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 1-6 of 6 results.

A193287 E.g.f.: A(x) = 1/(1 - 2*x^2)^(1 + 1/(2*x)).

Original entry on oeis.org

1, 1, 5, 19, 145, 981, 10141, 98575, 1289569, 16314121, 258568021, 4023553931, 74961787825, 1383475135069, 29636315118957, 632414472704071, 15316605861040321, 370875832116841105, 10021723060544059429, 271409166367070755843
Offset: 0

Views

Author

Paul D. Hanna, Jul 21 2011

Keywords

Comments

More generally, we have the identity:
Sum_{n>=0} (x^n/n!)*Product_{k=1..n} (1+k*y) = 1/(1 - x*y)^(1 + 1/y); here y=2*x.

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 19*x^3/3! + 145*x^4/4! + 981*x^5/5! +...
where A(x) satisfies:
A(x)^(2*x/(1+2*x)) = 1 + 2*x^2 + 4*x^4 + 8*x^6 + 16*x^8 + 32*x^10 +...
Also,
A(x) = 1 + x*(1+2*x) + x^2*(1+2*x)*(1+4*x)/2! + x^3*(1+2*x)*(1+4*x)*(1+6*x)/3! + x^4*(1+2*x)*(1+4*x)*(1+6*x)*(1+8*x)/4! +...
The logarithm begins:
log(A(x)) = x + 2*x^2 + 2*x^3/2 + 4*x^4/2 + 4*x^5/3 + 8*x^6/3 + 8*x^7/4 +...
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * 2^floor(k/2)/floor((k+1)/2) * a(n-k)/(n-k)!. - _Seiichi Manyama_, Apr 30 2022

Links

Crossrefs

Cf. A193281, A193288, A193289, A193290.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-2*x^2)^(1+1/(2*x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 25 2013 *)
  • PARI
    {a(n)=n!*polcoeff(1/(1 - 2*x^2 +x^2*O(x^n))^((1+2*x)/(2*x)),n)}
  • PARI
    {a(n)=n!*polcoeff(sum(m=0,n,x^m/m!*prod(k=1,m,1+2*k*x+x*O(x^n))),n)}
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, j*2^(j\2)/((j+1)\2)*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Apr 30 2022

Formula

E.g.f.: A(x) = Sum_{n>=0} x^n/n! * Product_{k=1..n} (1 + 2*k*x).
a(n) ~ n! * 2^(n/2-1/2-1/sqrt(2))*n^(1/sqrt(2))/Gamma(1/sqrt(2)). - Vaclav Kotesovec, Jun 25 2013

A193288 E.g.f.: A(x) = 1/(1 - 3*x^2)^(1 + 1/(3*x)).

Original entry on oeis.org

1, 1, 7, 28, 289, 2131, 29161, 316072, 5395993, 77326165, 1583326171, 28229026156, 674412621697, 14384156661343, 392879390385301, 9753823992141496, 299849358712509361, 8492478062686906057, 290226665437376352463, 9233909417529486840412
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2011

Keywords

Comments

More generally, we have the identity:
Sum_{n>=0} (x^n/n!)*Product_{k=1..n} (1+k*y) = 1/(1 - x*y)^(1 + 1/y); here y=3*x.

Examples

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 28*x^3/3! + 289*x^4/4! + 2131*x^5/5! +...
where A(x) satisfies:
A(x)^(3*x/(1+3*x)) = 1 + 3*x^2 + 9*x^4 + 27*x^6 + 81*x^8 + 243*x^10 +...
Also,
A(x) = 1 + x*(1+3*x) + x^2*(1+3*x)*(1+6*x)/2! + x^3*(1+3*x)*(1+6*x)*(1+9*x)/3! + x^4*(1+3*x)*(1+6*x)*(1+9*x)*(1+12*x)/4! +...
The logarithm begins:
log(A(x)) = x + 3*x^2 + 3*x^3/2 + 9*x^4/2 + 9*x^5/3 + 27*x^6/3 + 27*x^7/4 +...
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * 3^floor(k/2)/floor((k+1)/2) * a(n-k)/(n-k)!. - _Seiichi Manyama_, Apr 30 2022

