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.

A274804 The exponential transform of sigma(n).

Original entry on oeis.org

1, 1, 4, 14, 69, 367, 2284, 15430, 115146, 924555, 7991892, 73547322, 718621516, 7410375897, 80405501540, 914492881330, 10873902417225, 134808633318271, 1738734267608613, 23282225008741565, 323082222240744379, 4638440974576329923, 68794595993688306903
Offset: 0

Views

Author

Johannes W. Meijer, Jul 27 2016

Keywords

Comments

The exponential transform [EXP] transforms an input sequence b(n) into the output sequence a(n). The EXP transform is the inverse of the logarithmic transform [LOG], see the Weisstein link and the Sloane and Plouffe reference. This relation goes by the name of Riddell's formula. For information about the logarithmic transform see A274805. The EXP transform is related to the multinomial transform, see A274760 and the second formula.
The definition of the EXP transform, see the second formula, shows that n >= 1. To preserve the identity LOG[EXP[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the exponential transform, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the multinomial coefficients A178867 appear.
We observe that a(0) = 1 and provides no information about any value of b(n), this notwithstanding it is customary to start the a(n) sequence with a(0) = 1.
The Maple programs can be used to generate the exponential transform of a sequence. The first program uses a formula found by Alois P. Heinz, see A007446 and the first formula. The second program uses the definition of the exponential transform, see the Weisstein link and the second formula. The third program uses information about the inverse of the exponential transform, see A274805.
Some EXP transform pairs are, n >= 1: A000435(n) and A065440(n-1); 1/A000027(n) and A177208(n-1)/A177209(n-1); A000670(n) and A075729(n-1); A000670(n-1) and A014304(n-1); A000045(n) and A256180(n-1); A000290(n) and A033462(n-1); A006125(n) and A197505(n-1); A053549(n) and A198046(n-1); A000311(n) and A006351(n); A030019(n) and A134954(n-1); A038048(n) and A053529(n-1); A193356(n) and A003727(n-1).

Examples

			Some a(n) formulas, see A178867:
a(0) = 1
a(1) = x(1)
a(2) = x(1)^2 + x(2)
a(3) = x(1)^3 + 3*x(1)*x(2) + x(3)
a(4) = x(1)^4 + 6*x(1)^2*x(2) + 4*x(1)*x(3) + 3*x(2)^2 + x(4)
a(5) = x(1)^5 + 10*x(1)^3*x(2) + 10*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 5*x(1)*x(4) + 10*x(2)*x(3) + x(5)

References

  • Frank Harary and Edgar M. Palmer, Graphical Enumeration, 1973.
  • Robert James Riddell, Contributions to the theory of condensation, Dissertation, University of Michigan, Ann Arbor, 1951.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

Links

Crossrefs

Cf. A274805, A274760, A178867, A000203, A007446.
Cf. A177208, A177209, A006351, A197505, A144180, A256180, A033462, A198046, A134954, A145460, A188489, A005432, A029725, A124213, A002801.
Cf. A036040, another version.

Programs

  • Maple
    nmax:=21: with(numtheory): b := proc(n): sigma(n) end: a:= proc(n) option remember; if n=0 then 1 else add(binomial(n-1, j-1) * b(j) *a(n-j), j=1..n) fi: end: seq(a(n), n=0..nmax); # End first EXP program.
nmax:= 21: with(numtheory): b := proc(n): sigma(n) end: t1 := exp(add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n=0..nmax); # End second EXP program.
nmax:=21: with(numtheory): b := proc(n): sigma(n) end: f := series(log(1+add(q(n)*x^n/n!, n=1..nmax+1)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(0):=1: q(0):=1: a(1):=b(1): q(1):=b(1): for n from 2 to nmax+1 do q(n) := solve(d(n)-b(n), q(n)): a(n):=q(n): od: seq(a(n), n=0..nmax); # End third EXP program.
  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n-1, j-1]*DivisorSigma[1, j]*a[n-j], {j, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 22 2017 *)
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 08 2021 *)

Formula

a(n) = Sum_{j=1..n} (binomial(n-1,j-1) * b(j) * a(n-j)), n >= 1 and a(0) = 1, with b(n) = A000203(n) = sigma(n).
E.g.f.: exp(Sum_{n >= 1} b(n)*x^n/n!) with b(n) = sigma(n) = A000203(n).

A348468 Expansion of e.g.f. sqrt(exp(x)*(2-exp(x))).

