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 1-10 of 18 results. Next

A055601 Number of n X n binary matrices with no zero rows.

Original entry on oeis.org

1, 1, 9, 343, 50625, 28629151, 62523502209, 532875860165503, 17878103347812890625, 2375680873491867011912191, 1255325460068093790930770843649, 2644211984585174742731315532085090303, 22235498641774645581443610453175918212890625
Offset: 0

Views

Author

Vladeta Jovovic, Jun 01 2000

Keywords

Comments

More generally, Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n / n! = Sum_{n>=0} (m*q^n + b)^n * x^n / n! for all q, m, b. - Paul D. Hanna, Jan 02 2008

Examples

			A(x) = 1 + x + 3^2*x^2/2! + 7^3*x^3/3! + 15^4*x^4/4! +... + (2^n-1)^n*x^n/n! +...
A(x) = exp(-x) + 2*exp(-2x) + 2^4*exp(-4x)*x^2/2! + 2^9*exp(-8x)*x^3/3! +...+ 2^(n^2)*exp(-2^n*x)*x^n/n! +...
This is a special case of the more general statement: Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n! = Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b) where F(x) = exp(x), q=2, m=1, b=-1. - _Paul D. Hanna_, Jan 02 2008

Links

Crossrefs

Cf. A048291, A092477, A136516, A303560.

Programs

  • Maple
    a:= n-> mul(Stirling2(n+1, 2), j=1..n): seq(a(n), n=0..10); # Zerinvary Lajos, Jan 01 2009
  • Mathematica
    Join[{1},Table[(2^n-1)^n,{n,16}]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)
  • PARI
    a(n)=n!*polcoeff(sum(k=0,n,2^(k^2)*exp(-2^k*x)*x^k/k!),n) \\ Paul D. Hanna, Jan 02 2008
  • Python
    a = lambda n:((1<Kenny Lau, Jul 05 2016
  • Python
    N = 58
base = 0
a = []
for i in range(N):
    a += [base**i]
    base = (base<<1)|1 #base = base*2+1
print(a)
# Kenny Lau, Jul 05 2016

Formula

a(n) = A092477(n, n) for n>0.
a(n) = (2^n - 1 )^n. - Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001
a(n) = Sum_{k=0..n} (-1)^k*C(n, k)*2^((n-k)*n).
E.g.f.: A(x) = Sum_{n>=0} 2^(n^2) * exp(-2^n*x) * x^n/n!. - Paul D. Hanna, Jan 02 2008
O.g.f.: Sum_{n>=0} 2^(n^2)*x^n/(1 + 2^n*x)^(n+1). - Paul D. Hanna, Jan 20 2010
Sum_{n>=1} 1/a(n) = A303560. - Amiram Eldar, Nov 18 2020

A155201 G.f.: A(x) = exp( Sum_{n>=1} (2^n + 1)^n * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 3, 17, 285, 21747, 7894143, 12593691755, 84961748935779, 2379148487805445513, 273416748863491468927893, 128009274688933686165252807225, 242979449433397149030644307317592609, 1863847996727745781866688849374488247858333, 57652096246331953203644653244501049018464175026133
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2009

Keywords

Comments

More generally, it appears that for m integer, exp( Sum_{n>=1} (m^n + y)^n * x^n/n ) is a power series in x and y with integer coefficients.

Examples

			G.f.: A(x) = 1 + 3*x + 17*x^2 + 285*x^3 + 21747*x^4 + 7894143*x^5 +...
log(A(x)) = 3*x + 5^2*x^2/2 + 9^3*x^3/3 + 17^4*x^4/4 + 33^5*x^5/5 +...

Crossrefs

Cf. A136516, A155200, A155202, A155810 (triangle), variants: A155204, A155208.

