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 1-5 of 5 results.

A181070 Expansion of G.f.: A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^(k+1)*x^k)*x^n/n ).

Original entry on oeis.org

1, 1, 2, 4, 8, 23, 88, 379, 3044, 32116, 379279, 9160509, 237458908, 7651718328, 495105710770, 29747390685988, 2718143583980173, 436044028162542425, 61494671526637653928, 16346049663440380567782, 6106008029903796482509688
Offset: 0

Views

Author

Paul D. Hanna, Oct 02 2010

Keywords

Comments

Conjecture: this sequence consists entirely of integers.
Note that the following g.f. does NOT yield an integer series:
exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^k * x^k) * x^n/n ).

Examples

			G.f. A(x) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 23*x^5 + 88*x^6 + ...
The logarithm of g.f. A(x) begins:
  log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 15*x^4/4 + 66*x^5/5 + 357*x^6/6 + 1891*x^7/7    + ... + A181071(n)*x^n/n + ...
and equals the series:
  log(A(x)) = (1 + x)*x + (1 + 2^2*x + x^2)*x^2/2
  + (1 + 3^2*x +  3^3*x^2 +      x^3)*x^3/3
  + (1 + 4^2*x +  6^3*x^2 +  4^4*x^3 +      x^4)*x^4/4
  + (1 + 5^2*x + 10^3*x^2 + 10^4*x^3 +  5^5*x^4 +     x^5)*x^5/5
  + (1 + 6^2*x + 15^3*x^2 + 20^4*x^3 + 15^5*x^4 + 6^6*x^5 + x^6)*x^6/6 + ...

Links

Crossrefs

Cf. A181071(log), variants: A181072, A181074, A181080.

Programs

  • Magma
    m:=30;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( Exp( (&+[ (&+[ Binomial(n,k)^(k+1)*x^(n+k)/n : k in [0..m+2]]): n in [1..m+1]]) ) )); // G. C. Greubel, Apr 05 2021
  • Mathematica
    With[{m=30}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n, k]^(k+1)*x^(n+k)/n, {k, 0, m+2}], {n, m+1}]], {x,0,m}], x]] (* G. C. Greubel, Apr 05 2021 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m,k)^(k+1)*x^k)*x^m/m)+x*O(x^n)), n)}
  • Sage
    m=30;
def A181070_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( exp( sum( sum( binomial(n,k)^(k+1)*x^(n+k)/n for k in (0..m+2) ) for n in (1..m+1)) ) ).list()
A181070_list(m) # G. C. Greubel, Apr 05 2021

A181074 Expansion of g.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^(k+1) *x^k ] *x^n/n ).

Original entry on oeis.org

1, 1, 2, 5, 23, 231, 5405, 322799, 42761356, 12597156231, 9136063939651, 14655841196011960, 51639276405198967750, 449212631407010945983244, 8871353886432410987179493370, 378793180251425841753491012596531
Offset: 0

Views

Author

Paul D. Hanna, Oct 02 2010

Keywords

Comments

Conjecture: this sequence consists entirely of integers.
Note that the following g.f. does NOT yield an integer series:
exp( Sum_{n>=1} [Sum_{k>=0} C(n+k-1,k)^k * x^k] * x^n/n ).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 23*x^4 + 231*x^5 + 5405*x^6 +...
The logarithm begins:
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 71*x^4/4 + 1026*x^5/5 + 30912*x^6/6 +...+ A181075(n)*x^n/n +...
which equals the series:
  log(A(x)) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)*x
  + (1 + 2^2*x +  3^3*x^2 +  4^4*x^3 +   5^5*x^4 +   6^6*x^5 + ...)*x^2/2
  + (1 + 3^2*x +  6^3*x^2 + 10^4*x^3 +  15^5*x^4 +  21^6*x^5 + ...)*x^3/3
  + (1 + 4^2*x + 10^3*x^2 + 20^4*x^3 +  35^5*x^4 +  56^6*x^5 + ...)*x^4/4
  + (1 + 5^2*x + 15^3*x^2 + 35^4*x^3 +  70^5*x^4 + 126^6*x^5 + ...)*x^5/5
  + (1 + 6^2*x + 21^3*x^2 + 56^4*x^3 + 126^5*x^4 + 252^6*x^5 + ...)*x^6/6
  + (1 + 7^2*x + 28^3*x^2 + 84^4*x^3 + 210^5*x^4 + 462^6*x^5 + ...)*x^7/7 + ...

