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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 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 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-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.)