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 31-40 of 49 results. Next

A158097 G.f.: A(x) = exp( Sum_{n>=1} x^n/n * 2^(n^2)/(1 - 2^(n^2)*x^n) ).

Original entry on oeis.org

1, 2, 14, 204, 16982, 6746636, 11467009772, 80444425963128, 2306004014991374374, 268654794950955551450892, 126765597355485863873077402788, 241678070949320869650125781001909864
Offset: 0

Views

Author

Paul D. Hanna, Mar 26 2009

Keywords

Comments

Compare to g.f. of the partition numbers A000041:
exp( Sum_{n>=1} x^n/(1 - x^n)/n ) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 +...

Examples

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 204*x^3 + 16982*x^4 + 6746636*x^5 +...
log(A(x)) = 2*x + 24*x^2/2 + 536*x^3/3 + 66112*x^4/4 + 33554592*x^5/5 +...
log(A(x)) = 2*x/(1-2*x) + 2^4*x^2/(1-2^4*x^2)/2 + 2^9*x^3/(1-2^9*x^3)/3 +...

Crossrefs

Cf. A158096, A155200.

Programs

  • PARI
    {a(n)=if(n==0,1,polcoeff(exp(sum(k=1,n,(2^k*x)^k/(1-(2^k*x)^k +x*O(x^n))/k)),n))}
for(n=0, 15, print1(a(n), ", "))
  • PARI
    {a(n) = polcoeff( exp( sum(m=1, n, x^m/m * sumdiv(m, d, 2^(m*d) * m/d) ) +x*O(x^n)), n)}
for(n=0, 15, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 30 2015

Formula

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

A158110 G.f.: A(x) = exp( Sum_{n>=1} 2^(n^3)*x^n/n ).

Original entry on oeis.org

1, 2, 130, 44739500, 4611686018516874838, 8507059173023461595807737228465099196, 17552048611426197782986337964292523732529439672780432120964458900
Offset: 0

Views

Author

Paul D. Hanna, Mar 19 2009

Keywords

Comments

Conjecture: given q and m are nonnegative integers, then
exp( Sum_{n>=1} q^(n^m)*x^n/n )
is a power series in x with integer coefficients.

Examples

			G.f.: A(x) = 1 + 2*x + 130*x^2 + 44739500*x^3 +...
log(A(x)) = 2*x + 2^8*x^2/2 + 2^27*x^3/3 + 2^64*x^4/4 +...

Crossrefs

Cf. A155200.

Programs

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

A185385 G.f. satisfies: A(x) = exp( Sum_{n>=1} (2*A(x) - (-1)^n)^n * x^n/n ).

Original entry on oeis.org

1, 3, 11, 61, 381, 2527, 17559, 126265, 931321, 7007035, 53568131, 414929621, 3249392917, 25684315319, 204645707183, 1641910625009, 13253684541553, 107561523423731, 877109999610107, 7183095973808493, 59053492869471661, 487189276030904207, 4032100262853037127
Offset: 0

Views

Author

Paul D. Hanna, Dec 22 2011

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 11*x^2 + 61*x^3 + 381*x^4 + 2527*x^5 + 17559*x^6 +...
where
log(A(x)) = (2*A(x) + 1)*x + (2*A(x) - 1)^2*x^2/2 + (2*A(x) + 1)^3*x^3/3 + (2*A(x) - 1)^4*x^4/4 +...
log(A(x)*(1-2*x*A(x))) = 1/(1 + 2*x*A(x))*x + 1/(1 - 2*x*A(x))^2*x^2/2 + 1/(1 + 2*x*A(x))^3*x^3/3 + 1/(1 - 2*x*A(x))^4*x^4/4 +...

Crossrefs

Cf. A202519, A163138, A202668, A202630, A202518, A155200.

Programs

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

Formula

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

A202669 G.f. satisfies: A(x) = exp( Sum_{n>=1} (A(x) + (-1)^n)^n * x^n/n ).

