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 1-7 of 7 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

Views

Author

Vaclav Kotesovec, Aug 10 2018

Keywords

Examples

			3.9561842030261697545408021818783008332999988095...

Crossrefs

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

A302615 G.f.: Sum_{n>=0} (3 + (1+x)^n)^n / (4 + (1+x)^n)^(n+1).

Original entry on oeis.org

1, 1, 10, 130, 2390, 56714, 1644138, 56327820, 2226708772, 99761490536, 4995375316146, 276464859358474, 16757956600528786, 1104116777798713154, 78565751676021256606, 6004629888868350015506, 490572645247461234631946, 42665124626946741636482996, 3935474733572880332326074450, 383756013888633346483785849474
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 + 10*x^2 + 130*x^3 + 2390*x^4 + 56714*x^5 + 1644138*x^6 + 56327820*x^7 + 2226708772*x^8 + 99761490536*x^9 + ...
such that
A(x) = 1/5  +  (3 + (1+x))/(4 + (1+x))^2  +  (3 + (1+x)^2)^2/(4 + (1+x)^2)^3  +  (3 + (1+x)^3)^3/(4 + (1+x)^3)^4  +  (3 + (1+x)^4)^4/(4 + (1+x)^4)^5  +  (3 + (1+x)^5)^5/(4 + (1+x)^5)^6  +  (3 + (1+x)^6)^6/(4 + (1+x)^6)^7  + ...
Also,
A(x) = 1  +  ((1+x) - 1)/(4 - 3*(1+x))^2  +  ((1+x)^2 - 1)^2/(4 - 3*(1+x)^2)^3  +  ((1+x)^3 - 1)^3/(4 - 3*(1+x)^3)^4  +  ((1+x)^4 - 1)^4/(4 - 3*(1+x)^4)^5  +  ((1+x)^5 - 1)^5/(4 - 3*(1+x)^5)^6  +  ((1+x)^6 - 1)^6/(4 - 3*(1+x)^6)^7 + ...

Links

Crossrefs

Cf. A122400, A302598, A302614.

Programs

  • PARI
    {a(n) = my(A=1, o=x*O(x^n)); A = sum(m=0, n, ((1+x +o)^m - 1)^m / (4 - 3*(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 / (4 - 3*(1+x)^n)^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = 5.2709551504518355656831902094014170087... and c = 0.26621450180820822374221893929... - Vaclav Kotesovec, Aug 09 2018

A302614 G.f.: Sum_{n>=0} (2 + (1+x)^n)^n / (3 + (1+x)^n)^(n+1).

