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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 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 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.)