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.

Previous Showing 11-17 of 17 results.

A356564 Expansion of e.g.f. ( Product_{k>0} (1+x^k)^(1/k) )^x.

Original entry on oeis.org

1, 0, 2, 0, 28, -30, 888, -1260, 51728, -196560, 5293080, -22286880, 710229408, -4851269280, 138348035616, -1091188098000, 36482139114240, -379928382462720, 11812558481332992, -137793570801143040, 4609972759421554560, -67292912045817561600
Offset: 0

Views

Author

Seiichi Manyama, Aug 12 2022

Keywords

Crossrefs

Cf. A356565, A356566.
Cf. A007113, A048272, A338814, A356392.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^(1/k))^x))
  • PARI
    a048272(n) = sumdiv(n, d, (-1)^(n/d+1));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*a048272(j-1)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1, a(1) = 0; a(n) = Sum_{k=2..n} k! * A048272(k-1)/(k-1) * binomial(n-1,k-1) * a(n-k).

A356565 Expansion of e.g.f. ( Product_{k>0} (1+x^k) )^x.

Original entry on oeis.org

1, 0, 2, 3, 44, 90, 2034, 9240, 168944, 951048, 24042600, 185387400, 4411634952, 44020650960, 1166597641104, 14101322278680, 399099955203840, 5522583764698560, 169123038510919104, 2779010889700890240, 87888034148774728320, 1637061268780618450560
Offset: 0

Views

Author

Seiichi Manyama, Aug 12 2022

Keywords

Crossrefs

Cf. A356564, A356566.
Cf. A000593, A007113, A356393.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[(1 + x^k)^x, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 17 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, 1+x^k)^x))
  • PARI
    a000593(n) = sumdiv(n, d, (-1)^(n/d+1)*d);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*a000593(j-1)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1, a(1) = 0; a(n) = Sum_{k=2..n} k! * A000593(k-1)/(k-1) * binomial(n-1,k-1) * a(n-k).

A356566 Expansion of e.g.f. ( Product_{k>0} (1+x^k)^k )^x.

Original entry on oeis.org

1, 0, 2, 9, 92, 510, 7074, 68040, 1002224, 12529944, 228706920, 3565888920, 71035245192, 1348127454960, 30270949077264, 661700017709640, 16516072112482560, 408336559236083520, 11204399270843020224, 309489391954850336640, 9258803420755891835520
Offset: 0

Views

Author

Seiichi Manyama, Aug 12 2022

Keywords

Crossrefs

Cf. A356564, A356565.
Cf. A007113, A078306, A356394.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^k)^x))
  • PARI
    a078306(n) = sumdiv(n, d, (-1)^(n/d+1)*d^2);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*a078306(j-1)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1, a(1) = 0; a(n) = Sum_{k=2..n} k! * A078306(k-1)/(k-1) * binomial(n-1,k-1) * a(n-k).

A073478 Expansion of (1+x)^(1/(1-x)).

Original entry on oeis.org

1, 1, 2, 9, 44, 290, 2154, 19026, 186752, 2070792, 25119720, 334960560, 4824346152, 75100568088, 1250180063664, 22235660291880, 419595248663040, 8388866239417920, 176823515257447104, 3923498370610292544
Offset: 0

Views

Author

Vladeta Jovovic, Aug 26 2002

Keywords

Examples

			E.g.f.: (1+x)^(1/(1-x)) = 1 + x + 2*x^2/2! + 9*x^3/3! + 44*x^4/4! + 290*x^5/5! + 2154*x^6/6! + 19026*x^7/7! + 186752*x^8/8! + 2070792*x^9/9! + ...
which may be written as
(1+x)^(1/(1-x)) = exp(x + x^2*(1+x)/2 + x^3*(1+x+x^2)/3 + x^4*(1+x+x^2+x^3)/4 + x^5*(1+x+x^2+x^3+x^4)/5 + ... + x^n*((1-x^n)/(1-x))/n + ...).

Links

Crossrefs

Cf. A007113, A007120, A087761.

Programs

  • Mathematica
    CoefficientList[Series[(1+x)^(1/(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Apr 21 2014 *)
  • PARI
    {a(n)=n!*polcoeff((1+x +x*O(x^n))^(1/(1-x)),n)} \\ Paul D. Hanna, Jan 08 2014
  • PARI
    {a(n)=local(A);A=exp(sum(m=1,n,sum(k=1,m,-(-1)^k/k)*x^m)+x*O(x^n)); n!*polcoeff(A,n)} \\ Paul D. Hanna, Jan 08 2014

Formula

E.g.f.: exp( Sum_{n>=1} x^n * Sum_{k=1..n} -(-1)^k/k ). - Paul D. Hanna, Jan 08 2014
E.g.f.: exp( Sum_{n>=1} x^n * ((1-x^n)/(1-x)) / n ). - Paul D. Hanna, Nov 24 2024
a(n) ~ (log(2))^(1/4) * exp(2*sqrt(n*log(2)) - n - 1/2) * n^(n-1/4). - Vaclav Kotesovec, Apr 21 2014

Extensions

More terms from Robert G. Wilson v, Aug 28 2002

A073479 Expansion of e.g.f.: (1-x)^(-1-x).

