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.

Previous Showing 41-49 of 49 results.

A180591 G.f.: A(x) = exp( Sum_{n>=1} 2^[A001511(n)^2]*x^n/n ) where A001511(n) is the exponent in the highest power of 2 that divides 2n.

Original entry on oeis.org

1, 2, 10, 18, 178, 338, 1450, 2562, 23234, 43906, 186602, 329298, 2276914, 4224530, 16898506, 29572482, 191488770, 353405058, 1394069578, 2434734098, 14073489714, 25712245330, 97969052778, 170225860226, 938475356354
Offset: 0

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Author

Paul D. Hanna, Sep 10 2010

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 18*x^3 + 178*x^4 + 338*x^5 +...
log(A(x)) = 2^1*x + 2^4*x^2/2 + 2^1*x^3/3 + 2^9*x^4/4 + 2^1*x^5/5 + 2^4*x^6/6 + 2^1*x^7/7 + 2^16*x^8/8 +...+ 2^[A001511(n)^2]*x^n/n +...

Crossrefs

Cf. A155200, A001511, A000123.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n,2^(valuation(2*m,2)^2)*x^m/m)+x*O(x^n)),n)}

Extensions

Name corrected by Paul D. Hanna, Sep 19 2010

A202516 G.f.: exp( Sum_{n>=1} (2^n + 3^n)^n * x^n/n ).

Original entry on oeis.org

1, 5, 97, 14735, 22208431, 314664801905, 41448076127290195, 50905029765702161210225, 582983891132858366160979787245, 62080074367851800086180277369110042475, 61205889017397342360456211893643596980919936577
Offset: 0

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Author

Paul D. Hanna, Dec 20 2011

Keywords

Comments

More generally, for integers p and q, exp( Sum_{n>=1} (p^n + q^n)^n * x^n/n ) is a power series in x with integer coefficients.

Examples

			G.f.: A(x) = 1 + 5*x + 97*x^2 + 14735*x^3 + 22208431*x^4 +...
where
log(A(x)) = (2+3)*x + (2^2 + 3^2)^2*x^2/2 + (2^3 + 3^3)^3*x^3/3 + (2^4 + 3^4)^4*x^4/4 + (2^5 + 3^5)^5*x^5/5 +...
more explicitly,
log(A(x)) = 5*x + 13^2*x^2/2 + 35^3*x^3/3 + 97^4*x^4/4 + 275^5*x^5/5 +...

Crossrefs

Cf. A155200, A155201, A155202, A202517, A326555.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n,(2^m+3^m)^m*x^m/m)+x*O(x^n)),n)}

A202517 G.f.: exp( Sum_{n>=1} (3^n - 2^n)^n * x^n/n ).

Original entry on oeis.org

1, 1, 13, 2299, 4465027, 83649932869, 14413888012788031, 22412828378864422506133, 312169717565869706933620630009, 38865154523992131836783382601539858727, 43266472789023671032936589458127528396392744933
Offset: 0

Views

Author

Paul D. Hanna, Dec 20 2011

Keywords

Comments

More generally, for integers p and q, exp( Sum_{n>=1} (p^n - q^n)^n * x^n/n ) is a power series in x with integer coefficients.

Examples

			G.f.: A(x) = 1 + x + 13*x^2 + 2299*x^3 + 4465027*x^4 + 83649932869*x^5 +...
where
log(A(x)) = (3-2)*x + (3^2 - 2^2)^2*x^2/2 + (3^3 - 2^3)^3*x^3/3 + (3^4 - 2^4)^4*x^4/4 + (3^5 - 2^5)^5*x^5/5 +...
more explicitly,
log(A(x)) = x + 5^2*x^2/2 + 19^3*x^3/3 + 65^4*x^4/4 + 211^5*x^5/5 +...

Crossrefs

Cf. A202516, A155200, A155201, A155202.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n,(3^m-2^m)^m*x^m/m)+x*O(x^n)),n)}

A211897 G.f.: exp( Sum_{n>=1} (2^n + (-1)^n)^n * x^n/n ).

