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%I #30 Apr 22 2020 07:01:11
%S 1,1,1,3,4,7,10,15,22,34,46,65,93,123,175,245,324,425,592,764,1015,
%T 1352,1750,2266,2931,3793,4897,6259,7930,10080,12788,16047,20176,
%U 25482,31641,39630,49306,60932,75552,93432,114597,141013,173259,211595,258933,316375,384359,466927,566443
%N Expansion of Product_{k >= 0} (1 + A147954(k)*x^k).
%F a(n) = [x^n] Product_{k >= 0} (1 + A147954(k)*x^k).
%F a(n) = Sum_{(b_1,...,b_n)} f(1)^b_1 * f(2)^b_2 * ... * f(n)^b_n, where f(m) = A147954(m), and the sum is taken over all lists (b_1,...,b_n) with b_j in {0,1} and Sum_{j=1..n} j*b_j = n. - _Petros Hadjicostas_, Apr 21 2020
%e From _Petros Hadjicostas_, Apr 21 2020: (Start)
%e Let f(m) = A147954(m). Using the strict partitions of n (see A000009), we get:
%e a(1) = f(1) = 1,
%e a(2) = f(2) = 1,
%e a(3) = f(3) + f(1)*f(2) = 2 + 1*1 = 3,
%e a(4) = f(4) + f(1)*f(3) = 2 + 1*2 = 4,
%e a(5) = f(5) + f(1)*f(4) + f(2)*f(3) = 3 + 1*2 + 1*2 = 7,
%e a(6) = f(6) + f(1)*f(5) + f(2)*f(4) + f(1)*f(2)*f(3) = 3 + 1*3 + 1*2 + 1*1*2 = 10,
%e a(7) = f(7) + f(1)*f(6) + f(2)*f(5) + f(3)*f(4) + f(1)*f(2)*f(4) = 3 + 1*3 + 1*3 + 2*2 + 1*1*2 = 15. (End)
%p f := proc(n) local v; option remember;
%p if n = 0 then v := 0; end if;
%p if n = 1 or n = 2 then v := 1; end if;
%p if 3 <= n and n <= 5 then v := f(f(n - 1)) + f(n - f(n - 1)); end if;
%p if 6 <= n and 5 <> n mod 6 then v := f(f(n - 1)) + f(f(floor(n/6))); end if;
%p if 6 <= n and 5 = n mod 6 then v := f(f(n - 1)) + f(n - f(floor(n/6))); end if; v; end proc; # this gives sequence A147954
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-1) +`if`(i>n, 0, b(n-i, i-1)*f(i))))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..50); # _Petros Hadjicostas_, Apr 21 2020 (using _Alois P. Heinz_'s program from A147655)
%t f[0] = 0; f[1] = 1; f[2] = 1;
%t f[n_] := f[n] =
%t f[f[n - 1]] +
%t If[n < 6, f[n - f[n - 1]],
%t If[Mod[n, 6] == 0, f[f[n/6]],
%t If[Mod[n, 6] == 1, f[f[(n - 1)/6]],
%t If[Mod[n, 6] == 2, f[f[(n - 2)/6]],
%t If[Mod[n, 6] == 3, f[f[(n - 3)/6]],
%t If[Mod[n, 6] == 4, f[f[(n - 4)/6]], f[n - f[(n - 5)/6]]]]]]]];
%t P[x_, n_] := P[x, n] = Product[1 + f[m]*x^m, {m, 0, n}];
%t Take[CoefficientList[P[x, 45], x], 45]
%Y Cf. A000009, A004001, A147655, A147665, A147871, A147954.
%K nonn
%O 0,4
%A _Roger L. Bagula_, Nov 17 2008
%E Name, data, and Mathematica program edited and corrected by _Petros Hadjicostas_, Apr 21 2020
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