Original entry on oeis.org

1, 0, 2, 2, 12, 20, 96, 212, 898, 2354, 9266, 27070, 102094, 319930, 1177838, 3865762, 14050948, 47574460, 171886784, 594572676, 2143957648, 7528825924, 27156892364, 96412294088, 348314869652, 1246689890248, 4513958859208, 16257651642036, 59010423148052, 213586733348928
Offset: 0

Views

Author

Paul D. Hanna, Dec 22 2011

Keywords

Examples

			G.f.: A(x) = 1 + 2*x^2 + 2*x^3 + 12*x^4 + 20*x^5 + 96*x^6 + 212*x^7 +...
where
log(A(x)) = (A(x) - 1)*x + (A(x) + 1)^2*x^2/2 + (A(x) - 1)^3*x^3/3 + (A(x) + 1)^4*x^4/4 +...
log(A(x)*(1-x*A(x))) = -1/(1 + x*A(x))*x + 1/(1 - x*A(x))^2*x^2/2 - 1/(1 + x*A(x))^3*x^3/3 + 1/(1 - x*A(x))^4*x^4/4 +...

Crossrefs

Cf. A202668, A202629, A202519, A155200.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (A+(-1)^m+x*O(x^n))^m*x^m/m))); polcoeff(A, n)}

Formula

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

A292501 G.f.: exp( Sum_{n>=1} [ Sum_{k>=1} (2*k-1)^n * x^k ]^n * (1-x)^n / n ).

Original entry on oeis.org

1, 1, 3, 13, 91, 1119, 23235, 879361, 55447631, 6274018595, 1192773105789, 400761393446831, 231147252957096671, 231434829013884972151, 406000810484101907916927, 1216355994930424625967455929, 6474418584620388915674215696687, 58229572245447428847208518694227279, 936163501254507409972001699357677028097, 25330794407893091120626418701416294765820223, 1224635875718403110628189182372406488768960029317
Offset: 0

Views

Author

Paul D. Hanna, Sep 17 2017

Keywords

Comments

A060187(n,k) = Sum_{j=1..k} (-1)^(k-j) * binomial(n,k-j) * (2*j-1)^(n-1).
Note that exp( Sum_{n>=1} [ Sum_{k=0..n} A060187(n+1,k+1) * x^k ] / (1-x)^n * x^n/ n ) does not yield an integer series.
Conjecture: a(n)^(1/n^2) tends to 3^(1/4). - Vaclav Kotesovec, Oct 17 2020

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 91*x^4 + 1119*x^5 + 23235*x^6 + 879361*x^7 + 55447631*x^8 + 6274018595*x^9 + 1192773105789*x^10 + 400761393446831*x^11 + 231147252957096671*x^12 + 231434829013884972151*x^13 + 406000810484101907916927*x^14 + 1216355994930424625967455929*x^15 +...
RELATED SERIES.
log(A(x)) = x + 5*x^2/2 + 31*x^3/3 + 305*x^4/4 + 5041*x^5/5 + 131477*x^6/6 + 5973311*x^7/7 + 436089793*x^8/8 + 55949083681*x^9/9 + 11863792842885*x^10/10 + 4395111080551775*x^11/11 + 2768928615166879025*x^12/12 + 3005637312940054635857*x^13/13 + 5680764740993004611483477*x^14/14 + 18239242940612856315412499071*x^15/15 +...
The logarithm of g.f. A(x) equals the series:
log(A(x)) = Sum_{n>=1} (x + 3^n*x^2 + 5^n*x^3 +...+ (2*k-1)^n*x^k +...)^n * (1-x)^n/n,
or,
log(A(x)) = (x + 3*x^2 + 5*x^3 + 7*x^4 + 9*x^5 +...) * (1-x) +
(x + 3^2*x^2 + 5^2*x^3 + 7^2*x^4 + 9^2*x^5 +...)^2 * (1-x)^2/2 +
(x + 3^3*x^2 + 5^3*x^3 + 7^3*x^4 + 9^3*x^5 +...)^3 * (1-x)^3/3 +
(x + 3^4*x^2 + 5^4*x^3 + 7^4*x^4 + 9^4*x^5 +...)^4 * (1-x)^4/4 + ...
This logarithmic series can be written using the Eulerian numbers of type B like so:
log(A(x)) = (x + x^2) / (1-x) +
(x + 6*x^2 + x^3)^2 / (1-x)^4/2 +
(x + 23*x^2 + 23*x^3 + x^4)^3 / (1-x)^9/3 +
(x + 76*x^2 + 230*x^3 + 76*x^4 + x^5)^4 / (1-x)^16/4 +
(x + 237*x^2 + 1682*x^3 + 1682*x^4 + 237*x^5 + x^6)^5 / (1-x)^25/5 +
(x + 722*x^2 + 10543*x^3 + 23548*x^4 + 10543*x^5 + 722*x^6 + x^7)^6 / (1-x)^36/6 +
(x + 2179*x^2 + 60657*x^3 + 259723*x^4 + 259723*x^5 + 60657*x^6 + 2179*x^7 + x^8)^7 / (1-x)^49/7 +...+
[ Sum_{k=0..n} A060187(n+1,k+1) * x^k ]^n  / (1-x)^(n^2) * x^n/n +...