Original entry on oeis.org

1, 1, 13, 127, 21079, 5748277, 12575820727, 76137769800001, 2378969789430032869, 263966921383940194614823, 128008718415112846211347561597, 240383035701447602719960666753525867, 1863847508172945183054545696402414919578641
Offset: 0

Views

Author

Paul D. Hanna, Apr 25 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 13*x^2 + 127*x^3 + 21079*x^4 + 5748277*x^5 +...
such that
log(A(x)) = x + 5^2*x^2 + 7^3*x^3 + 17^4*x^4 + 31^5*x^5 + 65^6*x^6 + 127^7*x^7 +...+ (2^n + (-1)^n)^n*x^n/n +...

Crossrefs

Cf. A211898, A155200.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(k=1, n, (2^k+(-1)^k)^k*x^k/k)+x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) == 1 (mod 6).

A211898 G.f.: exp( Sum_{n>=1} (2^n - (-1)^n)^n * x^n/n ).

Original entry on oeis.org

1, 3, 9, 261, 13419, 7867287, 10444212819, 84955235950827, 2235017786095822257, 273416315791427558035965, 125533366255776787874473759857, 242979442003484538229530424638338553, 1852958949086213206247388599213928431454549
Offset: 0

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Author

Paul D. Hanna, Apr 25 2012

Keywords

Comments

CONJECTURE: the highest power of 3 dividing a(n) equals 3^A089792(n) for n>=0; that is, n!*a(n)/3^n is an integer not divisible by 3 for n>=0.
Given g.f. A(x), note that A(x)^(1/3) is not an integer series.

Examples

			G.f.: A(x) = 1 + 3*x + 9*x^2 + 261*x^3 + 13419*x^4 + 7867287*x^5 +...
such that
log(A(x)) = 3*x + 3^2*x^2/2 + 9^3*x^3/3 + 15^4*x^4/4 + 33^5*x^5/5 + 63^6*x^6/6 + 129^7*x^7/7 + 255^8*x^8/8 +...+ (2^n - (-1)^n)^n*x^n/n +...

Crossrefs

Cf. A211897, A155200, A089792.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(k=1, n, (2^k-(-1)^k)^k*x^k/k)+x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) == 3 (mod 6) for n>0.

A306063 O.g.f. A(x) satisfies: Sum_{n>=1} (2^n*x - A(x))^n / n = 0.

Original entry on oeis.org

2, 2, 68, 9398, 4833428, 9454918068, 72006088426248, 2165455076559211174, 259347215815106405220132, 124310299732163916975832447388, 239094057363977384232311742474570360, 1847535112178186477442381068773529944826172, 57378255829217503847229646446662951215946818659912, 7161580198468591866673993366959923994699839199088716021928
Offset: 1

Views

Author

Paul D. Hanna, Jun 19 2018

Keywords

Comments

It is remarkable that this sequence should consist entirely of integers.

Examples

			O.g.f.: A(x) = 2*x + 2*x^2 + 68*x^3 + 9398*x^4 + 4833428*x^5 + 9454918068*x^6 + 72006088426248*x^7 + 2165455076559211174*x^8 + 259347215815106405220132*x^9 + 124310299732163916975832447388*x^10 + 239094057363977384232311742474570360*x^11 + 1847535112178186477442381068773529944826172*x^12 + ...
such that
0 = (2*x - A(x)) + (2^2*x - A(x))^2/2 + (2^3*x - A(x))^3/3 + (2^4*x - A(x))^4/4 + (2^5*x - A(x))^5/5 + (2^6*x - A(x))^6/6 + (2^7*x - A(x))^7/7 + ...
RELATED SERIES.
exp( Sum_{n>=1} 2^(n^2)*x^n / n ) = 1 + 2*x + 10*x^2 + 188*x^3 + 16774*x^4 + 6745436*x^5 + 11466849412*x^6 + 80444398636280*x^7 + ... + A155200(n)*x^n + ...
exp( Sum_{n>=1} A(x)^n / n ) = 1/(1 - A(x)) = 1 + 2*x + 6*x^2 + 84*x^3 + 9714*x^4 + 4872228*x^5 + 9474410908*x^6 + 72043987279208*x^7 + 2165743253217563938*x^8 + ...
Sum_{n>=1} A(x)^n / n = -log(1 - A(x)) = 2*x + 8*x^2/2 + 224*x^3/3 + 38192*x^4/4 + 24263312*x^5/5 + 56787868688*x^6/6 + 504175196453504*x^7/7 + ...

