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 1-5 of 5 results.

A345969 Expansion of the e.g.f. 1 / sqrt(3 - 2 / ((1 - x)*exp(x))).

Original entry on oeis.org

1, 0, 1, 2, 18, 104, 1015, 9666, 116557, 1504856, 22300704, 358916480, 6373675825, 122332173300, 2540560235161, 56558354414870, 1346402030278050, 34093192112537888, 915570658175517151, 25983157665663651150, 777141557158947654637, 24430880483991543481580
Offset: 0

Views

Author

Mélika Tebni, Jul 01 2021

Keywords

Examples

			1/sqrt(3-2/((1-x)*exp(x))) =  1 + x^2/2! + 2*x^3/3! + 18*x^4/4! + 104*x^5/5! + 1015*x^6/6! + 9666*x^7/7! + 116557*x^8/8! + 1504856*x^9/9! + ...
a(17) = Sum_{k=1..8} A305404(k)*A008306(17,k) = 34093192112537888.
For k=1, A305404(1)*A008306(17,1) == -1 (mod 17), because A305404(1) = 1 and A008306(17,1) = (17-1)!
For k>=2, A305404(k)*A008306(17,k) == 0 (mod 17), because A008306(17,k) == 0 (mod 17), result a(17) == -1 (mod 17).

Crossrefs

Cf. A000166, A008306, A305404, A327006, A345697.

Programs

  • Maple
    A305404:= n-> add(Stirling2(n,k)*doublefactorial(2*k-1), k=0..n):
A008306 := proc(n, k): if k=1 then (n-1)! ; elif n<=2*k-1 then 0; else (n-1)*procname(n-1, k)+(n-1)*procname(n-2, k-1) ; end if; end proc:
a := n-> add((A305404(k)*A008306(n, k)), k=1..iquo(n,2)):a(0):=1 ; seq(a(n), n=0..24);
# second program:
a := series(1/sqrt(3-2/((1-x)*exp(x))), x=0, 25):seq(n!*coeff(a, x, n), n=0..24);
  • Mathematica
    CoefficientList[Series[1/Sqrt[3-2/((1-x)*E^x)], {x, 0, 24}], x] * Range[0, 24]!
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(1/sqrt(3 - 2 / ((1 - x)*exp(x))))) \\ Michel Marcus, Jul 01 2021

Formula

E.g.f. y(x) satisfies y' = exp(-x)*y^3*x/(1-x)^2.
a(0)=1, a(n) = Sum_{k=1..floor(n/2)} A305404(k)*A008306(n,k) for n > 0.
For all p prime, a(p) == -1 (mod p).
a(n) ~ sqrt(-2*LambertW(-2*exp(-1)/3)/3) * n^n / (exp(n) * (1 + LambertW(-2*exp(-1)/3))^(n+1)). - Vaclav Kotesovec, Jul 01 2021

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

A346119 Expansion of the e.g.f. sqrt(2*x*exp(x) - 2*exp(x) + 3).

Original entry on oeis.org

1, 0, 1, 2, 0, -16, -35, 342, 2779, -6424, -239382, -822460, 22393657, 278844084, -1553468891, -68399947042, -275025888900, 15302175612416, 243541868882077, -2463105309082902, -121649966081262521, -473088821582805820, 50905612811064360006, 945133249101683013812, -15321255878414345388335
Offset: 0

Views

Author

Mélika Tebni, Jul 05 2021

Keywords

Examples

			sqrt(2*x*exp(x)-2*exp(x)+3) = 1 + x^2/2! + 2*x^3/3! - 16*x^5/5! - 35*x^6/6! + 342*x^7/7! + 2779*x^8/8! - 6424*x^9/9! + ...
a(11) = Sum_{k=1..5} (-1)^(k-1)*A006677(k)*A008306(11,k) = -822460.
For k=1, (-1)^(1-1)*A006677(1)*A008306(11,1) == -1 (mod 11), because A006677(1) = 1 and A008306(11,1) = (11-1)!
For k>=2, (-1)^(k-1)*A006677(k)*A008306(11,k) == 0 (mod 11), because A008306(11,k) == 0 (mod 11), result a(11) == -1 (mod 11).
a(8) = Sum_{k=1..4} (-1)^(k-1)*A006677(k)*A008306(8,k) = 2779.
a(8) == 0 (mod (8-1)), because for k >= 1, A008306(8,k) == 0 (mod 7).

Crossrefs

Cf. A000166, A006677, A008306, A327006, A345652, A345697, A345969.

