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A258378 O.g.f. satisfies A^3(z) = 1/(1 - z)*( BINOMIAL(BINOMIAL(A(z))) )^2. 6
1, 5, 37, 385, 5417, 99421, 2296077, 64510617, 2142013137, 82103710517, 3566271497845, 173005328363057, 9265752053418233, 542783129304580237, 34511577062800532573, 2366512551126709790793, 174056559606294111346593, 13666923859188010833522789, 1140970414332381380968275653 (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 = 2.

From Peter Bala, Dec 06 2017: (Start)

a(n) appears to be of the form 4*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 8) = (1, 5, 5, 1, 1, 5, 5, 1,...) seems to be purely periodic with period length 4 and a(n) (mod 16) = (1, 5, 5, 1, 9, 13, 13, 9, 1, 5, 5, 1, 9, 13, 13, 9,...) seems to be purely periodic with period length 8 (both checked up to n = 1000). (End)

LINKS

Table of n, a(n) for n=0..18.

N. J. A. Sloane, Transforms.

FORMULA

a(0) = 1 and for n >= 1, a(n) = 1/n*Sum_{i = 0..n-1} R(i+1,2)*a(n-1-i), where R(n,x) denotes the n-th row polynomial of A145901.

O.g.f.: A(z) = 1 + 5*z + 37*z^2 + 385*z^3 + 5417*z^4 + ... satisfies A^3(z) = 1/(1 - z)*1/(1 - 2*z)^2*A^2(z/(1 - 2*z)).

O.g.f.: A(z) = exp( Sum_{k >= 1} R(k,2)*z^k/k ).

MAPLE

#A258378

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 = 2

seq(a(n, 2), n = 0..18);

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, 2];

a /@ Range[0, 18] (* Jean-Fran├žois Alcover, Oct 02 2019 *)

CROSSREFS

Cf. A019538, A145901, A258377 (N = 1), A258379 (N = 3), A258380 (N = 4), A258381 (N = 5).

Sequence in context: A055869 A208231 A112937 * A273954 A092649 A179923

Adjacent sequences:  A258375 A258376 A258377 * A258379 A258380 A258381

KEYWORD

nonn,easy

AUTHOR

Peter Bala, May 28 2015

STATUS

approved

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Last modified May 12 07:28 EDT 2021. Contains 343821 sequences. (Running on oeis4.)