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 1-7 of 7 results.

A345697 Expansion of the e.g.f. sqrt(1 / (2*exp(x) - 2*x*exp(x) - 1)).

Original entry on oeis.org

1, 0, 1, 2, 12, 64, 485, 4038, 39991, 441992, 5492322, 75171700, 1127989577, 18381446004, 323527186957, 6114296752718, 123513004310640, 2655648779976640, 60554669008300565, 1459559515622280282, 37079264125376670955, 990226180225789628660, 27733277682719819190246, 812818183963966524137332, 24880254143735238825011057
Offset: 0

Views

Author

Mélika Tebni, Jun 24 2021

Keywords

Examples

			sqrt(1/(2*exp(x)-2*x*exp(x)-1)) = 1 + x^2/2! + 2*x^3/3! + 12*x^4/4! + 64*x^5/5! + 485*x^6/6! + 4038*x^7/7! + 39991*x^8/8! + 441992*x^9/9! + ...
a(13) = Sum_{k=1..6} A014307(k)*A008306(13,k) = 18381446004.
A014307(1)*A008306(13,1) == -1 (mod 13), because A014307(1) = 1 and A008306(13,1) = (13-1)!
For k>=2, A008306(13,k) == 0 (mod 13), result a(13) == -1 (mod 13).

Crossrefs

Cf. A000166, A008306, A014307, A327006.

Programs

  • Maple
    A014307 := proc(n) option remember; `if`(n=0, 1 , 1+add((-1+binomial(n, k))*A014307(k), k=1..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((A014307(k)*A008306(n,k)), k=1..floor(n/2)):a(0):=1 ;
seq(a(n), n=0..24);
# second program:
a := series(sqrt((1/(2*exp(x)-2*x*exp(x)-1))), x=0, 25):
seq(n!*coeff(a, x, n), n=0..24);
  • Mathematica
    CoefficientList[Series[Sqrt[1/(2*E^x-2*x*E^x-1)], {x, 0, 24}], x] * Range[0, 24]!
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(sqrt(1 / (2*exp(x) - 2*x*exp(x) -1)))) \\ Michel Marcus, Jun 24 2021

Formula

E.g.f. y(x) satisfies y' = x*exp(x)*y^3.
a(0)=1, a(n) = Sum_{k=1..floor(n/2)} A014307(k)*A008306(n,k) for n >= 1.
For all p prime, a(p) == -1 (mod p).
a(n) ~ sqrt(2*c) * n^n / ((1-c)^(n+1) * exp(n)), where c = -LambertW(-exp(-1)/2). - Vaclav Kotesovec, Jun 25 2021

A345652 Expansion of the e.g.f. exp(-1 + (x + 1)*exp(-x)).

Original entry on oeis.org

1, 0, -1, 2, 0, -16, 65, -78, -749, 6232, -22068, -28920, 1004685, -7408740, 22263215, 157632230, -2874256740, 21590948480, -53087332675, -956539294506, 16344490525835, -132605481091060, 294656170409328, 9113173803517344, -167298122286332823
Offset: 0

Views

Author

Mélika Tebni, Jun 21 2021

Keywords

Comments

For all p prime, a(p)/(p-1) == 1 (mod p). - Mélika Tebni, Mar 21 2022

Examples

			exp(-1+(x+1)*exp(-x)) = 1 - x^2/2! + 2*x^3/3! - 16*x^5/5! + 65*x^6/6! - 78*x^7/7! - 749*x^8/8! + 6232*x^9/9! + ...

Links

Crossrefs

Cf. A003725, A014182, A327006.
Cf. A292935 (without 1+x: EGF e^(e^(-x)-1)), A000110 (absolute values: Bell numbers, EGF e^(e^x-1))
Cf. A003725, A106828, A347210, A347571.

