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A258380 O.g.f. satisfies A^5(z) = 1/(1 - z)*( BINOMIAL(BINOMIAL(A(z))) )^4. 6
1, 9, 121, 2289, 58561, 1954281, 82055449, 4190913201, 252934661569, 17620643974921, 1390978843729657, 122629436549879473, 11935272648323364097, 1270531043409588667753, 146799401794935250517017, 18292108113357605085295345, 2444763748582590165449000065 (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 = 4.

From Peter Bala, Dec 06 2017: (Start)

a(n) appears to be of the form 8*m + 1. Calculation suggests that for k = 1,2,3,..., the sequence a(n) (mod 2^k) is purely periodic with period length a divisor of 2^(k-1). For example, a(n) (mod 16) = (1,9,9,1,1,9,9,1,...) seems to be purely periodic with period length 4 and a(n) (mod 32) = (1,9,25,17,1,9,25,17,...) seems to be purely periodic with period length 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+1,4)*a(n-1-i), where R(n,x) denotes the n-th row polynomial of A145901.

O.g.f.: A(z) = 1 + 9*z + 121*z^2 + 2289*z^3 + 58561*z^4 + ... satisfies A^5(z) = 1/(1 - z)*1/(1 - 2*z)^4*A^4(z/(1 - 2*z)).

O.g.f.: A(z) = exp( Sum_{k >= 1} R(k,4)*z^k/k ).

EXAMPLE

a(n) (mod 5) = [1, 4, 1, 4, 1, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 0, 0, 0, 0, 0, 3, 2, 3, 2, 3, 4, 1, 4, 1, 4, 4, 1, 4, 1, 4, 3, 2, 3, 2, 3, 0, 0, 0, 0, 0, 2, 3, 2, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 1, 4, 1, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 0, 0, 0, 0, 0, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 1, 4, 1, 4, 1, 0, 0, 0, 0, 0, 4, 1, 4, 1, 4, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 1, 4, 1, 4, 1, 0, 0, 0, 0, 0, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 0, 0, 0, 0, 0, 3, 2, 3, 2, 3, ...]. - Peter Bala, Dec 06 2017

MAPLE

#A258380

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 = 4

seq(a(n, 4), n = 0..16);

MATHEMATICA

R[n_, x_] := R[n, x] = If[n == 0, 1, 1 + x*Sum[Binomial[n, i]*2^(n - i)*R[i, x], {i, 0, n - 1}]];

a[n_, k_] := a[n, k] = If[n == 0, 1, 1/n*Sum[R[i + 1, k]*a[n - 1 - i, k], {i, 0, n - 1}]];

a[n_] := a[n, 4];

a /@ Range[0, 16] (* Jean-Fran├žois Alcover, Oct 02 2019 *)

CROSSREFS

Cf. A019538, A145901, A258377 (N = 1), A258378 (N = 2), A258379 (N = 3), A258381 (N = 5).

Sequence in context: A321847 A050353 A112941 * A045976 A276256 A276095

Adjacent sequences:  A258377 A258378 A258379 * A258381 A258382 A258383

KEYWORD

nonn,easy

AUTHOR

Peter Bala, May 28 2015

STATUS

approved

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Last modified April 20 01:55 EDT 2021. Contains 343118 sequences. (Running on oeis4.)