Links

Crossrefs

Cf. A155200.

Programs

  • PARI
    {a(n) = my(A=[2]); for(i=1,n, A = concat(A,0); A[#A] = polcoeff(sum(m=1,#A,(2^m*x - x*Ser(A))^m/m), #A));A[n]}
for(n=1,20,print1(a(n),", "))

A165940 G.f.: Sum_{n>=0} a(n)*x^n/2^(n^2+n) = exp( Sum_{n>=1} x^n/[n*2^(n^2)] ).

Original entry on oeis.org

1, 2, 10, 152, 7684, 1352096, 852120928, 1960591940480, 16697154282192928, 531801639623740649984, 63854080509077223292639744, 29089348119991257994736112048128
Offset: 0

Views

Author

Paul D. Hanna, Oct 01 2009

Keywords

Comments

Conjectured to consist entirely of integers.

Examples

			G.f.: 1 + 2*x/2^2 + 10*x^2/2^6 + 152*x^3/2^12 + 7684*x^4/2^20 +...
= exp( x/2 + x^2/(2*2^4) + x^3/(3*2^9) + x^4/(4*2^16) +... ).
Evaluated at x=1:
Sum_{n>=0} a(n)/2^(n^2+n) = 1.7021716250154556344906565654972646...

Crossrefs

Cf. A155200.

Programs

  • PARI
    {a(n)=2^(n^2+n)*polcoeff(exp(sum(m=1, n+1, 2^(-m^2)*x^m/m)+x*O(x^n)), n)}

A260756 G.f.: exp( Sum_{n>=1} 2^(n^n) * x^n/n ).

Original entry on oeis.org

1, 2, 10, 44739260, 28948022309329048855892746252171976963317496166410141009864396001978371888518
Offset: 0

Views

Author

Paul D. Hanna, Jul 30 2015

Keywords

Comments

Does the g.f. describe an integer sequence?
The number of digits in the terms begin: [1, 1, 2, 8, 77, 941, 14045, 247911, ...].

Examples

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 44739260*x^3 +...
where
log(A(x)) = 2^1*x + 2^4*x^2/2 + 2^27*x^3/3 + 2^256*x^4/4 + 2^3125*x^5/5 + 2^46656*x^6/6 + 2^823543*x^7/7 + 2^16777216*x^8/8 +...+ 2^(n^n)*x^n/n +...

Crossrefs

Cf. A155200, A158110.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n+1, 2^(m^m)*x^m/m)+x*O(x^n)), n)}
for(n=0,7,print1(a(n),", "))

A381422 Expansion of g.f. = exp( Sum_{n>=1} A066802(n)*x^n/n ).

Original entry on oeis.org

1, 20, 662, 26780, 1205961, 58050204, 2924165436, 152231599628, 8125577046740, 442293253888592, 24457749066666142, 1370114821790970340, 77591333270514869230, 4434803157977731784808, 255492958449660158603448, 14820943641891118200315756, 864962304943085638764540396
Offset: 0

Views

Author

Karol A. Penson, Apr 22 2025

Keywords

Crossrefs

Cf. A066802, A155200, A255881, A229451, A229452, A156216.

Formula

G.f. = 64/((1 + sqrt(1 - 4*x^(1/3)))^2*(1 + sqrt(1 + 4*(-1)^(1/3)*x^(1/3)))^2*(1 + sqrt(1 - 4*(-1)^(2/3)*x^(1/3)))^2).
The above g.f. denoted by h satisfies algebraic equation of order eight:
1 + (8*x - 1)*h + 4*x*(7*x + 3)*h^2 + 7*x^2*(8*x - 1)*h^3 + x^2*(70*x^2 - 40*x + 1)*h^4 + 7*x^4*(8*x - 1)*h^5 + 4*x^5*(7*x + 3)*h^6 + x^6*(8*x - 1)*h^7 + x^8*h^8 = 0.
Previous Showing 41-49 of 49 results.

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