Links

Crossrefs

Cf. A193281, A193287, A193289, A193290.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-3*x^2)^(1+1/(3*x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 25 2013 *)
  • PARI
    {a(n)=n!*polcoeff(1/(1 - 3*x^2 +x^2*O(x^n))^((1+3*x)/(3*x)),n)}
  • PARI
    {a(n)=n!*polcoeff(sum(m=0,n,x^m/m!*prod(k=1,m,1+3*k*x+x*O(x^n))),n)}
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, j*3^(j\2)/((j+1)\2)*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Apr 30 2022

Formula

E.g.f.: A(x) = Sum_{n>=0} x^n/n! * Product_{k=1..n} (1 + 3*k*x).
a(n) ~ n! * n^(1/sqrt(3))*3^(n/2+1/2)/(2^(1+1/sqrt(3))*Gamma(1/sqrt(3))). - Vaclav Kotesovec, Jun 25 2013

A193289 E.g.f.: A(x) = 1/(1 - 6*x^2)^(1 + 1/(3*x)).

Original entry on oeis.org

1, 2, 16, 116, 1456, 18272, 315424, 5592512, 123304192, 2814746624, 75639399424, 2108241486848, 66872341633024, 2198914617257984, 80437062279012352, 3046047243283570688, 126259635313097506816, 5408763597941368291328, 250569314672586154835968
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2011

Keywords

Comments

More generally, we have the identity:
Sum_{n>=0} (x^n/n!)*Product_{k=1..n} (1+k*y) = 1/(1 - x*y)^(1 + 1/y); here x=2*x, y=3*x.

Examples

			E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 116*x^3/3! + 1456*x^4/4! + 18272*x^5/5! +...
where A(x) satisfies:
A(x)^(3*x/(1+3*x)) = 1 + 6*x^2 + 36*x^4 + 216*x^6 +...+ 6^n*x^(2*n) +...
Also,
A(x) = 1 + 2*x*(1+3*x) + 4*x^2*(1+3*x)*(1+6*x)/2! + 8*x^3*(1+3*x)*(1+6*x)*(1+9*x)/3! + 16*x^4*(1+3*x)*(1+6*x)*(1+9*x)*(1+12*x)/4! +...
The logarithm begins:
log(A(x)) = 2*x + 6*x^2 + 2*6*x^3/2 + 6^2*x^4/2 + 2*6^2*x^5/3 + 6^3*x^6/3 + 2*6^3*x^7/4 + 6^4*x^8/4 + 2*6^4*x^9/5 + 6^5*x^10/5 +...

Links

Crossrefs

Cf. A193281, A193287, A193288, A193290.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-6*x^2)^(1+1/(3*x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 22 2013 *)
  • PARI
    {a(n)=n!*polcoeff(1/(1 - 6*x^2 +x^2*O(x^n))^((1+3*x)/(3*x)),n)}
  • PARI
    {a(n)=n!*polcoeff(sum(m=0,n,2^m*x^m/m!*prod(k=1,m,1+3*k*x+x*O(x^n))),n)}

Formula

E.g.f.: A(x) = Sum_{n>=0} 2^n*x^n/n! * Product_{k=1..n} (1 + 3*k*x).
a(n) ~ n! * 6^(n/2)*(n/2)^sqrt(2/3)/(2*Gamma(1+sqrt(2/3))). - Vaclav Kotesovec, Sep 22 2013

A193290 E.g.f. satisfies: A(x) = 1/(1 - x*A(x))^(1 + 1/A(x)).

Original entry on oeis.org

1, 2, 10, 96, 1388, 26960, 659352, 19471984, 674425600, 26814697056, 1203912012000, 60251644584384, 3326134996826688, 200792710948417536, 13159474030202943744, 930524202271542658560, 70616227020854238216192, 5724780985202503068533760
Offset: 0

Views

Author

Paul D. Hanna, Jul 21 2011

Keywords

Comments

More generally, we have the identity:
Sum_{n>=0} (x^n/n!)*Product_{k=1..n} (1+k*y) = 1/(1 - x*y)^(1 + 1/y); here y=A(x).

Examples

			E.g.f.: A(x) = 1 + 2*x + 10*x^2/2! + 96*x^3/3! + 1388*x^4/4! + 26960*x^5/5! +...
where e.g.f. A = A(x) satisfies:
A = 1 + x*(1+A) + x^2*(1+A)*(1+2*A)/2! + x^3*(1+A)*(1+2*A)*(1+3*A)/3! + x^4*(1+A)*(1+2*A)*(1+3*A)*(1+4*A)/4! + x^5*(1+A)*(1+2*A)*(1+3*A)*(1+4*A)*(1+5*A)/5! +...

