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.

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A202365 G.f.: Sum_{n>=0} (n-x)^n * x^n / (1 + n*x - x^2)^n.

Original entry on oeis.org

1, 1, 2, 10, 54, 336, 2400, 19440, 176400, 1774080, 19595520, 235872000, 3073593600, 43110144000, 647610163200, 10374216652800, 176536039680000, 3180264062976000, 60466862776320000, 1210048630382592000, 25423825985445888000, 559567461880627200000, 12874917427270778880000
Offset: 0

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Author

Paul D. Hanna, Jan 09 2013

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 54*x^4 + 336*x^5 + 2400*x^6 +...
where
A(x) = 1 + (1-x)*x/(1+x-x^2) + (2-x)^2*x^2/(1+2*x-x^2)^2 + (3-x)^3*x^3/(1+3*x-x^2)^3 + (4-x)^4*x^4/(1+4*x-x^2)^4 + (5-x)^5*x^5/(1+5*x-x^2)^5 +...

Crossrefs

Cf. A187742, A187741, A187735, A187746.

Programs

  • Mathematica
    a[n_] := Switch[n, 0|1, 1, _, (n-1)*(n+2)/2*(n-1)!];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 24 2022 *)
  • PARI
    {a(n)=polcoeff( sum(m=0, n, (m-x)^m*x^m/(1+m*x-x^2 +x*O(x^n))^m), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=if(n==0||n==1, 1, (n-1)*(n+2)/2 * (n-1)!)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=n!*polcoeff(1/2 + 1/(2*(1-x)^2) + x + log(1-x +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) = (n-1)*(n+2)/2 * (n-1)!, for n>1 with a(0)=a(1)=1.
E.g.f.: 1/2 + 1/(2*(1-x)^2) + x + log(1-x).
E.g.f.: Sum_{n>=0} a(n+1)*x^n/n! = 1/(1-x)^3 - x/(1-x).
From Amiram Eldar, Dec 23 2022: (Start)
Sum_{n>=0} 1/a(n) = Pi^2/9 + 43/27.
Sum_{n>=0} (-1)^n/a(n) = Pi^2/18 - 4*log(2)/9 + 5/27. (End)

A230056 G.f.: Sum_{n>=0} (n+3)^n * x^n / (1 + (n+3)*x)^n.

Original entry on oeis.org

1, 4, 9, 30, 132, 720, 4680, 35280, 302400, 2903040, 30844800, 359251200, 4550515200, 62270208000, 915372057600, 14384418048000, 240612083712000, 4268249137152000, 80029671321600000, 1581386305314816000, 32844177110384640000, 715273190403932160000, 16298010552775311360000
Offset: 0

Views

Author

Paul D. Hanna, Oct 07 2013

Keywords

Examples

			O.g.f.: A(x) = 1 + 4*x + 9*x^2 + 30*x^3 + 132*x^4 + 720*x^5 + 4680*x^6 +...
where
A(x) = 1 + 4*x/(1+4*x) + 5^2*x^2/(1+5*x)^2 + 6^3*x^3/(1+6*x)^3 + 7^4*x^4/(1+7*x)^4 + 8^5*x^5/(1+8*x)^5 +...
E.g.f.: E(x) = 1 + 4*x + 9*x^2/2! + 30*x^3/3! + 132*x^4/4! + 720*x^5/5! +...
where
E(x) = 1 + 4*x + 9/2*x^2 + 5*x^3 + 11/2*x^4 + 6*x^5 + 13/2*x^6 + 7*x^7 +...
which is the expansion of: (2 + 4*x - 5*x^2) / (2 - 4*x + 2*x^2).

Crossrefs

Cf. A229039, A038720, A230056, A187735, A187738, A187739, A229039, A221160, A221161, A187740.

Programs

  • Maple
    a:=series(add((n+3)^n*x^n/(1+(n+3)*x)^n,n=0..100),x=0,23): seq(coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    a[n_] := (n + 7)*n!/2; a[0] = 1; Array[a, 25, 0] (* Amiram Eldar, Dec 11 2022 *)
  • PARI
    {a(n)=polcoeff( sum(m=0, n, ((m+3)*x)^m / (1 + (m+3)*x +x*O(x^n))^m), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=if(n==0, 1, (n+7) * n!/2 )}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = (n+7) * n!/2 for n>0 with a(0)=1.
E.g.f.: (2 + 4*x - 5*x^2)/(2*(1-x)^2).
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 530*e - 10075/7.
Sum_{n>=0} (-1)^n/a(n) = 10085/7 - 3914/e. (End)

A354327 Expansion of e.g.f. 1/(1 + x/4 * log(1 - 2 * x)).

Original entry on oeis.org

1, 0, 1, 3, 22, 180, 1902, 23730, 344872, 5706288, 105960600, 2181449160, 49311653616, 1214109056160, 32339248301808, 926527371653520, 28410493609687680, 928335829570087680, 32201658919855225728, 1181755749910942408320, 45744743939940787150080
Offset: 0

Views

Author

Seiichi Manyama, May 24 2022

Keywords

Crossrefs

Cf. A052830, A187735, A354325, A354328.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x/4*log(1-2*x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!*sum(j=2, i, 2^(j-3)/(j-1)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\2, 2^(n-3*k)*k!*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

a(0) = 1; a(n) = n! * Sum_{k=2..n} 2^(k-3)/(k-1) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-3*k) * k! * |Stirling1(n-k,k)|/(n-k)!.