Programs

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

Formula

Equals row sums of triangle A155810.
a(n) = (1/n)*Sum_{k=1..n} (2^k + 1)^k * a(n-k) for n>0, with a(0)=1.
a(n) = B_n( 0!*(2^1+1)^1, 1!*(2^2+1)^2, 2!*(2^3+1)^3, ..., (n-1)!*(2^n+1)^n ) / n!, where B_n() is the n-th complete Bell polynomial. - Max Alekseyev, Oct 10 2014

A165327 E.g.f: Sum_{n>=0} 2^(n(n-1)) * exp(2^n*x) * x^n/n!.

Original entry on oeis.org

1, 2, 9, 125, 6561, 1419857, 1291467969, 4902227890625, 76686282021340161, 4891005035897482905857, 1262332172765951010966606849, 1312086657801266767978668212890625
Offset: 0

Views

Author

Paul D. Hanna, Sep 15 2009

Keywords

Comments

More generally, Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n/n! = Sum_{n>=0} (m*q^n + b)^n * x^n/n! for all q, m, b.

Examples

			E.g.f: A(x) = 1 + 2*x + 3^2*x^2/2! + 5^3*x^3/3! + 9^4*x^4/4! +...
A(x) = exp(x) + exp(2x)*x + 2^2*exp(4x)*x^2/2! + 2^6*exp(8x)*x^3/3! +...
This is a special case of the more general statement:
Sum_{n>=0} m^n * F(q^n*x)^b * log(F(q^n*x) )^n / n! = Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b) where F(x) = exp(x), q=2, m=1/2, b=1.

Crossrefs

Cf. variants: A136516, A055601, A079491.

Programs

  • PARI
    {a(n,q=2,m=1/2,b=1)=n!*polcoeff(sum(k=0, n, m^k*q^(k^2)*exp(b*q^k*x+x*O(x^n))*x^k/k!), n)}

Formula

a(n) = (2^(n-1) + 1)^n.

A202989 E.g.f: Sum_{n>=0} 3^(n^2) * exp(3^n*x) * x^n/n!.

Original entry on oeis.org

1, 4, 100, 21952, 45212176, 864866612224, 151334226289000000, 240066313618039143841792, 3437872835498096514323500400896, 443629285048033016198674962874808664064, 515464807019389919369209932597753906250000000000
Offset: 0

Views

Author

Paul D. Hanna, Dec 26 2011

Keywords

Comments

E.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n/n! = Sum_{n>=0} (m*q^n + b)^n * x^n/n! for all q, m, b.
O.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * x^n/(1-b*q^n*x)^(n+1) = Sum_{n>=0} (m*q^n + b)^n * x^n for all q, m, b.

Examples

			E.g.f.: A(x) = 1 + 4*x + 100*x^2/2! + 21952*x^3/3! + 45212176*x^4/4! +..
By the series identity, the e.g.f.:
A(x) = exp(x) + 3*exp(3*x)*x + 3^4*exp(3^2*x)*x^2/2! + 3^9*exp(3^3*x)*x^3/3! +...
expands into:
A(x) = 1 + 4*x + 10^2*x^2/2! + 28^3*x^3/3! + 82^4*x^4/4! + 244^5*x^5/5! +...+ (3^n+1)^n*x^n/n! +...

Crossrefs

Cf. A165327, A060613, A244004, A136516, A326013.

Programs

  • PARI
    {a(n, q=3, m=1, b=1)=(m*q^n + b)^n}
  • PARI
    {a(n, q=3, m=1, b=1)=n!*polcoeff(sum(k=0, n, m^k*q^(k^2)*exp(b*q^k*x+x*O(x^n))*x^k/k!), n)}
  • PARI
    {a(n, q=3, m=1, b=1)=polcoeff(sum(k=0,n,m^k*q^(k^2)*x^k/(1-b*q^k*x +x*O(x^n))^(k+1)),n)}

Formula

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

A243918 a(n) = Sum_{k=0..n} binomial(n,k) * (1 + 2^k)^k.