Original entry on oeis.org

1, 1, 4, 15, 80, 490, 3534, 28938, 266048, 2710440, 30311640, 369127440, 4862219592, 68881435896, 1044331262688, 16872336545400, 289380447338880, 5251237965683520, 100519388543098944, 2024241909160239936, 42780009017657888640, 946724781741392908800
Offset: 0

Views

Author

Vladeta Jovovic, Aug 26 2002

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 15*x^3/3! + 80*x^4/4! + 490*x^5/5! +...
Explicit expressions for the e.g.f.:
(1-x)^(-1-x) = 1 + (1+x)*x + (1+x)(2+x)*x^2/2! + (1+x)(2+x)(3+x)*x^3/3! +... - _Paul D. Hanna_, Nov 01 2010
(1-x)^(-1-x) = exp(x + 3*x^2/2 + 5*x^3/6 + 7*x^4/12 + 9*x^5/20 + 11*x^6/30 +...). - _Paul D. Hanna_, Sep 27 2014

Links

Crossrefs

Cf. A007113, A007120.

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1-x)^(-1-x) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 30 2018
  • Maple
    S:= series((1-x)^(-1-x),x,51):
seq(coeff(S,x,j)*j!,j=0..50); # Robert Israel, Apr 20 2017
  • Mathematica
    CoefficientList[ Series[(1 - x)^(-1 - x), {x, 0, 19}], x]*Table[(n - 1)!, {n, 1, 20}]
  • PARI
    {a(n)=n!*polcoeff(sum(m=0,n,prod(k=1,m,k+x)*x^m/m!)+x*O(x^n),n)} \\ Paul D. Hanna, Nov 01 2010
  • PARI
    {a(n)=n!*polcoeff((1-x+x*O(x^n))^(-1-x),n)} \\ Paul D. Hanna, Nov 01 2010

Formula

(1-x)^(-1-x) = Sum_{n>=0} (Product_{k=1..n} (k+x)) * x^n/n!. [Paul D. Hanna, Nov 01 2010]
E.g.f.: (1-x)^(-1-x) = 1+(x*(1+x))/(Q(0)-x*(1+x)); Q(k)=(1+x)*k+1+x+(x^2)-x*(k+1)*(k+2+x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 27 2011
E.g.f.: 1 + x*(Q(0) - 1)/(x-1) where Q(k) = 1 - (1+x/(k+1))/(1 - x/(x - 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 05 2013
a(n) ~ n! * (n - log(n) + 1 - gamma), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Sep 29 2013
E.g.f.: exp( x + Sum_{n>=2} (2*n-1) * x^n / (n*(n-1)) ). - Paul D. Hanna, Sep 27 2014

A347726 Expansion of e.g.f.: exp(x / (1-x)^x).

Original entry on oeis.org

1, 1, 1, 7, 37, 221, 1801, 15709, 157641, 1775521, 21898801, 296379931, 4346295757, 68682481141, 1163591541113, 21024039322441, 403559222086801, 8199021870113985, 175746277620520417, 3963144157484213359, 93778090395227788021, 2323080237693908254381
Offset: 0

Views

Author

Seiichi Manyama, Sep 11 2021

Keywords

Crossrefs

Cf. A007113, A202152.

Programs

  • Mathematica
    a[0] = 1; a[n_] := n! * Sum[(-1)^(n - i) * Sum[i^j * StirlingS1[n - i - j, j]/(n - i - j)!, {j, 0, n - i}]/i!, {i, 1, n}]; Array[a, 20, 0] (* Amiram Eldar, Sep 11 2021 *)
  • PARI
    a(n) = n!*sum(i=0, n, (-1)^(n-i)*sum(j=0, n-i, i^j*stirling(n-i-j, j, 1)/(n-i-j)!)/i!);
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^x)))

Formula

a(n) = n! * Sum_{i=0..n} (-1)^(n-i) * ( Sum_{j=0..n-i} i^j * Stirling1(n-i-j,j)/(n-i-j)! )/i!.

A354613 Expansion of e.g.f. 1/(2 - (1 + x)^x).

Original entry on oeis.org

1, 0, 2, -3, 44, -210, 2694, -23520, 330672, -4168584, 67622040, -1095648840, 20621674776, -403514963280, 8734659594192, -199049377658040, 4894304369356800, -126907901533425600, 3501394314254828352, -101643840316833194880, 3112491474764866339200
Offset: 0

Views

Author

Seiichi Manyama, Jul 08 2022

Keywords

Crossrefs

Cf. A007113, A354612.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(2-(1+x)^x)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j!*sum(k=0, j\2, stirling(j-k, k, 1)/(j-k)!)*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A007113(k) * binomial(n,k) * a(n-k).
Previous Showing 11-17 of 17 results.

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