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

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.

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.

A181083 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^n * n/(n-k).

Original entry on oeis.org

1, 3, 13, 111, 1686, 88737, 14355265, 3583775847, 1789371713317, 4311992850152298, 23667113846872049808, 185391762466214524964649, 4305238471804328835068596175, 468653724243371951619336632177235
Offset: 1

Views

Author

Paul D. Hanna, Oct 02 2010

Keywords

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 13*x^3/3 + 111*x^4/4 + 1686*x^5/5 + ...
which equals the series:
  L(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 + ...
Exponentiation yields the g.f. of A181082:
  exp(L(x)) = 1 + x + 2*x^2 + 6*x^3 + 34*x^4 + 375*x^5 + 15200*x^6 + 2066401*x^7 + ...

Links

Crossrefs

Variants: A166895, A181071, A181081.
Cf. A181082 (exp).

Programs

  • Magma
    [(&+[Binomial(n-k,k)^n*(n/(n-k)): k in [0..Floor(n/2)]]): n in [1..25]]; // G. C. Greubel, Apr 05 2021
  • Maple
    A181083:= n-> add(binomial(n-k,k)^n*(n/(n-k)), k=0..floor(n/2)); seq(A181083(n), n=1..20); # G. C. Greubel, Apr 05 2021
  • Mathematica
    Table[Sum[Binomial[n-k,k]^n n/(n-k),{k,0,Floor[n/2]}],{n,20}] (* Harvey P. Dale, Jun 24 2015 *)
  • PARI
    a(n)=sum(k=0, n\2, binomial(n-k, k)^n*n/(n-k))
  • PARI
    {a(n)=n*polcoeff(sum(m=1, n, sum(k=0, m, binomial(m,k)^(m+k)*x^k)*x^m/m)+x*O(x^n), n)}
  • Sage
    [sum(binomial(n-k,k)^n*(n/(n-k)) for k in (0..n//2)) for n in (1..25)] # G. C. Greubel, Apr 05 2021

Formula

L.g.f.: L(x) = Sum_{n>=1} [Sum_{k=0..n} binomial(n,k)^(n+k)*x^k] * x^n/n.
Logarithmic derivative of A181082.

A181085 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^(n+1) * n/(n-k).

Original entry on oeis.org

1, 3, 25, 327, 6336, 513657, 142074241, 52903930911, 36806786795365, 148308705637730728, 1318954828711012426638, 15279013243159345043036553, 534104982404807772659968455891, 97749134742042348389685885848315523
Offset: 1

Views

Author

Paul D. Hanna, Oct 28 2010

Keywords

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 25*x^3/3 + 327*x^4/4 + 6336*x^5/5 + ...
which equals the series:
L(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 + ...
Exponentiation yields the g.f. of A181084:
exp(L(x)) = 1 + x + 2*x^2 + 10*x^3 + 92*x^4 + 1367*x^5 + 87090*x^6 + ...

Links

Crossrefs

Variants: A166895, A181071, A181081, A181083.
Cf. A181084 (exp).

Programs

  • Magma
    [(&+[Binomial(n-j,j)^(n+1)*(n/(n-j)): j in [0..Floor(n/2)]]): j in [1..20]]; // G. C. Greubel, Apr 04 2021
  • Mathematica
    Table[Sum[Binomial[n-k, k]^(n+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+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+1)*(n/(n-k)) for k in (0..n//2)) for n in (1..20)] # G. C. Greubel, Apr 04 2021
Showing 1-5 of 5 results.

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