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.

A347210 Expansion of the e.g.f. (1 - 2*x - 2*log(1 - x) - exp(2*x)*(1 - x)^2) / 4 - 1.

Original entry on oeis.org

-1, 0, 1, 2, 3, 4, 20, 216, 2072, 18880, 177984, 1805440, 19935872, 239445504, 3113377280, 43588830208, 653836446720, 10461393240064, 177843710148608, 3201186844016640, 60822550184493056, 1216451004043755520, 25545471085755629568, 562000363888584687616
Offset: 0

Views

Author

Mélika Tebni, Aug 23 2021

Keywords

Comments

For all p prime, a(p) == -1 (mod p).
For n > 1, a(n) == 0 (mod (n-1)).

Examples

			E.g.f.: -1 + x^2/2! + 2*x^3/3! + 3*x^4/4! + 4*x^5/5! + 20*x^6/6! + 216*x^7/7! + 2072*x^8/8! + 18880*x^9/9! + ...
a(19) = Sum_{k=1..9} (-1)^(k-1)*ceiling(2^(k-2))*A106828(19, k) = 3201186844016640.
For k = 1, (-1)^(1-1)*ceiling(2^(1-2))*A106828(19, 1) == -1 (mod 19), because (-1)^(1-1)*ceiling(2^(1-2)) = 1 and A106828(19, 1) = (19-1)!
For k >= 2, (-1)^(k-1)*ceiling(2^(k-2))*A106828(19, k) == 0 (mod 19), because A106828(19, k) == 0 (mod 19), result a(19) == -1 (mod 19).
a(10) = Sum_{k=1..5}  (-1)^(k-1)*ceiling(2^(k-2))*A106828(10, k) = 177984.
a(10) == 0 (mod (10-1)), because for k >= 1, A106828(10, k) == 0 (mod 9).

Crossrefs

Cf. A106828, A343482, A345697, A345969, A346119.

Programs

  • Maple
    a := series((1-2*x-2*log(1-x)-exp(2*x)*(1-x)^2)/4-1, x=0, 25):
seq(n!*coeff(a, x, n), n=0..23);
# second program:
a := n -> add((-1)^(k-1)*ceil(2^(k-2))*A106828(n, k), k=0..iquo(n, 2)):
seq(a(n), n=0..23);
  • Mathematica
    CoefficientList[Series[(1 - 2*x - 2*Log[1 - x] - E^(2*x)*(1 - x)^2)/4 - 1, {x, 0, 23}], x]*Range[0, 23]!
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace((1-2*x-2*log(1-x)-exp(2*x)*(1-x)^2)/4 - 1)) \\ Michel Marcus, Aug 23 2021

Formula

a(n) = Sum_{k=0..floor(n/2)} (-1)^(k-1)*ceiling(2^(k-2))*A106828(n, k).
a(n) ~ (n-1)!/2. - Vaclav Kotesovec, Dec 09 2021

A347571 Expansion of the e.g.f. (-1 - 2*x - 2*log(1 - x) + exp(-2*x) / (1 - x)^2) / 4 + 1.

Original entry on oeis.org

1, 0, 1, 2, 9, 44, 280, 2064, 17528, 167488, 1777536, 20721920, 263055232, 3610443264, 53256280064, 839974309888, 14103897738240, 251146689069056, 4726795773018112, 93746994502828032, 1954053073794596864, 42702893781890498560, 976276451410488066048, 23303485413254033309696
Offset: 0

Views

Author

Mélika Tebni, Sep 07 2021

Keywords

Comments

For all p prime, a(p) == -1 (mod p).
For n > 1, a(n) == 0 (mod (n-1)).

Examples

			E.g.f.: 1 + x^2/2! + 2*x^3/3! + 9*x^4/4! + 44*x^5/5! + 280*x^6/6! + 2064*x^7/7! + 17528*x^8/8! + 167488*x^9/9! + ...
a(11) = Sum_{k=0..5} ceiling(2^(k-2))*A106828(11, k) = 20721920.
For k = 0, A106828(11,0) = 0.
For k = 1, ceiling(2^(1-2))*A106828(11, 1) == -1 (mod 11), because ceiling(2^(1-2)) = 1 and A106828(11, 1) = (11-1)!
For k >= 2, ceiling(2^(k-2))*A106828(11, k) == 0 (mod 11), because A106828(11, k) == 0 (mod 11), result a(11) == -1 (mod 11).
a(10) = Sum_{k=0..5} ceiling(2^(k-2))*A106828(10, k) = 1777536.
a(10) == 0 (mod (10-1)), because for k >= 0, A106828(10, k) == 0 (mod 9).

Crossrefs

Cf. A106828, A343482, A345697, A345969, A346119, A347210.

Programs

  • Maple
    a := series((-1-2*x-2*log(1-x)+exp(-2*x)/(1-x)^2)/4+1, x=0, 24):
seq(n!*coeff(a, x, n), n=0..23);
# second program:
a := n -> add(ceil(2^(k-2))*A106828(n, k), k=0..iquo(n, 2)):
seq(a(n), n=0..23);
  • Mathematica
    CoefficientList[Series[(-1 - 2*x - 2*Log[1 - x] + Exp[-2*x]/(1 - x)^2)/4 + 1, {x, 0, 23}], x]*Range[0, 23]!
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace((-1-2*x-2*log(1-x)+exp(-2*x)/(1-x)^2)/4 + 1)) \\ Michel Marcus, Sep 07 2021

Formula

a(n) = Sum_{k=0..floor(n/2)} ceiling(2^(k-2))*A106828(n, k).
a(n) ~ n * n! / (4*exp(2)). - Vaclav Kotesovec, Sep 10 2021
Showing 1-2 of 2 results.

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