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.

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).

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

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.

A264871 Array read by antidiagonals: T(n,m) = (1+2^n)^m; n,m>=0.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 9, 5, 1, 16, 27, 25, 9, 1, 32, 81, 125, 81, 17, 1, 64, 243, 625, 729, 289, 33, 1, 128, 729, 3125, 6561, 4913, 1089, 65, 1, 256, 2187, 15625, 59049, 83521, 35937, 4225, 129, 1, 512, 6561, 78125, 531441, 1419857, 1185921, 274625, 16641, 257
Offset: 0

Views

Author

R. J. Mathar, Nov 27 2015

Keywords

Examples

			       1,       2,       4,       8,      16,      32,
       1,       3,       9,      27,      81,     243,
       1,       5,      25,     125,     625,    3125,
       1,       9,      81,     729,    6561,   59049,
       1,      17,     289,    4913,   83521, 1419857,
       1,      33,    1089,   35937, 1185921,39135393,

Crossrefs

Cf. A000079 (row 0), A000244 (row 1), A000351 (row 2), A001019 (row 3), A001026 (row 4), A009977 (row 5), A000051 (column 1), A028400 (column 2), A136516 (main diagonal), A165327 (upper subdiagonal).

Programs

  • Mathematica
    Reverse /@ Table[(1 + 2^(n - m))^m, {n, 0, 9}, {m, 0, n}] // Flatten (* Michael De Vlieger, Nov 27 2015 *)

Formula

G.f. for row n: 1/(1-(1+2^n)*x). - R. J. Mathar, Dec 15 2015

A337851 a(n) = (2^n + 2)^n.

Original entry on oeis.org

1, 4, 36, 1000, 104976, 45435424, 82653950016, 627485170000000, 19631688197463081216, 2504194578379511247798784, 1292628144912333835229805413376, 2687153475176994340820312500000000000, 22431765115399782718874449007331506546282496
Offset: 0

Views

Author

Paul D. Hanna, Sep 26 2020

Keywords

Comments

In general, we have the o.g.f. 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 ; here, q=2, m=1, b=2.
In general, we have the e.g.f. 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! ; here, q=2, m=1, b=2.

Examples

			O.g.f.: A(x) = 1 + 4*x + 36*x^2 + 1000*x^3 + 104976*x^4 + 45435424*x^5 + 82653950016*x^6 + 627485170000000*x^7 + 19631688197463081216*x^8 + ...
where
A(x) = 1/(1 - 2*x) + 2*x/(1 - 2^2*x)^2 + 2^4*x^2/(1 - 2^3*x)^3 + 2^9*x^3/(1 - 2^4*x)^4 + 2^16*x^4/(1 - 2^5*x)^5 + 2^25*x^5/(1 - 2^6*x)^6 + ...

Crossrefs

Cf. A165327, A055601, A251657, A337852, A136516.

Programs

  • PARI
    {a(n,q,m,b) = (m*q^n + b)^n}
for(n=0,15, print1(a(n,q=2,m=1,b=2),", "))
  • PARI
    /* E.g.f. formula: */
{a(n,q,m,b) = 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=2),", "))
  • PARI
    /* E.g.f. formula: */
{a(n,q,m,b) = 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=2),", "))

Formula

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

A202990 E.g.f: Sum_{n>=0} 3^n * 2^(n^2) * exp(-2*2^n*x) * x^n/n!.

Original entry on oeis.org

1, 4, 100, 10648, 4477456, 7339040224, 47045881000000, 1186980379913527168, 118530511097526559703296, 47035767668340696232372862464, 74367598058372171073462490000000000, 469253945833810205185008441288962454059008
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! + 10648*x^3/3! + 4477456*x^4/4! +..
By the series identity, the e.g.f.:
A(x) = exp(-2*x) + 3*2*exp(-2*2*x)*x + 3^2*2^4*exp(-2*2^2*x)*x^2/2! + 3^3*2^9*exp(-2*2^3*x)*x^3/3! +...
expands into:
A(x) = 1 + 4*x + 10^2*x^2/2! + 22^3*x^3/3! + 46^4*x^4/4! + 94^5*x^5/5! +...+ (3*2^n-2)^n*x^n/n! +...

Crossrefs

Cf. A180602, A165327, A202989, A060613, A055601.

Programs

  • Mathematica
    Table[(3*2^n-2)^n,{n,0,12}] (* Harvey P. Dale, Jul 16 2023 *)
  • PARI
    {a(n, q=2, m=3, b=-2)=(m*q^n + b)^n}
  • PARI
    {a(n, q=2, m=3, b=-2)=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=2, m=3, b=-2)=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*2^n - 2)^n.
O.g.f.: Sum_{n>=0} 3^n * 2^(n^2) * x^n/(1 + 2*2^n*x)^(n+1).

A202991 E.g.f: Sum_{n>=0} 3^(n^2) * exp(-2*3^n*x) * x^n/n!.

Original entry on oeis.org

1, 1, 49, 15625, 38950081, 812990017201, 147640825624179889, 237771659632917369765625, 3425319186561140076700951192321, 443021141828981570872668681812345111521, 515202988063835984513918825523304657054713360049
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 + x + 49*x^2/2! + 15625*x^3/3! + 38950081*x^4/4! +...
By the series identity, the g.f.:
A(x) = exp(-2*x) + 3*exp(-2*3*x)*x + 3^4*exp(-2*3^2*x)*x^2/2! + 3^9*exp(-2*3^3*x)*x^3/3! + 3^16*exp(-2*3^4*x)*x^4/4! +...
expands into:
A(x) = 1 + x + 7^2*x^2/2! + 25^3*x^3/3! + 79^4*x^4/4! + 241^5*x^5/5! +...+ (3^n-2)^n*x^n/n! +...

Crossrefs

Cf. A180602, A165327, A202990, A202989, A060613, A055601.

Programs

  • PARI
    {a(n, q=3, m=1, b=-2)=(m*q^n + b)^n}
  • PARI
    {a(n, q=3, m=1, b=-2)=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=-2)=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 - 2)^n.
O.g.f.: Sum_{n>=0} 3^(n^2)*x^n/(1 + 2*3^n*x)^(n+1).
Showing 1-7 of 7 results.

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