Links

Crossrefs

Cf. A292500, A156170, A276751, A276752, A156171, A155200, A060187.

Programs

  • PARI
    {a(n) = polcoeff( exp( sum(m=1, n+1, sum(k=1, n, (2*k-1)^m * x^k +x*O(x^n))^m*(1-x)^m/m ) ), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {A060187(n, k) = sum(j=1, k, (-1)^(k-j) * binomial(n, k-j) * (2*j-1)^(n-1))}
{a(n) = my(A=1, Oxn=x*O(x^n));
A = exp( sum(m=1,n+1, sum(k=0, m, A060187(m+1, k+1)*x^k)^m /(1-x +Oxn)^(m^2) * x^m/m ) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f.: exp( Sum_{n>=1} [ Sum_{k=0..n} A060187(n+1,k+1) * x^k ]^n / (1-x)^(n^2) * x^n/n ), where A060187 are the Eulerian numbers of type B.

A156334 G.f.: A(x) = exp( Sum_{n>=1} 2^[n^2/2+1]*x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 2, 6, 20, 166, 1980, 91612, 4980968, 1083899526, 246514209900, 225675208005684, 210073940172966552, 787481680820307364188, 2977392786568558334126040, 45279192083837920124027862264
Offset: 0

Views

Author

Paul D. Hanna, Feb 10 2009

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 166*x^4 + 1980*x^5 + 91612*x^6 +...
log(A(x)) = 2*x + 2^3*x^2/2 + 2^5*x^3/3 + 2^9*x^4/4 + 2^13*x^5/5 + 2^19*x^6/6 +...

Crossrefs

Cf. A156340, A155200.

Programs

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

Formula

a(n) = (1/n)*Sum_{k=1..n} 2^floor(k^2/2+1) * a(n-k) for n>0, with a(0)=1.

A156910 G.f.: A(x) = exp( Sum_{n>=1} 2^(n^2)/(1 - 2^n*x)^n * x^n/n ).

Original entry on oeis.org

1, 2, 14, 268, 21462, 7872396, 12585797612, 84949155244024, 2379063526056509734, 273414369715003663482380, 128009001272184822673783879332, 242979321424122460096958142064785384
Offset: 0

Views

Author

Paul D. Hanna, Feb 17 2009

Keywords

Comments

An example of this logarithmic identity at q=2:
Sum_{n>=1} [q^(n^2)/(1 - q^n*x)^n]*x^n/n = Sum_{n>=1} [(1 + q^n)^n - 1]*x^n/n.

Examples

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 268*x^3 + 21462*x^4 +...
log(A(x)) = 2/(1-2*x)*x + 2^4/(1-2^2*x)^2*x^2/2 + 2^9/(1-2^3*x)^3*x^3/3 +...
log(A(x)) = (3-1)*x + (5^2-1)*x^2/2 + (9^3-1)*x^3/3 + (17^4-1)*x^4/4 +...

Crossrefs

Cf. A155201, A155200.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n+1, 2^(m^2)/(1-2^m*x)^m*x^m/m)+x*O(x^n)), n)}
  • PARI
    /* As First Differences of A155201: */
{a(n)=polcoeff((1-x)*exp(sum(m=1, n+1, (2^m+1)^m*x^m/m)+x*O(x^n)), n)}

Formula

G.f.: A(x) = (1-x)*exp( Sum_{n>=1} (1 + 2^n)^n * x^n/n );
Equals the first differences of A155201.

A164764 G.f. satisfies: the coefficient of x^n in A(x)^n = 2^(n^2) for n>=1 with A(0)=1.

