A255928 -id:A255928 - cp's OEIS frontend

A094088 E.g.f. 1/(2-cosh(x)) (even coefficients).

1, 1, 7, 121, 3907, 202741, 15430207, 1619195761, 224061282907, 39531606447181, 8661323866026007, 2307185279184885001, 734307168916191403507, 275199311597682485597221, 119956934012963778952439407

Offset: 0

Ralf Stephan, Apr 30 2004

With alternating signs, e.g.f.: 1/(2-cos(x)).
7 divides a(3n+2). Ira Gessel remarks: For any odd prime p, the coefficients of 1/(2-cosh(x)) as e.g.f. are periodic with period dividing p-1.
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.
a(n) is the number of ordered set partitions of {1,2,...,2n} into even size blocks. - Geoffrey Critzer, Dec 03 2012
Except for a(0), row sums of A241171. - Peter Bala, Aug 20 2014
Exp( Sum_{n >= 1} a(n)*x^n/n) is the o.g.f. for A255928. - Peter Bala, Mar 13 2015
Also the 2-packed words of degree n; cf. A011782, A000670, A094088, A243664, A243665, A243666 for k-packed words for 0<=k<=5. - Peter Luschny, Jul 06 2015

Seiichi Manyama, Table of n, a(n) for n = 0..235 (terms 0..40 from Peter Luschny)
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020.

Cf. A241171, A210676, A210657, A028296, A210672, A210674, A255928.

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(1); # N. J. A. Sloane, Mar 28 2012

nn=30;Select[Range[0,nn]!CoefficientList[Series[1/(2-Cosh[x]),{x,0,nn}],x],#>0&]  (* Geoffrey Critzer, Dec 03 2012 *)
a[0]=1; a[n_] := Sum[1/2*(1+(-1)^(2*n))*Sum[((-1)^(k-j)*Binomial[k, j]*Sum[(j-2*i )^(2*n)*Binomial[j, i], {i, 0, j}])/2^j, {j, 1, k}], {k, 1, n}]; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Apr 03 2015, after Vladimir Kruchinin *)

a(n):=b(2*n+2);
b(n):=sum(((sum(((sum((j-2*i)^n*binomial(j,i),i,0,j))*(-1)^(k-j)*binomial(k,j))/2^(j),j,1,k))*((-1)^n+1))/2,k,1,n/2); /* Vladimir Kruchinin, Apr 23 2011 */

a(n):=sum(sum((i-k)^(2*n)*binomial(2*k,i)*(-1)^(i),i,0,k-1)/(2^(k-1)),k,1,2*n); /* Vladimir Kruchinin, Oct 05 2012 */

a(n) = if (n == 0, 1, sum(k=1, n, binomial(2*n, 2*n-2*k)*a(n-k)));

def A094088(n) :
    @CachedFunction
    def intern(n) :
        if n == 0 : return 1
        if n % 2 != 0 : return 0
        return add(intern(k)*binomial(n,k) for k in range(n)[::2])
    return intern(2*n)
[A094088(n) for n in (0..14)]  # Peter Luschny, Jul 14 2012

1/(2-cosh(x)) = Sum_{n>=0} a(n)x^(2n)/(2n)! = 1 + x^2/2 + 7x^4/24 + 121x^6/720 + ...
Recurrence: a(0)=1, a(n) = Sum_{k=1..n} C(2n, 2n-2k)*a(n-k).
a(0)=1 and, for n>=1, a(n)=b(2*n) where b(n) = sum(k=1..n/2,((sum(j=1..k, ((sum(i=0..j,(j-2*i)^n*binomial(j,i)))*(-1)^(k-j)*binomial(k,j))/2^(j)))*((-1)^n+1))/2). - Vladimir Kruchinin, Apr 23 2011
E.g.f.: 1/(2-cosh(x))=8*(1-x^2)/(8 - 12*x^2 + x^4*U(0))  where U(k)= 1 + 4*(k+1)*(k+2)/(2*k+3 - x^2*(2*k+3)/(x^2 + 8*(k+1)*(k+2)*(k+3)/U(k+1))) ; (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Sep 30 2012
a(n) = Sum_{k=1..2*n} ( Sum_{i=0..k-1} (i-k)^(2*n) * binomial(2*k,i) * (-1)^i )/2^(k-1), n>0, a(0)=1. - Vladimir Kruchinin, Oct 05 2012
a(n) ~ 2*(2*n)! /(sqrt(3) * (log(2+sqrt(3)))^(2*n+1)). - Vaclav Kotesovec, Oct 19 2013

Corrected definition, Joerg Arndt, Apr 26 2011

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

Peter Bala, Mar 11 2015

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.

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

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);

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.

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

Peter Bala, Mar 11 2015

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.

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

#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);

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.

A255930 Expansion of exp( Sum_{n >= 1} A210674(n)*x^n/n ).

Original entry on oeis.org

1, 3, 33, 991, 63060, 7018860, 1206748720, 295775068680, 97835325011235, 41970842737399345, 22655642596496388759, 15025240474194493147857, 12008582230377080862401692, 11382727559611560650861409564, 12625404970864692720119281536900, 16199644066580777034289339157904220

Offset: 0

Peter Bala, Mar 11 2015

It appears that this sequence is integer valued.
The o.g.f. A(x) = 1 + 3*x + 33*x^2 + 991*x^3 + ... for this sequence is such that 1 + x*d/dx( log(A(x)) ) is the o.g.f. for A210674.
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 A255926(m = -3), A255882(m = -2), A255881(m = -1), A255928 (m = 1) and A255929(m = 2).
Note that u(n), as a polynomial in the variable m, is the n-th row generating polynomial of A241171.

Cf. A210674, A241171, A255926(m = -3), A255882(m = -2), A255881(m = -1), A255928(m = 1), A255929(m = 2).

#A255930
A210674 := proc (n) option remember; if n = 0 then 1 else 3*add(binomial(2*n, 2*k)*A210674(k), k = 0 .. n-1) end if; end proc:
A255930 := proc (n) option remember; if n = 0 then 1 else add(A210674(n-k)*A255930(k), k = 0 .. n-1)/n end if; end proc:
seq(A255930(n), n = 0 .. 15);

O.g.f.: exp(3*x + 57*x^2/2 + 2703*x^3/3 + 239277*x^4/4 + ...) = 1 + 3*x + 33*x^2 + 991*x^3 + 63060*x^4 + ....

a(0) = 1 and a(n) = 1/n*Sum_{k = 0..n-1} A210674(n-k)*a(k) for n >= 1.

cp's OEIS Frontend

A094088 E.g.f. 1/(2-cosh(x)) (even coefficients).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

Maxima

PARI

Sage

Formula

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Formula

A255930 Expansion of exp( Sum_{n >= 1} A210674(n)*x^n/n ).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Formula