Original entry on oeis.org

1, 4, 32, 814, 86600, 39560554, 75654970772, 594996059517934, 19035905851947436400, 2460857798358946973785234, 1280109151917797032199865564812, 2672783800502564772495577135824089014, 22366199286781599568269093307412768076442280
Offset: 0

Views

Author

Paul D. Hanna, Jun 17 2014

Keywords

Examples

			O.g.f.: A(x) = 1 + 4*x + 32*x^2 + 814*x^3 + 86600*x^4 + 39560554*x^5 +...
where the g.f. may be expressed by the series identity:
A(x) = 1/(1-x) + 3*x/(1-x)^2 + 5^2*x^2/(1-x)^3 + 9^3*x^3/(1-x)^4 + 17^4*x^4/(1-x)^5 + 33^5*x^5/(1-x)^6 + 65^6*x^6/(1-x)^7 +...
A(x) = 1/(1-2*x) + 2*x/(1-3*x)^2 + 2^4*x^2/(1-5*x)^3 + 2^9*x^3/(1-9*x)^4 + 2^16*x^4/(1-17*x)^5 + 2^25*x^5/(1-33*x)^6 + 2^36*x^6/(1-65*x)^7 +...
Illustration of initial terms:
a(0) = 1;
a(1) = 1 + (1+2);
a(2) = 1 + 2*(1+2) + (1+2^2)^2;
a(3) = 1 + 3*(1+2) + 3*(1+2^2)^2 + (1+2^3)^3;
a(4) = 1 + 4*(1+2) + 6*(1+2^2)^2 + 4*(1+2^3)^3 + (1+2^4)^4;
a(5) = 1 + 5*(1+2) + 10*(1+2^2)^2 + 10*(1+2^3)^3 + 5*(1+2^4)^4 + (1+2^5)^5; ...
Also, by a binomial identity we have
a(0) = 1;
a(1) = 2 + 2;
a(2) = 2^2 + 2*(1+2)*2 + 2^4;
a(3) = 2^3 + 3*(1+2)^2*2 + 3*(1+2^2)*2^4 + 2^9;
a(4) = 2^4 + 4*(1+2)^3*2 + 6*(1+2^2)^2*2^4 + 4*(1+2^3)*2^9 + 2^16;
a(5) = 2^5 + 5*(1+2)^4*2 + 10*(1+2^2)^3*2^4 + 10*(1+2^3)^2*2^9 + 5*(1+2^4)*2^16 + 2^25; ...

Crossrefs

Cf. A244004, A136516.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]*(1+2^k)^k,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 18 2014 *)
  • PARI
    {a(n)=sum(k=0, n, binomial(n, k)*(1+2^k)^k)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=sum(k=0, n, binomial(n, k)*(1+2^k)^(n-k)*2^(k^2))}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * (1 + 2^k)^(n-k) * 2^(k^2).
O.g.f.: Sum_{n>=0} (1 + 2^n)^n * x^n / (1-x)^(n+1).
O.g.f.: Sum_{n>=0} 2^(n^2) * x^n / (1 - (1+2^n)*x)^(n+1).
E.g.f.: exp(x) * Sum_{n>=0} (1 + 2^n)^n * x^n / n!.
a(n) ~ 2^(n^2). - Vaclav Kotesovec, Jun 18 2014

A180602 a(n) = (2^(n+1) - 1)^n.

Original entry on oeis.org

1, 3, 49, 3375, 923521, 992436543, 4195872914689, 70110209207109375, 4649081944211090042881, 1227102111503512992112190463, 1291749870339606615892191271170049, 5429914198235566686555216227881787109375
Offset: 0

Views

Author

Paul D. Hanna, Sep 11 2010

Keywords

Comments

More generally, we have the following identities:
(1) Sum_{n>=0} m^n* F(q^n*x)^b* log( F(q^n*x) )^n/n! = Sum_{n>=0} x^n* [y^n] F(y)^(m*q^n + b);
(2) Sum_{n>=0} m^n* q^(n^2)* exp(b*q^n*x)*x^n/n! = Sum_{n>=0} (m*q^n + b)^n*x^n/n! for all q, m, b.
This sequence results from (2) when q=2, m=2, b=-1.
For n >= 2, a(n) is the first number in a set of three powerful numbers in arithmetic progression with difference a(n)*(2^n - 1). - Arkadiusz Wesolowski, Aug 26 2013

