k - k*y^n)^(n+1).
This sequence is an example of (C2) when y = 1+x and k = 3.

Examples

			G.f.: A(x) = 1 + 3*x + 54*x^2 + 1494*x^3 + 58248*x^4 + 2921346*x^5 + 179119836*x^6 + 12981530772*x^7 + 1085678924472*x^8 + ...
such that
A(x) = 1  +  ((1+x) - 1)*3/(4 - 3*(1+x))^2  +  ((1+x)^2 - 1)^2*3^2/(4 - 3*(1+x)^2)^3  +  ((1+x)^3 - 1)^3*3^3/(4 - 3*(1+x)^3)^4  +  ((1+x)^4 - 1)^4*3^4/(4 - 3*(1+x)^4)^5  +  ((1+x)^5 - 1)^5*3^5/(4 - 3*(1+x)^5)^6  +  ((1+x)^6 - 1)^6*3^6/(4 - 3*(1+x)^6)^7 + ...
Also,
A(x) = 1/7  +  ((1+x) + 1)*3/(4 + 3*(1+x))^2  +  ((1+x)^2 + 1)^2*3^2/(4 + 3*(1+x)^2)^3  +  ((1+x)^3 + 1)^3*3^3/(4 + 3*(1+x)^3)^4  +  ((1+x)^4 + 1)^4*3^4/(4 + 3*(1+x)^4)^5  +  ((1+x)^5 + 1)^5*3^5/(4 + 3*(1+x)^5)^6  +  ((1+x)^6 + 1)^6*3^6/(4 + 3*(1+x)^6)^7 + ...

Links

Crossrefs

Cf. A302598, A317662, A317664, A302615.

Programs

  • PARI
    {a(n) = my(A=1); A = sum(m=0, n, ( (1+x)^m - 1 +x*O(x^n) )^m * 3^m / (4 - 3*(1+x)^m +x*O(x^n) )^(m+1) ); ;polcoeff(A,n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} ( (1+x)^n - 1 )^n * 3^n / (4 - 3*(1+x)^n)^(n+1).
(2) A(x) = Sum_{n>=0} ( (1+x)^n + 1 )^n * 3^n / (4 + 3*(1+x)^n)^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = 11.154788564351081167494585241180262193438722530344791058752757035461192417... and c = 0.321897864665202841967234839159770976446040882710871129852558... - Vaclav Kotesovec, Aug 09 2018
Showing 1-3 of 3 results.

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