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 1-4 of 4 results.

A210674 a(0)=1; thereafter a(n) = 3*Sum_{k=1..n} binomial(2n,2k)*a(n-k).

Original entry on oeis.org

1, 3, 57, 2703, 239277, 34041603, 7103141697, 2043564786903, 775293596155317, 375019773885750603, 225270492555606688137, 164517775480287009524703, 143555042043378357951428157, 147502150365016885913874781203, 176273363579960990244526939543377, 242422256082395157286909073370272103
Offset: 0

Views

Author

N. J. A. Sloane, Mar 28 2012

Keywords

Comments

Consider the sequence defined by a(0) = 1; thereafter a(n) = c*Sum_{k = 1..n} binomial(2n,2k)*a(n-k). For c = -3, -2, -1, 1, 2, 3, 4 this is A210676, A210657, A028296, A094088, A210672, A210674, A249939.
Exp( Sum_{n >= 1} a(n)*x^n/n) is the o.g.f. for A255930. - Peter Bala, Mar 13 2015
In general, for c > 0 is a(n) ~ sqrt(Pi/(2*c+1)) * 2^(2*n+2) * n^(2*n+1/2) / (exp(2*n) * (log((c + 1 + sqrt(2*c+1)) / c))^(2*n+1)) = 2*(2*n)!/(sqrt(2*c+1)*(arccosh((c+1)/c))^(2*n+1)). - Vaclav Kotesovec, Mar 13 2015
For c < 0 is a(n) ~ (-1)^n * (2*n)! / (sqrt(-2*c-1) * 2^(2*n) * arccos(sqrt((2*c + 1)/(2*c)))^(2*n+1)). - Vaclav Kotesovec, Mar 14 2015

Links

Crossrefs

Cf. A210676 (c=-3), A210657 (c=-2), A028296 (c=-1), A094088 (c=1), A210672 (c=2), A249939 (c=4).
Cf. A255930.

Programs

  • Maple
    f:=proc(n,k) option remember;  local i;
if n=0 then 1
else k*add(binomial(2*n,2*i)*f(n-i,k),i=1..floor(n)); fi; end;
g:=k->[seq(f(n,k),n=0..40)];
g(3);
  • Mathematica
    nmax=20; Table[(CoefficientList[Series[1/(4-3*Cosh[x]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[2*n+1]], {n,0,nmax}] (* Vaclav Kotesovec, Mar 14 2015 *)

Formula

a(n) ~ sqrt(Pi/7) * 2^(2*n+2) * n^(2*n+1/2) / (exp(2*n) * (log((4 + sqrt(7)) / 3))^(2*n+1)). - Vaclav Kotesovec, Mar 13 2015
E.g.f.: 1/(4-3*cosh(x)) (even coefficients). - Vaclav Kotesovec, Mar 14 2015

A255926 Expansion of exp( Sum_{n >= 1} A210676(n)*x^n/n ).

Original entry on oeis.org

1, -3, 30, -802, 45414, -4508190, 692197470, -151610017950, 44827810930305, -17193060505570335, 8298004578522898140, -4920774627129981351120, 3516683319021255757053900, -2980761698101283167670391780, 2956463734237276273792194346560, -3392220222832838757465019626175680
Offset: 0

Views

Author

Peter Bala, Mar 11 2015

Keywords

Comments

It appears that this sequence is integer valued.
The o.g.f. A(x) = 1 - 3*x + 30*x^2 - 802*x^3 + ... for this sequence is such that 1 + x*d/dx( log(A(x)) ) is the o.g.f. for A210676.
This sequence is the particular case m = -3 of the following general conjecture.
Let m be an integer and consider the sequence u(n) defined by the recurrence u(n) = m*Sum_{k = 0..n-1} binomial(2*n,2*k)*u(k) with the initial condition u(0) = 1. Then the expansion of exp( Sum_{n >= 1} u(n)*x^n/n ) has integer coefficients.
For cases see A255882(m = -2), A255881(m = -1), A255928(m = 1), A255929(m = 2) and A255930(m = 3).
Note that u(n), as a polynomial in the variable m, is the n-th row polynomial of A241171.

Crossrefs

Cf. A210676, A241171, A255882(m = -2), A255881(m = -1), A255928(m = 1), A255929(m = 2), A255930(m = 3).

Programs

  • Maple
    A210676 := proc (n) option remember; if n = 0 then 1 else -3*add(binomial(2*n, 2*k)*A210676(k), k = 0 .. n-1) end if; end proc:
A255926 := proc (n) option remember; if n = 0 then 1 else add(A210676(n-k)*A255926(k), k = 0 .. n-1)/n end if; end proc:
seq(A255926(n), n = 0 .. 16);

Formula

O.g.f.: exp(-3*x + 51*x^2/2 - 2163*x^3/3 + 171231*x^4/4 + ...) = 1 - 3*x + 30*x^2 - 802*x^3 + 45414*x^4 - ....
a(0) = 1 and a(n) = (1/n)*Sum_{k = 0..n-1} A210676(n-k)*a(k) for n >= 1.