Original entry on oeis.org

1, 0, -1, -3, -10, -45, -271, -2058, -18775, -199335, -2410516, -32683563, -490870315, -8087188200, -144994236661, -2810079139143, -58536519252130, -1304198088413265, -30946462816602331, -779104979758256298, -20742005411397108595, -582214473250983046155, -17184302765073000634276
Offset: 0

Views

Author

Michel Marcus, Oct 19 2021

Keywords

Links

Crossrefs

Cf. A000629, A000918, A014304, A014307, A260701, A260902.

Programs

  • Mathematica
    m = 22; Range[0, m]! * CoefficientList[Series[Sqrt[Exp[x]*(2 - Exp[x])], {x, 0, m}], x] (* Amiram Eldar, Oct 19 2021 *)
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(sqrt(exp(x)*(2-exp(x)))))

Formula

a(n) ~ -sqrt(2) * n^(n-1) / (log(2)^(n - 1/2) * exp(n)). - Vaclav Kotesovec, Oct 21 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

A280939 Expansion of e.g.f.: 2*sinh(x/2) / sqrt(2 - exp(x)).

Original entry on oeis.org

1, 1, 4, 19, 121, 946, 8779, 94249, 1148746, 15667741, 236396029, 3909054304, 70297156021, 1365847397461, 28512838809004, 636437585232559, 15125744356058821, 381337518656892106, 10164860714961807079, 285635253778131491389, 8438962752941736017146, 261512261403795336646801, 8481542634943973943517129, 287325556922319462615912544, 10148442521179099638781764121
Offset: 1

Views

Author

Paul D. Hanna, Jan 11 2017

Keywords

Examples

			E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 19*x^4/4! + 121*x^5/5! + 946*x^6/6! + 8779*x^7/7! + 94249*x^8/8! + 1148746*x^9/9! + 15667741*x^10/10! + 236396029*x^11/11! + 3909054304*x^12/12! + ...

Links

Crossrefs

Cf. A014304, A014307.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(2*Sinh(x/2)/Sqrt(2 - Exp(x)))); [Factorial(n)*b[n]: n in [1..m-1]]; // G. C. Greubel, Oct 10 2018
  • Maple
    seq(coeff(series(factorial(n)*(2*sinh(x/2)/sqrt(2-exp(x))),x,n+1), x, n), n = 1 .. 25); # Muniru A Asiru, Oct 11 2018
  • Mathematica
    Rest[With[{nmax = 50}, CoefficientList[Series[2*Sinh[x/2]/Sqrt[2 - Exp[x]], {x, 0, nmax}], x]*Range[0, nmax]!]] (* G. C. Greubel, Oct 10 2018 *)
  • PARI
    {a(n) = my(X=x+x*O(x^n)); n!*polcoeff( 2*sinh(X/2) / sqrt(2 - exp(X)),n)}
for(n=1,20,print1(a(n),", "))

Formula

a(n) ~ n^n / (sqrt(2) * log(2)^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Jan 11 2017

A309473 a(n) = (-1)^n + Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 0, 1, 1, 3, 11, 43, 195, 1063, 6395, 42371, 311883, 2501159, 21672355, 202544323, 2028522067, 21658255431, 245738583307, 2952508103651, 37440976938875, 499785548010759, 7005210659040979, 102862231664567651, 1579045889274408259, 25294106622048460903
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 11 2020

Keywords

Crossrefs

Cf. A006677, A014304, A054687, A332776, A335441.

Programs

  • Mathematica
    a[n_] := a[n] = (-1)^n + Sum[Binomial[n - 1, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 24}]
terms = 24; A[] = 1; Do[A[x] = Normal[Integrate[A[x]^2 - Exp[-x], x] + O[x]^(terms + 1)], terms]; CoefficientList[A[x], x] Range[0, terms]!

Formula

E.g.f. A(x) satisfies: A'(x) = A(x)^2 - exp(-x).
From Vaclav Kotesovec, Jun 11 2020: (Start)
E.g.f.: exp(-x/2) * (BesselI(1, 2*exp(-x/2)) * (BesselK(0, 2) + BesselK(1, 2)) + (BesselI(0, 2) - BesselI(1, 2)) * BesselK(1, 2*exp(-x/2))) / ((BesselI(1, 2) - BesselI(0, 2)) * BesselK(0, 2*exp(-x/2)) + BesselI(0, 2*exp(-x/2)) * (BesselK(0, 2) + BesselK(1, 2))).
a(n) ~ n! / r^(n+1), where r = 1.4982609322383959128764444062824740935658895762... is the real root of the equation (BesselI(0, 2) - BesselI(1, 2)) * BesselK(0, 2*exp(-r/2)) = (BesselK(0, 2) + BesselK(1, 2)) * BesselI(0, 2*exp(-r/2)). (End)
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.