A258380 - OEIS

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A258380

O.g.f. satisfies A^5(z) = 1/(1 - z)*( BINOMIAL(BINOMIAL(A(z))) )^4.

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/nSum_{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 + 9z + 121z^2 + 2289z^3 + 58561z^4 + ... satisfies A^5(z) = 1/(1 - z)1/(1 - 2z)^4A^4(z/(1 - 2z)).` `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 + xadd(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/nadd(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 + xSum[Binomial[n, i]2^(n - i)R[i, x], {i, 0, n - 1}]];` `a[n_, k_] := a[n, k] = If[n == 0, 1, 1/nSum[R[i + 1, k]a[n - 1 - i, k], {i, 0, n - 1}]];` `a[n_] := a[n, 4];` `a /@ Range[0, 16] ( Jean-François 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.)