A298854 Characteristic polynomials of Jacobi coordinates. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 6, 11, 11, 6, 24, 50, 61, 50, 24, 120, 274, 379, 379, 274, 120, 720, 1764, 2668, 3023, 2668, 1764, 720, 5040, 13068, 21160, 26193, 26193, 21160, 13068, 5040, 40320, 109584, 187388, 248092, 270961, 248092, 187388, 109584, 40320, 362880, 1026576, 1836396, 2565080, 2995125, 2995125, 2565080, 1836396, 1026576, 362880
Offset: 0

Views

Author

F. Chapoton, Jan 27 2018

Keywords

Comments

This is just a different normalization of A223256 and A223257.

Examples

			For n = 3, the polynomial is 6*x^3 + 11*x^2 + 11*x + 6.
The first few polynomials, as a table:
[  1],
[  1,   1],
[  2,   3,   2],
[  6,  11,  11,   6],
[ 24,  50,  61,  50,  24],
[120, 274, 379, 379, 274, 120]

Crossrefs

Closely related to A223256 and A223257.
Row sums are A002720.
Leftmost and rightmost columns are A000142.
Alternating row sums are A177145.
Absolute value of evaluation at x = exp(2*i*Pi/3) is A080171.
Evaluation at x=2 gives A187735.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n<1, n+1, expand(
      n*(x+1)*b(n-1)-(n-1)^2*x*b(n-2)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Apr 01 2021
  • Mathematica
    P[0] = 1 ; P[1] = x + 1;
P[n_] := P[n] = n (x + 1) P[n - 1] - (n - 1)^2 x P[n - 2];
Table[CoefficientList[P[n], x], {n, 0, 9}] // Flatten (* Jean-François Alcover, Mar 16 2020 *)
  • Sage
    @cached_function
def poly(n):
    x = polygen(ZZ, 'x')
    if n < 0:
        return x.parent().zero()
    elif n == 0:
        return x.parent().one()
    else:
        return n * (x + 1) * poly(n - 1) - (n - 1)**2 * x * poly(n - 2)
A298854_row = lambda n: list(poly(n))
for n in (0..7): print(A298854_row(n))

Formula

P(0)=1 and P(n) = n * (x + 1) * P(n - 1) - (n - 1)^2 * x * P(n - 2).

A229036 G.f.: Sum_{n>=0} (3*n-1)^n * x^n / (1 + (3*n-1)*x)^n.

Original entry on oeis.org

1, 2, 21, 270, 4212, 77760, 1662120, 40415760, 1102248000, 33331979520, 1107097891200, 40069801094400, 1569793384051200, 66185883219456000, 2988292627358438400, 143855017177487616000, 7355369573944584192000, 398090614491857903616000, 22737098558477268725760000
Offset: 0

Views

Author

Paul D. Hanna, Sep 11 2013

Keywords

Comments

More generally,
if Sum_{n>=0} a(n)*x^n = Sum_{n>=0} (b*n+c)^n * x^n / (1 + (b*n+c)*x)^n,
then Sum_{n>=0} a(n)*x^n/n! = (2 - 2*(b-c)*x + b*(b-2*c)*x^2)/(2*(1-b*x)^2)
so that a(n) = (b*n + (b+2*c)) * b^(n-1) * n!/2 for n>0 with a(0)=1.

Examples

			O.g.f.: A(x) = 1 + 2*x + 21*x^2 + 270*x^3 + 4212*x^4 + 77760*x^5 +...
where
A(x) = 1 + 2*x/(1+2*x) + 5^2*x^2/(1+5*x)^2 + 8^3*x^3/(1+8*x)^3 + 11^4*x^4/(1+11*x)^4 + 14^5*x^5/(1+14*x)^5 +...
E.g.f.: E(x) = 1 + 2*x + 21*x^2/2! + 270*x^3/3! + 4212*x^4/4! + 77760*x^5/5! +...
where
E(x) =  1 + 2*x + 21/2*x^2 + 45*x^3 + 351/2*x^4 + 648*x^5 + 4617/2*x^6 +...
which is the expansion of: (2 - 8*x + 15*x^2) / (2 - 12*x + 18*x^2).

Crossrefs

Cf. A229039, A187735, A187738, A187739, A221160, A221161, A187740.

Programs

  • Mathematica
    Join[{1},Table[(3n+1)3^(n-1) n!/2,{n,20}]] (* Harvey P. Dale, Feb 10 2015 *)
  • PARI
    {a(n)=polcoeff( sum(m=0, n, ((3*m-1)*x)^m / (1 + (3*m-1)*x +x*O(x^n))^m), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n) = if(n==0,1,(3*n+1)*3^(n-1)*n!/2)}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = (3*n+1) * 3^(n-1) * n!/2 for n>0 with a(0)=1.
E.g.f.: (2 - 8*x + 15*x^2)/(2*(1-3*x)^2).
Previous Showing 11-15 of 15 results.

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