Programs

  • Maple
    a := series(exp(-1+(x+1)*exp(-x)), x=0, 25): seq(n!*coeff(a, x, n), n=0..24);
a := proc(n) option remember; `if`(n=0, 1, add((n-1)*binomial(n-2, k)*(-1)^(n-1-k)*a(k), k=0..n-2)) end: seq(a(n), n=0..24);
# third program:
A345652 := n -> add((-1)^(n-k)*combinat[bell](k)*A106828(n, k), k=0..iquo(n, 2)):
seq(A345652(n), n=0..24); # Mélika Tebni, Sep 21 2021
  • Mathematica
    nmax = 24; CoefficientList[Series[Exp[-1+(x+1)*Exp[-x]], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    seq(n) = {Vec(serlaplace(exp(-1+(x+1)*exp(-x + O(x*x^n)))))} \\ Andrew Howroyd, Jun 21 2021
  • PARI
    a(n) = if(n==0, 1, sum(k=2, n, (-1)^(k-1)*(k-1)*binomial(n-1, k-1)*a(n-k))); \\ Seiichi Manyama, Mar 15 2022

Formula

The e.g.f. y(x) satisfies y' = -x*y*exp(-x).
a(n) = Sum_{k=0..n-2} (n-1)*binomial(n-2, k)*(-1)^(n-1-k)*a(k) for n > 0.
Conjecture: a(n) = 0 for only n = 1 and n = 4.
Conjecture: For all p prime, a(p)^2 == 1 (mod p).
Stronger conjecture: For n > 1, a(n) == -1 (mod n) iff n is a prime or 6. - M. F. Hasler, Jun 23 2021
a(n) = Sum_{k=0..floor(n/2)} (-1)^(n-k)*Bell(k)*A106828(n, k). - Mélika Tebni, Sep 21 2021
a(n) = Sum_{k=0..n} (-1)^k*A003725(n-k)*Bell(k)*binomial(n, k). - Mélika Tebni, Mar 21 2022

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

A005387 Number of partitional matroids on n elements.

Original entry on oeis.org

1, 2, 5, 16, 62, 276, 1377, 7596, 45789, 298626, 2090910, 15621640, 123897413, 1038535174, 9165475893, 84886111212, 822648571314, 8321077557124, 87648445601429, 959450073912136, 10894692556576613, 128114221270929646
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • Recski, A.; Enumerating partitional matroids. Stud. Sci. Math. Hungar. 9 (1974), 247-249 (1975).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A327006.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp((x-1)*Exp(x) + 2*x + 1) ))); // G. C. Greubel, Nov 16 2022
  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[(x-1)E^x+2x+1],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Nov 22 2012 *)
  • SageMath
    def A005387_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp((x-1)*exp(x) + 2*x + 1) ).egf_to_ogf().list()
A005387_list(40) # G. C. Greubel, Nov 16 2022

Formula

E.g.f.: exp( (x-1)*exp(x) + 2*x + 1 ).
a(n) = Sum_{j=0..n} binomial(n, j) * 2^(n-j) * A327006(j+1). - G. C. Greubel, Nov 16 2022

Extensions

More terms from James Sellers, Aug 21 2000

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.

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

A327005 T(n, k) = Sum_{i=1..n} BM[k][i] where BM is the BellMatrix(x -> x mod n) as defined in A264428. Square array read by ascending antidiagonals for n >= 1 and k >= 1.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 2, 4, 0, 1, 0, 1, 2, 3, 0, 0, 1, 0, 1, 2, 6, 21, 31, 0, 1, 0, 1, 2, 6, 20, 57, 0, 0, 1, 0, 1, 2, 6, 24, 101, 231, 379, 0, 1, 0, 1, 2, 6, 24, 100, 422, 1394, 0, 0, 1, 0, 1, 2, 6, 24, 105, 505, 2201, 5476, 6556, 0
Offset: 1

Views

Author

Peter Luschny, Aug 13 2019

Keywords

Comments

Rows converge to the main diagonal A327006.

Examples

			[1] 1, 0, 0, 0, 0,  0,   0,   0,    0,     0,      0,      0, ...
[2] 1, 0, 1, 0, 4,  0,  31,   0,  379,     0,   6556,      0, ...
[3] 1, 0, 1, 2, 3, 21,  57, 231, 1394,  5476,  32616, 203105, ...
[4] 1, 0, 1, 2, 6, 20, 101, 422, 2201, 12560,  76846, 483892, ...
[5] 1, 0, 1, 2, 6, 24, 100, 505, 2620, 15383,  97480, 657305, ...
[6] 1, 0, 1, 2, 6, 24, 105, 504, 2759, 16186, 103494, 710384, ...

Links

Crossrefs

A005046 is a bisection of row 2. Main diagonal is A327006.
Cf. A264428.

Programs

  • Maple
    # BellMatrix is defined in A264428.
T := proc(n, k) BellMatrix(x -> modp(x, n), k): add(i, i in %[k]) end:
seq(seq(T(n-k+1,k), k=1..n), n=1..12);
Showing 1-7 of 7 results.

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