Examples

			E.g.f: A(x) = 1 + 3*x + 7^2*x^2/2! + 15^3*x^3/3! + 31^4*x^4/4! +...
A(x) = exp(-x) + 2^2*exp(-2*x)*x + 2^6*exp(-4*x)*x^2/2! + 2^12*exp(-8*x)*x^3/3! +...

Crossrefs

Cf. A086459 (signed, offset 1), variants: A055601, A079491, A136516, A165327.
Cf. A001694.

Programs

  • Magma
    [(2^(n+1)-1)^n : n in [0..11]]; // Arkadiusz Wesolowski, Aug 26 2013
  • Maple
    A180602:=n->(2^(n+1) - 1)^n: seq(A180602(n), n=0..10); # Wesley Ivan Hurt, Oct 09 2014
  • Mathematica
    Table[(2^(n + 1) - 1)^n, {n, 0, 10}] (* Wesley Ivan Hurt, Oct 09 2014 *)
  • PARI
    {a(n)=n!*polcoeff(sum(k=0, n, 2^(k^2+k)*exp(-2^k*x+x*O(x^n))*x^k/k!), n)}
  • Python
    def A180602(n): return ((1<Chai Wah Wu, Sep 13 2024

Formula

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

Extensions

Name changed by Arkadiusz Wesolowski, Aug 26 2013

A171800 a(n) = ((n+1)*2^n + 1)*(2^n + 1)^(n-1).

Original entry on oeis.org

1, 5, 65, 2673, 397953, 228882753, 520970490625, 4723480504289025, 170687922720157732865, 24563695027660686202250241, 14068441356460459384918212890625, 32058887942708146080692278858371608577, 290694663888102785007861162394348756446314497
Offset: 0

Views

Author

Paul D. Hanna, Jan 20 2010

Keywords

Examples

			G.f.: A(x) = 1 + 5*x + 65*x^2 + 2673*x^3 + 397953*x^4 +...
A(x) = 1/(1-x) + 2*2*x/(1-2*x)^2 + 3*2^4*x^2/(1-2^2*x)^3 + 4*2^9*x^3/(1-2^3*x)^4 +...

Links

Crossrefs

Cf. A136516, A171801, A171799.

Programs

  • Mathematica
    Table[((n + 1)*2^n + 1)*(2^n + 1)^(n - 1), {n, 0, 15}] (* Wesley Ivan Hurt, Jan 19 2017 *)
  • PARI
    {a(n)=polcoeff(sum(m=0,n,(m+1)*2^(m^2)*x^m/(1-2^m*x+x*O(x^n))^(m+1)),n)}
  • PARI
    {a(n)=n!*polcoeff(sum(k=0, n, (k+1)*2^(k^2)*exp(2^k*x)*x^k/k!), n)}
  • PARI
    {a(n)=((n+1)*2^n+1)*(2^n+1)^(n-1)}

Formula

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

A014050 a(n) = (n^2+1)^n.

Original entry on oeis.org

1, 2, 25, 1000, 83521, 11881376, 2565726409, 781250000000, 318644812890625, 167619550409708032, 110462212541120451001, 89116503268220597577728, 86380562306022715087890625, 99045780329059370000000000000, 132595005683061140853836321918569
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A136516.

Programs

  • Mathematica
    Table[(n^2+1)^n,{n,0,20}] (* Harvey P. Dale, Sep 20 2023 *)

Formula

a(n) = Sum_{k=0..n} binomial(n,k)*n^(2*k). - Vaclav Kotesovec, Oct 16 2016

A171799 O.g.f.: Sum_{n>=0} 2^(n^2)*x^n/(1 - 2^n*x)^n.