Original entry on oeis.org

1, 1, 8, 91, 1474, 30765, 785053, 23682833, 824522797, 32537599175, 1435199414014, 69973425937141, 3736662443907962, 216901789032691605, 13598124265965160130, 915670842666691879191, 65913110467411283181409, 5050836914009172555862713, 410501468976427335127369669, 35269929119728622895198302033, 3194195105084750546987502710855
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 + 8*x^2 + 91*x^3 + 1474*x^4 + 30765*x^5 + 785053*x^6 + 23682833*x^7 + 824522797*x^8 + 32537599175*x^9 + ...
such that
A(x) = 1/4  +  (2 + (1+x))/(3 + (1+x))^2  +  (2 + (1+x)^2)^2/(3 + (1+x)^2)^3  +  (2 + (1+x)^3)^3/(3 + (1+x)^3)^4  +  (2 + (1+x)^4)^4/(3 + (1+x)^4)^5  +  (2 + (1+x)^5)^5/(3 + (1+x)^5)^6  +  (2 + (1+x)^6)^6/(3 + (1+x)^6)^7  + ...
Also,
A(x) = 1  +  ((1+x) - 1)/(3 - 2*(1+x))^2  +  ((1+x)^2 - 1)^2/(3 - 2*(1+x)^2)^3  +  ((1+x)^3 - 1)^3/(3 - 2*(1+x)^3)^4  +  ((1+x)^4 - 1)^4/(3 - 2*(1+x)^4)^5  +  ((1+x)^5 - 1)^5/(3 - 2*(1+x)^5)^6  +  ((1+x)^6 - 1)^6/(3 - 2*(1+x)^6)^7 + ...
RELATED INFINITE SERIES.
(1) At x = -1/3: the following sums are equal
S1 = Sum_{n>=0} 3^n * (2*3^n + 2^n)^n / (3^(n+1) + 2^n)^(n+1),
S1 = Sum_{n>=0} (-3)^n * (3^n - 2^n)^n / (3^(n+1) - 2^(n+1))^(n+1).
Explicitly,
S1 = 1/4 + 3*8/11^2 + 9*22^2/31^3 + 27*62^3/89^4 + 81*178^4/259^5 + 243*518^5/761^6 + 729*1522^6/2251^7 + 2187*4502^7/6689^8 + 6561*13378^8/19939^9 + 19683*39878^9/59561^10 + ...
S1 = 1 - 3*1/5^2 + 9*5^2/19^3 - 27*19^3/65^4 + 81*65^4/211^5 - 243*211^5/665^6 + 729*665^6/2059^7 - 2187*2059^7/6305^8 + 6561*6305^8/19171^9 - 19683*19171^9/58025^10 + ...
where S1 = 0.90501051059439877583104471171480036033530856741889530664913...
(2) At x = -1/2: the following sums are equal
S2 = Sum_{n>=0} 2^n * (2^(n+1) + 1)^n / (3*2^n + 1)^(n+1),
S2 = Sum_{n>=0} (-2)^n * (2^n - 1)^n / (3*2^n - 2)^(n+1).
Explicitly,
S2 = 1/4 + 2*5/7^2 + 4*9^2/13^3 + 8*17^3/25^4 + 16*33^4/49^5 + 32*65^5/97^6 + 64*129^6/193^7 + 128*257^7/385^8 + 256*513^8/769^9 + 512*1025^9/1537^10 + ...
S2 = 1 - 2*1/4^2 + 4*3^2/10^3 - 8*7^3/22^4 + 16*15^4/46^5 - 32*31^5/94^6 + 64*63^6/190^7 - 128*127^7/382^8 + 256*255^8/766^9 - 512*511^9/1534^10 + ...
where S2 = 0.90222608896798122564942421232120719521782835530371831680447...

Links

Crossrefs

Cf. A122400, A302598, A302615.

Programs

  • PARI
    {a(n) = my(A=1, o=x*O(x^n)); A = sum(m=0, n, ((1+x +o)^m - 1)^m / (3 - 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 / (3 - 2*(1+x)^n)^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = 4.64471605501103711823541367464... and c = 0.270134222044915506270113032... - Vaclav Kotesovec, Aug 10 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

A317664 G.f.: Sum_{n>=0} ( (1+x)^n - 1 )^n * 4^n / (5 - 4*(1+x)^n)^(n+1).

Original entry on oeis.org

1, 4, 96, 3520, 181584, 12046208, 976817408, 93618157824, 10353263884352, 1297682198608960, 181792547403610112, 28148715766252519424, 4773717142486206475264, 879979421777903153737728, 175192929827140711780067328, 37462651348142346656294109184, 8563418069261195349710481467648, 2083773631690873034841394464054272
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 = 4.

Examples

			G.f.: A(x) = 1 + 4*x + 96*x^2 + 3520*x^3 + 181584*x^4 + 12046208*x^5 + 976817408*x^6 + 93618157824*x^7 + 10353263884352*x^8 + ...
such that
A(x) = 1  +  ((1+x) - 1)*4/(5 - 4*(1+x))^2  +  ((1+x)^2 - 1)^2*4^2/(5 - 4*(1+x)^2)^3  +  ((1+x)^3 - 1)^3*4^3/(5 - 4*(1+x)^3)^4  +  ((1+x)^4 - 1)^4*4^4/(5 - 4*(1+x)^4)^5  +  ((1+x)^5 - 1)^5*4^5/(5 - 4*(1+x)^5)^6  +  ((1+x)^6 - 1)^6*4^6/(5 - 4*(1+x)^6)^7 + ...
Also,
A(x) = 1/9  +  ((1+x) + 1)*4/(5 + 4*(1+x))^2  +  ((1+x)^2 + 1)^2*4^2/(5 + 4*(1+x)^2)^3  +  ((1+x)^3 + 1)^3*4^3/(5 + 4*(1+x)^3)^4  +  ((1+x)^4 + 1)^4*4^4/(5 + 4*(1+x)^4)^5  +  ((1+x)^5 + 1)^5*4^5/(5 + 4*(1+x)^5)^6  +  ((1+x)^6 + 1)^6*4^6/(5 + 4*(1+x)^6)^7 + ...

