A147955 Expansion of Product_{k >= 0} (1 + A147954(k)*x^k).

1, 1, 1, 3, 4, 7, 10, 15, 22, 34, 46, 65, 93, 123, 175, 245, 324, 425, 592, 764, 1015, 1352, 1750, 2266, 2931, 3793, 4897, 6259, 7930, 10080, 12788, 16047, 20176, 25482, 31641, 39630, 49306, 60932, 75552, 93432, 114597, 141013, 173259, 211595, 258933, 316375, 384359, 466927, 566443

Offset: 0

Author

Roger L. Bagula, Nov 17 2008

Keywords

• nonn

Examples

			From _Petros Hadjicostas_, Apr 21 2020: (Start)
Let f(m) = A147954(m). Using the strict partitions of n (see A000009), we get:
a(1) = f(1) = 1,
a(2) = f(2) = 1,
a(3) = f(3) + f(1)*f(2) = 2 + 1*1 = 3,
a(4) = f(4) + f(1)*f(3) = 2 + 1*2 = 4,
a(5) = f(5) + f(1)*f(4) + f(2)*f(3) = 3 + 1*2 + 1*2 = 7,
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,
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)
		

Crossrefs

Cf. A000009, A004001, A147655, A147665, A147871, A147954.

Programs

Maple

f := proc(n) local v; option remember;
if n = 0 then v := 0; end if;
if n = 1 or n = 2 then v := 1; end if;
if 3 <= n and n <= 5 then v := f(f(n - 1)) + f(n - f(n - 1)); end if;
if 6 <= n and 5 <> n mod 6 then v := f(f(n - 1)) + f(f(floor(n/6))); end if;
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
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, b(n-i, i-1)*f(i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50); # Petros Hadjicostas, Apr 21 2020 (using Alois P. Heinz's program from A147655)

Mathematica

f[0] = 0; f[1] = 1; f[2] = 1;
f[n_] := f[n] =
   f[f[n - 1]] +
    If[n < 6, f[n - f[n - 1]],
     If[Mod[n, 6] == 0, f[f[n/6]],
      If[Mod[n, 6] == 1, f[f[(n - 1)/6]],
       If[Mod[n, 6] == 2, f[f[(n - 2)/6]],
        If[Mod[n, 6] == 3, f[f[(n - 3)/6]],
         If[Mod[n, 6] == 4, f[f[(n - 4)/6]], f[n - f[(n - 5)/6]]]]]]]];
P[x_, n_] := P[x, n] = Product[1 + f[m]*x^m, {m, 0, n}];
Take[CoefficientList[P[x, 45], x], 45]

Formula

a(n) = [x^n] Product_{k >= 0} (1 + A147954(k)*x^k).

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

Extensions

Name, data, and Mathematica program edited and corrected by Petros Hadjicostas, Apr 21 2020