Original entry on oeis.org

1, 2, 20, 648, 78608, 37949472, 74258600000, 589859028828288, 18957096840069579008, 2455889836782322072945152, 1278835681226410156250000000000, 2671465293024628033252951422140418048
Offset: 0

Views

Author

Paul D. Hanna, Jan 20 2010

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 20*x^2 + 648*x^3 + 78608*x^4 +...
A(x) = 1 + 2*x/(1-2*x) + 2^4*x^2/(1-2^2*x)^2 + 2^9*x^3/(1-2^3*x)^3 +...

Crossrefs

Cf. A171801, A171800, A136516.

Programs

  • PARI
    {a(n)=polcoeff(sum(m=0,n,2^(m^2)*x^m/(1-2^m*x+x*O(x^n))^m),n)}
  • PARI
    {a(n)=if(n==0,1,2^n*(2^n+1)^(n-1))}

Formula

a(n) = 2^n*(2^n + 1)^(n-1) for n>0 with a(0)=1.

A251657 a(n) = (2^n + 3)^n.

Original entry on oeis.org

1, 5, 49, 1331, 130321, 52521875, 90458382169, 662062621900811, 20248745068443234721, 2548385124666493326171875, 1305282261160894865367626964649, 2701607566979638625212777041914285051, 22497539334127167666989016452232087989410801, 751859086636251929847496735809485838154930419921875
Offset: 0

Views

Author

Paul D. Hanna, Jan 29 2015

Keywords

Comments

This is a special case of the more general statement:
Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n! = Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b) where F(x) = exp(x), q=2, m=1, b=3.

Examples

			E.g.f.: A(x) = 4^0 + 5^1*x + 7^2*x^2/2! + 11^3*x^3/3! + 19^4*x^4/4! + 35^5*x^5/5! + 67^6*x^6/6! + 131^7*x^7/7! +...+ (2^n+3)^n*x^n/n! +...
such that
A(x) = exp(3*x) + 2*exp(3*2*x) + 2^4*exp(3*4*x)*x^2/2! + 2^9*exp(3*8*x)*x^3/3! + 2^16*exp(3*16*x)*x^4/4! +...+ 2^(n^2)*exp(3*2^n*x)*x^n/n! +...

Crossrefs

Cf. A055601, A136516, A165327, A155810.

Programs

  • Mathematica
    Table[(2^n+3)^n,{n,0,20}] (* Harvey P. Dale, Mar 16 2016 *)
  • PARI
    {a(n,q=2,m=1,b=3) =( m*q^n + b)^n}
for(n=0,15,print1(a(n,q=2,m=1,b=3),", "))
  • PARI
    {a(n,q=2,m=1,b=3) = sum(k=0,n, binomial(n,k) * b^k * m^(n-k) * (q^n)^(n-k))}
for(n=0,15,print1(a(n,q=2,m=1,b=3),", "))
  • PARI
    {a(n,q=2,m=1,b=3) = n!*polcoeff(sum(k=0, n, m^k * q^(k^2) * exp(b*q^k*x +x*O(x^n)) * x^k/k!), n)}
for(n=0,15,print1(a(n,q=2,m=1,b=3),", "))
  • PARI
    {a(n,q=2,m=1,b=3) = polcoeff(sum(k=0, n, m^k * q^(k^2) * x^k / (1 - b*q^k*x  +x*O(x^n))^(k+1) ), n)}
for(n=0,15,print1(a(n,q=2,m=1,b=3),", "))

Formula

E.g.f.: Sum_{n>=0} 2^(n^2) * exp(3*2^n*x) * x^n/n! = Sum_{n>=0} (2^n + 3)^n * x^n/n!.
O.g.f.: Sum_{n>=0} 2^(n^2) * x^n / (1 + 3*2^n*x)^(n+1).
a(n) = Sum_{k=0..n} binomial(n, k) * 3^k * (2^n)^(n-k).
a(n) = Sum_{k=0..n} A155810(k)*3^k.
Showing 1-10 of 18 results. Next

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