Links

Crossrefs

Cf. A302598, A317662, A317663, A302615.

Programs

  • PARI
    {a(n) = my(A=1); A = sum(m=0, n, ( (1+x)^m - 1 +x*O(x^n) )^m * 4^m / (5 - 4*(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 * 4^n / (5 - 4*(1+x)^n)^(n+1).
(2) A(x) = Sum_{n>=0} ( (1+x)^n + 1 )^n * 4^n / (5 + 4*(1+x)^n)^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = 14.74821884963947298733792887778672923688310694846410198271766770874395484... and c = 0.329067655604412806858767072708083473088299024445... - Vaclav Kotesovec, Aug 09 2018

A302700 E.g.f.: Sum_{n>=0} (exp(n*x) + 1)^n / (2 + exp(n*x))^(n+1).

Original entry on oeis.org

1, 1, 13, 385, 21325, 1898401, 247841293, 44611568065, 10589093387725, 3204648461107681, 1204384753185644173, 550313048077989740545, 300436578515074737333325, 193139598305033634851120161, 144410707207961955130172624653, 124258444226932649355925701301825, 121911793079671988588136925596434125, 135284324089583933279712302959420767841
Offset: 0

Views

Author

Paul D. Hanna, Apr 14 2018

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 13*x^2/2! + 385*x^3/3! + 21325*x^4/4! + 1898401*x^5/5! + 247841293*x^6/6! + 44611568065*x^7/7! + 10589093387725*x^8/8! + 3204648461107681*x^9/9! + ...
such that
A(x) = 1/3 + (exp(x)+1)/(2+exp(x))^2 + (exp(2*x)+1)^2/(2+exp(2*x))^3 + (exp(3*x)+1)^3/(2+exp(3*x))^4 + (exp(4*x)+1)^4/(2+exp(4*x))^5 + (exp(5*x)+1)^5/(2+exp(5*x))^6 + (exp(6*x)+1)^6/(2+exp(6*x))^7 + ...
Also,
A(x) = 1 + (exp(x)-1)/(2-exp(x))^2 + (exp(2*x)-1)^2/(2-exp(2*x))^3 + (exp(3*x)-1)^3/(2-exp(3*x))^4 + (exp(4*x)-1)^4/(2-exp(4*x))^5 + (exp(5*x)-1)^5/(2-exp(5*x))^6 + (exp(6*x)-1)^6/(2-exp(6*x))^7 + ...

Links

Crossrefs

Cf. A302598.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Sum[(E^(k*x) - 1)^k / (2 - E^(k*x))^(k+1), {k, 0, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 11 2018 *)
  • PARI
    {a(n) = my(A=1); A = sum(m=0,n+1, (exp(m*x + x*O(x^n)) - 1)^m / (2 - exp(m*x + x*O(x^n)))^(m+1) ); n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! equals:
(1) Sum_{n>=0} (exp(n*x) + 1)^n / (2 + exp(n*x))^(n+1).
(2) Sum_{n>=0} (exp(n*x) - 1)^n / (2 - exp(n*x))^(n+1).
(3) Sum_{n>=0} 2^n*exp(n^2*x/2)*cosh(n*x/2)^n/(1 + 2*exp(n*x/2)*cosh(n*x/2))^(n+1).
(4) Sum_{n>=0} 2^n*exp(n^2*x/2)*sinh(n*x/2)^n/(1 - 2*exp(n*x/2)*sinh(n*x/2))^(n+1).
a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = A317904 = 3.9561842030261697545408... and c = 0.31165774853025500197969363638844... - Vaclav Kotesovec, Aug 10 2018
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