Links

Crossrefs

Cf. A094638, A193289, A193281, A193287, A193288.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1/(1-x*A +x*O(x^n))^(1+1/A));n!*polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,x^m/m!*prod(k=1,m,1+k*A+x*O(x^n))));n!*polcoeff(A,n)}

Formula

E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * Product_{k=1..n} (1 + k*A(x)).
a(n) ~ s*sqrt(r*(1+s)*(r*s-1)/(1-4*r*s+r^2*s*(2*s-1))) * n^(n-1) / (exp(n) * r^n), where s = 2.4590533113276368838... is the root of the equation (1+s)*(1+2*s) = s^(s/(1+s))*(1+s)^2 - s*log(s) and r = (1 - s^(-s/(1+s)))/s = 0.1921573821382919835... - Vaclav Kotesovec, Jan 11 2014

A246689 Expansion of e.g.f. 1/(1 - x^3)^(1 + 1/x + 1/x^2).

Original entry on oeis.org

1, 1, 3, 13, 61, 381, 2791, 22513, 210393, 2183401, 24575851, 305067621, 4097726293, 58876485253, 910581818511, 15005958062761, 261751577640241, 4844661893762193, 94564968066402643, 1938366513866527741, 41760228574294689261, 941821175462309114701
Offset: 0

Views

Author

Peter Bala, Sep 01 2014

Keywords

Comments

Compare with A193281.

Links

Crossrefs

Cf. A008620, A193281.

Programs

  • Maple
    seq(coeftayl(n!/(1-x^3)^(1+1/x+1/x^2), x = 0, n), n = 0..10);
  • Mathematica
    CoefficientList[Series[1/(1 - x^3)^(1 + 1/x + 1/x^2), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 01 2014 *)
  • PARI
    my(x='x+O('x^66)); Vec(serlaplace(1/(1 - x^3)^(1 + 1/x + 1/x^2))) \\ Joerg Arndt, Sep 01 2014
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, j/((j+2)\3)*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Apr 30 2022

Formula

E.g.f.: A(x) = 1/(1 - x^3)^(1 + 1/x + 1/x^2) = exp( Sum_{n>=1} x^n/A008620(n-1) ) = 1 + x + 3*x^2/2! + 13*x^3/3! + 61*x^4/4! + ....
A(x) = Sum_{n>=0} (x^n/n!)*Product {k = 1..n} (1 + x + k*x^2).
It appears that a(n) == 1 (mod n*(n-1)).
a(n) ~ n! * (n^2 / 54) * (1 + 6*log(n)/n). - Vaclav Kotesovec, Sep 01 2014
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k/A008620(k-1) * a(n-k)/(n-k)!. - Seiichi Manyama, Apr 30 2022

A353204 Expansion of e.g.f. (1 - x^2)^(1 + 1/x).

Original entry on oeis.org

1, -1, -1, 2, 1, 9, 1, 146, -167, 5363, -16109, 355354, -1844831, 37142117, -279336147, 5615638874, -55537087439, 1157104637831, -14174652825017, 311217052899986, -4538156701549279, 105770066665097729, -1785320722016719271, 44287095132343348482
Offset: 0

Views

Author

Seiichi Manyama, Apr 30 2022

Keywords

Links

Crossrefs

Cf. A110654, A193281, A353205.

Programs

  • Maple
    S:=series((1-x^2)^(1+1/x),x,31):
seq(coeff(S,x,i)*i!,i=0..30); # Robert Israel, Nov 01 2022
  • Mathematica
    nmax = 25; CoefficientList[Series[(1 - x^2)^(1 + 1/x), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, May 09 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^2)^(1+1/x)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, x^k/((k+1)\2)))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-(i-1)!*sum(j=1, i, j/((j+1)\2)*v[i-j+1]/(i-j)!)); v;

Formula

E.g.f.: exp( -Sum{k >= 1} x^k/A110654(k) ).
a(0) = 1; a(n) = -(n-1)! * Sum_{k=1..n} k/A110654(k) * a(n-k)/(n-k)!.
a(n) ~ -(-1)^n * n! / n^2 * (1 - 2*log(n)/n). - Vaclav Kotesovec, May 09 2022
Showing 1-6 of 6 results.

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