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.

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

A181075 a(n) = Sum_{k=0..n-1} C(n-1,k)^(k+1) * n/(n-k).

Original entry on oeis.org

1, 3, 10, 71, 1026, 30912, 2219946, 339460991, 112986526834, 91234232847938, 161113616883239406, 619495336824891912596, 5839092706931985694730356, 124192664709851995516427897172, 5681764626723349386531457243004370
Offset: 1

Views

Author

Paul D. Hanna, Oct 02 2010

Keywords

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 71*x^4/4 + 1026*x^5/5 + ...
which equals the series:
  L(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 + ...
Exponentiation yields the g.f. of A181074:
  exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 23*x^4 + 231*x^5 + 5405*x^6 + ...

Links

Crossrefs

Cf. A181076 (exp), variants: A181077, A181079.

Programs

  • Magma
    [(&+[Binomial(n-1, k)^(k+1)*n/(n-k): k in [0..n-1]]): n in [1..25]]; // G. C. Greubel, Apr 05 2021
  • Mathematica
    Table[Sum[Binomial[n-1, k]^(k+1)*n/(n-k), {k,0,n-1}], {n,25}] (* G. C. Greubel, Apr 05 2021 *)
  • PARI
    {a(n)=sum(k=0, n-1, binomial(n-1, k)^(k+1)*n/(n-k))}
  • PARI
    {a(n)=n*polcoeff(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)}
  • Sage
    [sum(binomial(n-1,k)^(k+1)*n/(n-k) for k in (0..n-1)) for n in (1..25)] # G. C. Greubel, Apr 05 2021

Formula

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

A181077 a(n) = Sum_{k=0..n-1} C(n-1,k)^(n-k) * n/(n-k).

Original entry on oeis.org

1, 3, 10, 59, 726, 20832, 1405566, 202357171, 66675848266, 52415395776938, 88820554918533846, 339849475991699902472, 3175234567292428864024192, 65420235446121559438182151848, 2970041251569931717805628420162750
Offset: 1

Views

Author

Paul D. Hanna, Oct 02 2010

Keywords

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 59*x^4/4 + 726*x^5/5 +...
which equals the series:
  L(x) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)*x
  + (1 + 2^2*x +  3^2*x^2 +  4^2*x^3 +   5^2*x^4 +   6^2*x^5 + ...)*x^2/2
  + (1 + 3^3*x +  6^3*x^2 + 10^3*x^3 +  15^3*x^4 +  21^3*x^5 + ...)*x^3/3
  + (1 + 4^4*x + 10^4*x^2 + 20^4*x^3 +  35^4*x^4 +  56^4*x^5 + ...)*x^4/4
  + (1 + 5^5*x + 15^5*x^2 + 35^5*x^3 +  70^5*x^4 + 126^5*x^5 + ...)*x^5/5
  + (1 + 6^6*x + 21^6*x^2 + 56^6*x^3 + 126^6*x^4 + 252^6*x^5 + ...)*x^6/6
  + (1 + 7^7*x + 28^7*x^2 + 84^7*x^3 + 210^7*x^4 + 462^7*x^5 + ...)*x^7/7 + ...
Exponentiation yields the g.f. of A181076:
  exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 20*x^4 + 168*x^5 + 3659*x^6 + ... + A181076(n)*x^n + ...

Links

Crossrefs

Cf. A181076 (exp), variants: A181073, A181075, A181079.

Programs

  • Magma
    [(&+[Binomial(n-1, k)^(n-k)*n/(n-k): k in [0..n-1]]): n in [1..25]]; // G. C. Greubel, Apr 05 2021
  • Mathematica
    Table[Sum[Binomial[n-1, k]^(n-k)*n/(n-k), {k,0,n-1}], {n,25}] (* G. C. Greubel, Apr 05 2021 *)
  • PARI
    {a(n)=sum(k=0, n-1, binomial(n-1, k)^(n-k)*n/(n-k))}
  • PARI
    {a(n)=n*polcoeff(sum(m=1, n, sum(k=0, n, binomial(m+k-1,k)^m*x^k)*x^m/m)+x*O(x^n), n)}
  • Sage
    [sum(binomial(n-1,k)^(n-k)*n/(n-k) for k in (0..n-1)) for n in (1..25)] # G. C. Greubel, Apr 05 2021

Formula

L.g.f.: L(x) = Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^n *x^k ] *x^n/n.
Logarithmic derivative of A181076.

A181078 Expansion of g.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^(n+k-1) *x^k ] *x^n/n ).