Links

Crossrefs

Variants: A181070, A181076, A181078, A181080.
Cf. A181075 (log).

Programs

  • Magma
    m:=30;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( Exp( (&+[ (&+[ Binomial(n+k-1,k)^(k+1)*x^(n+k)/n : k in [0..m+2]]): n in [1..m+1]]) ) )); // G. C. Greubel, Apr 05 2021
  • Mathematica
    With[{m=30}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n+k-1,k]^(k+1)*x^(n+k)/n, {k,0,m+2}], {n, m+1}]], {x,0,m}], x]] (* G. C. Greubel, Apr 05 2021 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, n, binomial(m+k-1,k)^(k+1)*x^k)*x^m/m)+x*O(x^n)), n)}
  • SageMath
    m=30;
def A181066_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( exp( sum( sum( binomial(n+k-1,k)^(k+1)*x^(n+k)/n for k in (0..m+2) ) for n in (1..m+1)) ) ).list()
A181066_list(m) # G. C. Greubel, Apr 05 2021

A181081 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^(n-2k*+1) * n/(n-k).

Original entry on oeis.org

1, 3, 7, 39, 336, 4077, 68461, 1955295, 129385141, 17371664728, 3501431925168, 947675920795833, 355261887514210899, 236156938257380344851, 390707976511340699319417, 1324768245535417597286345871
Offset: 1

Views

Author

Paul D. Hanna, Oct 02 2010

Keywords

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 39*x^4/4 + 336*x^5/5 + ...
which equals the series:
log(A(x)) = (1 + x)*x + (1 + 2^2*x + x^2)*x^2/2
+ (1+ 3^3*x + 3^2*x^2 + x^3)*x^3/3
+ (1+ 4^4*x + 6^3*x^2 + 4^2*x^3 + x^4)*x^4/4
+ (1+ 5^5*x + 10^4*x^2 + 10^3*x^3 + 5^2*x^4 + x^5)*x^5/5
+ (1+ 6^6*x + 15^5*x^2 + 20^4*x^3 + 15^3*x^4 + 6^2*x^5 + x^6)*x^6/6 + ...
Exponentiation yields the g.f. of A181080:
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 14*x^4 + 83*x^5 + 774*x^6 + 10641*x^7 + ...

Links

Crossrefs

Cf. A181080 (exp), variants: A181071, A166895.

Programs

  • Magma
    [(&+[Binomial(n-j,j)^(n-2*j+1)*(n/(n-j)): j in [0..Floor(n/2)]]): n in [1..20]]; // G. C. Greubel, Apr 04 2021
  • Mathematica
    Table[Sum[Binomial[n-k, k]^(n-2*k+1)*(n/(n-k)), {k, 0, Floor[n/2]}], {n, 20}] (* G. C. Greubel, Apr 04 2021 *)
  • PARI
    a(n)=sum(k=0, n\2, binomial(n-k, k)^(n-2*k+1)*n/(n-k))
  • PARI
    {a(n)=n*polcoeff(sum(m=1, n, sum(k=0, m, binomial(m,k)^(m-k+1)*x^k)*x^m/m)+x*O(x^n), n)}
  • Sage
    [sum( binomial(n-k, k)^(n-2*k+1)*(n/(n-k)) for k in (0..n//2)) for n in (1..20)] # G. C. Greubel, Apr 04 2021

Formula

L.g.f.: L(x) = Sum_{n>=1} ( Sum_{k=0..n} binomial(n,k)^(n-k+1)*x^k ) * x^n/n.
Logarithmic derivative of A181080.

A181082 Expansion of g.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^(n+k) * x^k] * x^n/n ).

