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A145513 Number of partitions of 10^n into powers of 10.

%I A145513 #21 Feb 01 2017 12:46:02
%S A145513 1,2,12,562,195812,515009562,10837901390812,1899421190329234562,
%T A145513 2851206628197445401265812,37421114946843687272702534859562,
%U A145513 4362395890943439751990308572939648140812,4573514084633441973328831327010967245403925484562,43557001521047571730475817291330175020887917015964570015812
%N A145513 Number of partitions of 10^n into powers of 10.
%C A145513 a(n) = A179051(10^n); for n>0: a(n) = A179052(10^(n-1)). - _Reinhard Zumkeller_, Jun 27 2010
%H A145513 Alois P. Heinz, <a href="/A145513/b145513.txt">Table of n, a(n) for n = 0..45</a>
%F A145513 a(n) = [x^(10^n)] 1/Product_{j>=0} (1-x^(10^j)).
%e A145513 a(1) = 2, because there are 2 partitions of 10^1 into powers of 10: [1,1,1,1,1,1,1,1,1,1], [10].
%p A145513 g:= proc(b,n,k) option remember; local t; if b<0 then 0 elif b=0 or n=0 or k<=1 then 1 elif b>=n then add(g(b-t, n, k) *binomial(n+1, t) *(-1)^(t+1), t=1..n+1); else g(b-1, n, k) +g(b*k, n-1, k) fi end: a:= n-> g(1,n,10): seq(a(n), n=0..13);
%t A145513 g[b_, n_, k_] := g[b, n, k] = Module[{t}, Which[b < 0, 0, b == 0 || n == 0 || k <= 1, 1, b >= n, Sum[g[b - t, n, k]*Binomial[n + 1, t] *(-1)^(t + 1), {t, 1, n + 1}], True, g[b - 1, n, k] + g[b*k, n - 1, k]]]; a[n_] := g[1, n, 10]; Table[a[n], {n, 0, 13}] (* _Jean-François Alcover_, Feb 01 2017, after _Alois P. Heinz_ *)
%o A145513 (Haskell)
%o A145513 import Data.MemoCombinators (memo2, list, integral)
%o A145513 a145513 n = a145513_list !! n
%o A145513 a145513_list = f [1] where
%o A145513    f xs = (p' xs $ last xs) : f (1 : map (* 10) xs)
%o A145513    p' = memo2 (list integral) integral p
%o A145513    p _ 0 = 1; p [] _ = 0
%o A145513    p ks'@(k:ks) m = if m < k then 0 else p' ks' (m - k) + p' ks m
%o A145513 -- _Reinhard Zumkeller_, Nov 27 2015
%Y A145513 Cf. 10th column of A145515, A007318.
%Y A145513 Cf. A011557, A002577, A078125.
%K A145513 nonn
%O A145513 0,2
%A A145513 _Alois P. Heinz_, Oct 11 2008