Programs

  • Maple
    stirtr:= proc(p) proc(n) add(p(k)*Stirling2(n, k), k=0..n) end end: f:= n-> `if`(n=0, 1, (2*n-2)!/ (n-1)!/ 2^(n-1)): A006677:= stirtr(f): # Alois P. Heinz, 2008.
A008306 := proc(n, k): if k=1 then (n-1)! ; elif n<=2*k-1 then 0; else (n-1)*procname(n-1, k)+(n-1)*procname(n-2, k-1) ; end if; end proc:
a:= n-> add(((-1)^(k-1)*A006677(k)*A008306(n,k)), k=1..iquo(n,2)):a(0):=1 ; seq(a(n), n=0..24);
# second program:
a := series(sqrt(2*x*exp(x)-2*exp(x)+3), x=0, 25):seq(n!*coeff(a, x, n), n=0..24);
  • Mathematica
    CoefficientList[Series[Sqrt(2*x*E^x-2*E^x+3), {x, 0, 24}], x] * Range[0, 24]!
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(sqrt(2*x*exp(x) - 2*exp(x) + 3))) \\ Michel Marcus, Jul 05 2021

Formula

E.g.f. y(x) satisfies y*y' = x*exp(x).
a(0)=1, a(n) = Sum_{k=1..floor(n/2)} (-1)^(k-1)*A006677(k)*A008306(n,k) for n > 0.
For all p prime, a(p) == -1 (mod p).
For n > 1, a(n) == 0 (mod (n-1)).
Conjecture: a(n) = 0 for only n = 1 and n = 4.

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

A343482 Expansion of the e.g.f. sqrt(-1 + 2 / (1 - x) / exp(x)).

Original entry on oeis.org

1, 0, 1, 2, 6, 24, 135, 930, 7105, 59192, 549360, 5746080, 66713361, 839528052, 11308954657, 163038260294, 2520332282910, 41640324943968, 730119174449151, 13507292654421390, 263004450921933817, 5385277610047242620, 115775314245285797256, 2606072891349667903152, 61248210450060537498321
Offset: 0

Views

Author

Mélika Tebni, Jul 06 2021

Keywords

Examples

			sqrt(-1+2/(1-x)/exp(x)) =  1 + x^2/2! + 2*x^3/3! + 6*x^4/4! + 24*x^5/5! + 135*x^6/6! + 930*x^7/7! + 7105*x^8/8! + 59192*x^9/9! + ...
a(23) = Sum_{k=1..11} (-1)^(k-1)*A014304(k-1)*A008306(23,k) = 2606072891349667903152.
For k=1, (-1)^(1-1)*A014304(1-1)*A008306(23,1) == -1 (mod 23), because A014304(0) = 1 and A008306(23,1) = (23-1)!
For k>=2, (-1)^(k-1)*A014304(k-1)*A008306(23,k) == 0 (mod 23), because A008306(23,k) == 0 (mod 23), result a(23) == -1 (mod 23).
a(18) = Sum_{k=1..9} (-1)^(k-1)*A014304(k-1)*A008306(18,k) = 730119174449151.
a(18) == 0 (mod (18-1)), because for k >= 1, A008306(18,k) == 0 (mod 17).

Crossrefs

Cf. A000166, A008306, A014304, A327006, A345697, A345969, A346119.

Programs

  • Maple
    A014304:= proc(n) option remember; `if`(n=0, 1, (-1)^n + add(binomial(n,k)*A014304(k)* A014304(n-k-1), k=0..n-1)) end:
A008306 := proc(n, k): if k=1 then (n-1)! ; elif n<=2*k-1 then 0; else (n-1)*procname(n-1, k)+(n-1)*procname(n-2, k-1) ; end if; end proc:
a:= n-> add(((-1)^(k-1)*A014304(k-1)*A008306(n,k)), k=1..iquo(n,2)):a(0):=1 ; seq(a(n), n=0..24);
# second program:
a := series(sqrt(-1+2/(1-x)/exp(x)), x=0, 25):seq(n!*coeff(a, x, n), n=0..24);
  • Mathematica
    CoefficientList[Series[Sqrt[-1+2/(1-x)/E^x], {x, 0, 24}], x] * Range[0, 24]!
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(sqrt(-1 + 2 / (1 - x) / exp(x)))) \\ Michel Marcus, Jul 06 2021

Formula

E.g.f. y(x) satisfies y*y' = exp(-x)*x/(1-x)^2.
a(0)=1, a(n) = Sum_{k=1..floor(n/2)} (-1)^(k-1)*A014304(k-1)*A008306(n,k) for n > 0.
For all p prime, a(p) == -1 (mod p).
For n > 1, a(n) == 0 (mod (n-1)).
a(n) ~ 2 * n^n / exp(n + 1/2). - Vaclav Kotesovec, Jul 06 2021
Showing 1-5 of 5 results.

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