Original entry on oeis.org

1, 1, 2, 6, 34, 375, 15200, 2066401, 450054919, 199271253643, 431399012916702, 2151987403947457136, 15451465958263071713102, 331187643758039140349444047, 33475597220485400781283541412048
Offset: 0

Views

Author

Paul D. Hanna, Oct 02 2010

Keywords

Comments

Conjecture: this sequence consists entirely of integers.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 34*x^4 + 375*x^5 + 15200*x^6 +...
The logarithm of g.f. A(x) begins:
  log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 111*x^4/4 + 1686*x^5/5 + 88737*x^6/6 + ... + A181083(n)*x^n/n + ...
and equals the series:
  log(A(x)) = (1 + x)*x + (1 + 2^3*x + x^2)*x^2/2
  + (1 + 3^4*x +  3^5*x^2 +      x^3)*x^3/3
  + (1 + 4^5*x +  6^6*x^2 +  4^7*x^3 +       x^4)*x^4/4
  + (1 + 5^6*x + 10^7*x^2 + 10^8*x^3 +   5^9*x^4 +      x^5)*x^5/5
  + (1 + 6^7*x + 15^8*x^2 + 20^9*x^3 + 15^10*x^4 + 6^11*x^5 + x^6)*x^6/6 + ...

Links

Crossrefs

Cf. A181083 (log), variants: A181080, A181084.

Programs

  • Magma
    m:=20;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( Exp( (&+[ (&+[ Binomial(n,k)^(n+k)*x^(n+k)/n : k in [0..n]]): n in [1..m+1]]) ) )); // G. C. Greubel, Apr 05 2021
  • Mathematica
    With[{m=20}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n, k]^(n+k)*x^(n+k)/n, {k, 0,n}], {n,m+1}]], {x,0,m}], x]] (* G. C. Greubel, Apr 05 2021 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m,k)^(m+k)*x^k)*x^m/m)+x*O(x^n)), n)}
  • Sage
    m=20;
def A181082_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( exp( sum( sum( binomial(n,k)^(n+k)*x^(n+k)/n for k in (0..n) ) for n in (1..m+1)) ) ).list()
A181082_list(m) # G. C. Greubel, Apr 05 2021

A181084 Expansion of g.f.: exp( Sum_{n>=1} [Sum_{k=0..n} binomial(n,k)^(n+k+1) * x^k] * x^n/n ).

Original entry on oeis.org

1, 1, 2, 10, 92, 1367, 87090, 20385333, 6633475836, 4096297538926, 14834973644512627, 119919823546238898903, 1273371038284317852447990, 41086272137585936052959008420, 6982122140549374036504235218052104
Offset: 0

Views

Author

Paul D. Hanna, Oct 28 2010

Keywords

Comments

Conjecture: this sequence consists entirely of integers.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 92*x^4 + 1367*x^5 + 87090*x^6 + ...
The logarithm of g.f. A(x) begins:
  log(A(x)) = x + 3*x^2/2 + 25*x^3/3 + 327*x^4/4 + 6336*x^5/5 + 513657*x^6/6 + ... + A181085(n)*x^n/n + ...
and equals the series:
  log(A(x)) = (1 + x)*x + (1 + 2^4*x + x^2)*x^2/2
  + (1 + 3^5*x +  3^6*x^2 +       x^3)*x^3/3
  + (1 + 4^6*x +  6^7*x^2 +   4^8*x^3 +       x^4)*x^4/4
  + (1 + 5^7*x + 10^8*x^2 +  10^9*x^3 +  5^10*x^4 +      x^5)*x^5/5
  + (1 + 6^8*x + 15^9*x^2 + 20^10*x^3 + 15^11*x^4 + 6^12*x^5 + x^6)*x^6/6 + ...

Links

Crossrefs

Cf. A181085 (log), variants: A181080, A181082.

Programs

  • Magma
    m:=20;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( Exp( (&+[ (&+[ Binomial(n,k)^(n+k+1)*x^(n+k)/n : k in [0..n]]): n in [1..m+1]]) ) )); // G. C. Greubel, Apr 05 2021
  • Mathematica
    With[{m=20}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n, k]^(n+k+1)*x^(n+k)/n, {k,0,n}], {n, m+1}]], {x,0,m}], x]] (* G. C. Greubel, Apr 05 2021 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m,k)^(m+k+1)*x^k)*x^m/m) + x*O(x^n)), n)}
  • Sage
    m=20;
def A181084_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( exp( sum( sum( binomial(n,k)^(n+k+1)*x^(n+k)/n for k in (0..n) ) for n in (1..m+1)) ) ).list()
A181084_list(m) # G. C. Greubel, Apr 05 2021
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