Original entry on oeis.org

1, 1, 2, 5, 29, 657, 61207, 22168009, 29875987984, 155804714312491, 3016989471632014921, 229552430038667549657248, 64995077386747098368845127628, 73163996832774559516266954450479682
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2010

Keywords

Comments

Conjecture: this sequence consists entirely of integers.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 29*x^4 + 657*x^5 + 61207*x^6 +...
The logarithm begins:
  log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 95*x^4/4 + 3126*x^5/5 + 363132*x^6/6 + ... + A181079(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 + 3^3*x +  6^4*x^2 + 10^5*x^3 +   15^6*x^4 +   21^7*x^5 + ...)*x^3/3
  + (1 + 4^4*x + 10^5*x^2 + 20^6*x^3 +   35^7*x^4 +   56^8*x^5 + ...)*x^4/4
  + (1 + 5^5*x + 15^6*x^2 + 35^7*x^3 +   70^8*x^4 +  126^9*x^5 + ...)*x^5/5
  + (1 + 6^6*x + 21^7*x^2 + 56^8*x^3 +  126^9*x^4 + 252^10*x^5 + ...)*x^6/6
  + (1 + 7^7*x + 28^8*x^2 + 84^9*x^3 + 210^10*x^4 + 462^11*x^5 + ...)*x^7/7 + ...

Links

Crossrefs

Cf. A181079 (log), variants: A181070, A181074, A181076.

Programs

  • Magma
    m:=30;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( Exp( (&+[ (&+[ Binomial(n+k-1,k)^(n+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=20}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n+k-1,k]^(n+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)^(m+k-1)*x^k)*x^m/m)+x*O(x^n)), n)}
  • Sage
    m=30;
def A181078_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( exp( sum( sum( binomial(n+k-1,k)^(n+k-1)*x^(n+k)/n for k in (0..m+2) ) for n in (1..m+1)) ) ).list()
A181078_list(m) # G. C. Greubel, Apr 05 2021

A181079 a(n) = Sum_{k=0..n-1} binomial(n-1,k)^(n-1) * n/(n-k).

Original entry on oeis.org

1, 3, 10, 95, 3126, 363132, 154742736, 238830058287, 1401973344195850, 30168336369959767298, 2525043541826640689536056, 779938173975597096091742711900, 951131113887078985926203597341181404
Offset: 1

Views

Author

Paul D. Hanna, Oct 03 2010

Keywords

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 95*x^4/4 + 3126*x^5/5 + ...
which equals the series:
L(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 + 3^3*x + 6^4*x^2 + 10^5*x^3 + 15^6*x^4 + 21^7*x^5 + ...)*x^3/3
+ (1 + 4^4*x + 10^5*x^2 + 20^6*x^3 + 35^7*x^4 + 56^8*x^5 + ...)*x^4/4
+ (1 + 5^5*x + 15^6*x^2 + 35^7*x^3 + 70^8*x^4 + 126^9*x^5 + ...)*x^5/5
+ (1 + 6^6*x + 21^7*x^2 + 56^8*x^3 + 126^9*x^4 + 252^10*x^5 + ...)*x^6/6
+ (1 + 7^7*x + 28^8*x^2 + 84^9*x^3 + 210^10*x^4 + 462^11*x^5 + ...)*x^7/7 + ...
Exponentiation yields the g.f. of A181078:
exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 29*x^4 + 657*x^5 + 61207*x^6 + … + A181078(n)*x^n + ...

Links

Crossrefs

Cf. A181078 (exp), variants: A181071, A181075, A181077.

Programs

  • Magma
    [(&+[Binomial(n-1,j)^(n-1)*(n/(n-j)): j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Apr 04 2021
  • Mathematica
    Table[Sum[Binomial[n-1,k]^(n-1) n/(n-k),{k,0,n-1}],{n,20}] (* Harvey P. Dale, Jun 13 2013 *)
  • PARI
    {a(n)=sum(k=0, n-1, binomial(n-1, k)^(n-1)*n/(n-k))}
  • PARI
    {a(n)=n*polcoeff(sum(m=1, n, sum(k=0, n, binomial(m+k-1,k)^(m+k-1)*x^k)*x^m/m)+x*O(x^n), n)}
  • Sage
    [sum( binomial(n-1, k)^(n-1)*(n/(n-k)) for k in (0..n-1)) for n in (1..20)] # G. C. Greubel, Apr 04 2021

Formula

L.g.f.: L(x) = Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^(n+k-1)*x^k ] *x^n/n.
Logarithmic derivative of A181076.
Showing 1-5 of 5 results.

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