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 1-2 of 2 results.

A344108 Expansion of Product_{k>=1} 1 / (1 - x^k)^binomial(2*k,k).

Original entry on oeis.org

1, 2, 9, 36, 154, 644, 2744, 11608, 49267, 208610, 882963, 3731640, 15754327, 66426946, 279766063, 1176920484, 4945739292, 20761707824, 87069433162, 364802647912, 1527072152856, 6386873581244, 26690795165394, 111453873957936, 465055114353616, 1939114409985956
Offset: 0

Views

Author

Ilya Gutkovskiy, May 09 2021

Keywords

Comments

Euler transform of A000984.

Links

Crossrefs

Cf. A000984, A005430, A088327, A194353, A344130.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d*
      binomial(2*d, d), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 13 2023
  • Mathematica
    nmax = 25; CoefficientList[Series[Product[1/(1 - x^k)^Binomial[2 k, k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d Binomial[2 d, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 25}]
nmax = 25; CoefficientList[Series[Exp[Sum[(1/Sqrt[1 - 4*x^j] - 1)/j, {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 10 2021 *)

Formula

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} A005430(d) ) * a(n-k).
a(n) ~ 2^(2*n - 1/3) * exp(3*n^(1/3)/2^(2/3) - 1 + c) / (sqrt(3*Pi) * n^(5/6)), where c = Sum_{k>=2} (1/sqrt(1 - 4^(1-k)) - 1)/k = 0.0907540019413286886324751305813463657179452545... - Vaclav Kotesovec, May 10 2021

A305256 Expansion of exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*sqrt(1 - 4*x^k))).

Original entry on oeis.org

1, 1, 2, 8, 27, 103, 389, 1497, 5786, 22556, 88230, 346576, 1365119, 5390585, 21327913, 84527939, 335477433, 1333079925, 5302763618, 21112688376, 84125853415, 335443149005, 1338370995240, 5342843332758, 21339341267983, 85266832981905, 340840044333836, 1362936812554758
Offset: 0

Views

Author

Ilya Gutkovskiy, May 28 2018

Keywords

Comments

Weigh transform of the central binomial coefficients 1, 2, 6, 20, 70, ... (A000984).

Links

Crossrefs

Cf. A000984, A194353.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(binomial(2*i-2, i-1), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 28 2018
  • Mathematica
    nmax = 27; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) x^k/(k Sqrt[1 - 4 x^k]), {k, 1, nmax}]], {x, 0, nmax}], x]
nmax = 27; CoefficientList[Series[Product[(1 + x^k)^Binomial[2 k - 2, k - 1], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d Binomial[2 d - 2, d - 1], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 27}]

Formula

G.f.: Product_{k>=1} (1 + x^k)^binomial(2*k-2,k-1).
Showing 1-2 of 2 results.

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