A255928 Expansion of exp( Sum_{n >= 1} A094088(n)*x^n/n ).

Original entry on oeis.org

1, 1, 4, 44, 1025, 41693, 2617128, 234091692, 28251572652, 4421489003700, 870650503128708, 210629395976568828, 61405707768736724472, 21231253444779700476672, 8589776776743377081599500, 4020181599664131540547091076, 2155088041310451318611119556661
Offset: 0

Views

Author

Peter Bala, Mar 11 2015

Keywords

Comments

It appears that this sequence is integer valued.
The o.g.f. A(x) = 1 + x + 4*x^2 + 44*x^3 + ... for this sequence is such that 1 + x*d/dx( log(A(x)) ) is the o.g.f. for A094088.
This sequence is the particular case m = 1 of the following general conjecture.
Let m be an integer and consider the sequence u(n) defined by the recurrence u(n) = m*Sum_{k = 0..n-1} binomial(2*n,2*k)*u(k) with the initial condition u(0) = 1. Then the expansion of exp( Sum_{n >= 1} u(n)*x^n/n ) has integer coefficients.
For cases see A255926(m = -3), A255882(m = -2), A255881(m = -1), A255929(m = 2) and A255930(m = 3).
Note that u(n), as a polynomial in the variable m, is the n-th row generating polynomial of A241171.

Crossrefs

Cf. A094088, A241171, A255926(m = -3), A255882(m = -2), A255881(m = -1), A255929(m = 2), A255930(m = 3).

Programs

  • Maple
    A094088 := proc (n) option remember; if n = 0 then 1 else add(binomial(2*n, 2*k)*A094088(k), k = 0 .. n-1) end if; end proc:
A255928 := proc (n) option remember; if n = 0 then 1 else add(A094088(n-k)*A255928(k), k = 0 .. n-1)/n end if; end proc:
seq(A255928(n), n = 0 .. 16);

Formula

O.g.f.: exp(x + 7*x^2/2 + 121*x^3/3 + 3907*x^4/4 + ...) = 1 + x + 4*x^2 + 44*x^3 + 1025*x^4 + ....
a(0) = 1 and a(n) = (1/n)*Sum_{k = 0..n-1} A094088(n-k)*a(k) for n >= 1.

A255929 Expansion of exp( Sum_{n >= 1} A210672(n)*x^n/n ).

Original entry on oeis.org

1, 2, 15, 308, 13399, 1019106, 119698377, 20039968920, 4527610159068, 1326616296092984, 489092182592254708, 221537815033845709776, 120928125204565597029220, 78286897353506845258973144, 59305342759674536454338570652, 51970719684035315747385128783808
Offset: 0

Views

Author

Peter Bala, Mar 11 2015

Keywords

Comments

It appears that this sequence is integer valued.
The o.g.f. A(x) = 1 + 2*x + 15*x^2 + 308*x^3 + ... for this sequence is such that 1 + x*d/dx( log(A(x)) ) is the o.g.f. for A210672.
This sequence is the particular case m = 2 of the following general conjecture.
Let m be an integer and consider the sequence u(n) defined by the recurrence u(n) = m*Sum_{k = 0..n-1} binomial(2*n,2*k)*u(k) with the initial condition u(0) = 1. Then the expansion of exp( Sum_{n >= 1} u(n)*x^n/n ) has integer coefficients.
For cases see A255926(m = -3), A255882(m = -2), A255881(m = -1), A255928(m = 1) and A255930(m = 3).
Note that u(n), as a polynomial in the variable m, is the n-th row generating polynomial of A241171.

Crossrefs

A210672, A241171, A255926(m = -3), A255882(m = -2), A255881(m = -1), A255928(m = 1), A255930(m = 3).

Programs

  • Maple
    #A255929
A210672 := proc (n) option remember; if n = 0 then 1 else 2*add(binomial(2*n, 2*k)*A210672(k), k = 0 .. n-1) end if; end proc:
A255929 := proc (n) option remember; if n = 0 then 1 else add(A210672(n-k)*A255929(k), k = 0 .. n-1)/n end if; end proc:
seq(A255929(n), n = 0 .. 15);

Formula

O.g.f.: exp(2*x + 26*x^2/2 + 842*x^3/3 + 50906*x^4/4 + ...) = 1 + 2*x + 15*x^2 + 308*x^3 + 13399*x^4 + ....
a(0) = 1 and a(n) = 1/n*Sum_{k = 0..n-1} A210672(n-k)*a(k) for n >= 1.
Showing 1-4 of 4 results.

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

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