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A258381 O.g.f. satisfies A^6(z) = 1/(1 - z)*( BINOMIAL(BINOMIAL(A(z))) )^5. 6
1, 11, 181, 4191, 131241, 5360883, 275510493, 17223156423, 1272268864593, 108480982129883, 10481174173743109, 1130938869235448879, 134719322898080187129, 17552325198110327173059, 2482129971814696069384749, 378542038806168341351484567, 61920836368469049844434420897 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The binomial transform of an o.g.f. A(z) is given by BINOMIAL(A(z)) = 1/(1 - z)*A(z/(1 - z)). For general remarks on a solution to the functional equation A^(N+1)(z) = 1/(1 - z)*( BINOMIAL(BINOMIAL(A(z))) )^N for integer N, and the connection with triangle A145901 see A258377 (case N = 1). This is the case N = 5.

From Peter Bala, Dec 06 2017: (Start)

a(n) appears to be always odd. Calculation suggests that for k = 1,2,3,... the sequence a(n) (mod 2^k) is purely periodic with period 2^(k-1). For example, a(n) (mod 4) = (1,3,1,3,...) seems to be purely periodic with period 2 and a(n) (mod 8) = (1,3,5,7,1,3,5,7,...) seems to be purely periodic with period 4 (both checked up to n = 1000).

The sequences a(n) (mod k), for other values of k, appear to have interesting but more complicated patterns. An example is given below.

(End)

LINKS

Table of n, a(n) for n=0..16.

N. J. A. Sloane, Transforms.

FORMULA

a(0) = 1 and for n >= 1, a(n) = 1/n*Sum_{i = 0..n-1} R(i+1,5)*a(n-1-i), where R(n,x) denotes the n-th row polynomial of A145901.

O.g.f.: A(z) = 1 + 11*z + 181*z^2 + 4191*z^3 + 131241*z^4 + ... satisfies A^6(z) = 1/(1 - z)*1/(1 - 2*z)^5*A^5(z/(1 - 2*z)).

O.g.f.: A(z) = exp( Sum_{k >= 1} R(k,5)*z^k/k ).

EXAMPLE

a(n) (mod 5) begins [1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1,...]. - Peter Bala, Dec 06 2017

MAPLE

#A258381

with(combinat):

#recursively define the row polynomials R(n, x) of A145901

R := proc (n, x) option remember; if n = 0 then 1 else 1 + x*add(binomial(n, i)*2^(n-i)*R(i, x), i = 0..n-1) end if; end proc:

#define a family of sequences depending on an integer parameter k

a := proc (n, k) option remember; if n = 0 then 1 else 1/n*add(R(i+1, k)*a(n-1-i, k), i = 0..n-1) end if; end proc:

# display the case k = 5

seq(a(n, 5), n = 0..16);

CROSSREFS

Cf. A019538, A145901, A258377 (N = 1), A258378 (N = 2), A258379 (N = 3), A258380 (N = 4).

Sequence in context: A009118 A321848 A112943 * A057618 A270663 A068648

Adjacent sequences:  A258378 A258379 A258380 * A258382 A258383 A258384

KEYWORD

nonn,easy

AUTHOR

Peter Bala, May 28 2015

STATUS

approved

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Last modified March 26 04:32 EDT 2019. Contains 321481 sequences. (Running on oeis4.)