Original entry on oeis.org

1, 2, 6, 144, 15390, 6580224, 11386265292, 80284132772352, 2304717583810291830, 268613293782939614576640, 126760224526971269877523841364, 241675282146473482949215936098066432
Offset: 0

Views

Author

Paul D. Hanna, Aug 25 2009

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 144*x^3 + 15390*x^4 + 6580224*x^5 +...
The coefficients in the successive powers of A(x) begin:
[1,(2), 6, 144, 15390, 6580224, 11386265292, 80284132772352,...];
[1, 4,(16), 312, 31392, 13223736, 22799056896, 160613894000880,...];
[1, 6, 30,(512), 48078, 19932480, 34238570076, 240989363896320,...];
[1, 8, 48, 752,(65536), 26708592, 45705005568, 321410623205088,...];
[1, 10, 70, 1040, 83870,(33554432), 57198570060, 401877753223680,...];
[1, 12, 96, 1384, 103200, 40472616,(68719476736), 482390835814224,...];
[1, 14, 126, 1792, 123662, 47466048, 80267945884,(562949953421312),...];
[1, 16, 160, 2272, 145408, 54537952, 91844205568, 643555189090240,...];
...
The above terms in parenthesis = [x^n] A(x)^n = 2^(n^2) for n=1,2,3,...
The main diagonal = [x^n] A(x)^(n+1) = (n+1)*A155200(n):
[1, 2*2, 3*10, 4*188, 5*16774, 6*6745436, 7*11466849412, ...].

Crossrefs

Cf. A155200, A156214, A156631.

Programs

  • PARI
    {a(n)=local(G=exp(sum(m=1,n,2^(m^2)*x^m/m)+x*O(x^n)));polcoeff(x/serreverse(x*G),n)}
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(sum(m=1,n,2^(m^2)*(x/A)^m/m)));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = exp( Sum_{n>=1} 2^(n^2)*[x/A(x)]^n/n ).
Let G(x) = exp(Sum_{n>=1} 2^(n^2)*x^n/n) = g.f. of A155200, then:
(1) A(x) = G(x/A(x)) and A(x*G(x)) = G(x) ;
(2) A(x) = x/Series_Reversion[x*G(x)] ;
(3) [x^n] A(x)^(n+1)/(n+1) = [x^n] G(x) = A155200(n) ;
(4) [x^n] A(x)^(n+m)*m/(n+m) = [x^n] G(x)^m for all m.

A171776 E.g.f.: A(x) = exp( Sum_{n>=1} 2^(n(n-1)) * x^n/n ).

Original entry on oeis.org

1, 1, 5, 141, 25161, 25295385, 129002055885, 3167498196303525, 363195624958803434385, 190409085693362565632615985, 449225585595812339036501379506325
Offset: 0

Views

Author

Paul D. Hanna, Jan 20 2010

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 141*x^3/3! + 25161*x^4/4! +...
log(A(x)) = x + 4*x^2/2 + 64*x^3/3 + 4096*x^4/4 + 1048576*x^5/5 +..

Crossrefs

Cf. A155200.

Programs

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

Formula

a(n) = A155200(n)*n!/2^n and is odd for n>=0.

A171777 E.g.f.: A(x) = exp( Sum_{n>=1} 2^(n(n-1)/2) * x^n/n ).

Original entry on oeis.org

1, 1, 3, 23, 473, 27057, 4102027, 1539365191, 1365364095921, 2783117747148641, 12795599930746180499, 130882205973999096722679, 2946911413331842739385098377, 144807670567304192694224250060817, 15419384323650924141916096692523710747
Offset: 0

Views

Author

Paul D. Hanna, Jan 23 2010

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 23*x^3/3! + 473*x^4/4! +...
log(A(x)) = x + 2*x^2/2 + 2^3*x^3/3 + 2^6*x^4/4 + 2^10*x^5/5 +...

Links

Crossrefs

Cf. A171776, A155200.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      binomial(n-1, j-1)*(j-1)!*2^(j*(j-1)/2), j=1..n))
    end:
seq(a(n), n=0..14);  # Alois P. Heinz, Mar 15 2023
  • PARI
    {a(n)=n!*polcoeff(exp(sum(m=1, n+1, 2^(m*(m-1)/2)*x^m/m)+x*O(x^n)), n)}
Previous Showing 31-40 of 49 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.