cp's OEIS Frontend

A306727 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) is the number of partitions of 3*n into powers of 3 less than or equal to 3^k.

%I A306727 #42 Apr 27 2020 06:22:47
%S A306727 1,1,1,1,2,1,1,2,3,1,1,2,3,4,1,1,2,3,5,5,1,1,2,3,5,7,6,1,1,2,3,5,7,9,
%T A306727 7,1,1,2,3,5,7,9,12,8,1,1,2,3,5,7,9,12,15,9,1,1,2,3,5,7,9,12,15,18,10,
%U A306727 1,1,2,3,5,7,9,12,15,18,22,11,1,1,2,3,5,7,9,12,15,18,23,26,12,1
%N A306727 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) is the number of partitions of 3*n into powers of 3 less than or equal to 3^k.
%C A306727 Column sequences converge to A005704.
%H A306727 Serguei Zolotov, <a href="/A306727/b306727.txt">Table of n, a(n) for n = 0..10584</a>
%F A306727 G.f. of column k: 1/(1-x) * 1/Product_{j=0..k-1} (1 - x^(3^j)).
%e A306727 A(3,3) = 5, because there are 5 partitions of 3*3=9 into powers of 3 less than or equal to 3^3=9: [9], [3,3,3], [3,3,1,1,1], [3,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1].
%e A306727 Square array A(n,k) begins:
%e A306727   1, 1, 1, 1, 1, 1,  ...
%e A306727   1, 2, 2, 2, 2, 2,  ...
%e A306727   1, 3, 3, 3, 3, 3,  ...
%e A306727   1, 4, 5, 5, 5, 5,  ...
%e A306727   1, 5, 7, 7, 7, 7,  ...
%e A306727   1, 6, 9, 9, 9, 9,  ...
%t A306727 nmax = 12;
%t A306727 f[k_] := f[k] = 1/(1-x) 1/Product[1-x^(3^j), {j, 0, k-1}] + O[x]^(nmax+1) // CoefficientList[#, x]&;
%t A306727 A[n_, k_] := f[k][[n+1]];
%t A306727 Table[A[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Nov 20 2019 *)
%o A306727 (Python)
%o A306727 def aseq(p, x, k):
%o A306727     # generic algorithm for any p - power base, p=3 for this sequence
%o A306727     if x < 0:
%o A306727         return 0
%o A306727     if x < p:
%o A306727         return 1
%o A306727     # coefficients
%o A306727     arr = [0]*(x+1)
%o A306727     arr[0] = 1
%o A306727     m = p**k
%o A306727     while m > 0:
%o A306727         for i in range(m, x+1, m):
%o A306727             arr[i] += arr[i-m]
%o A306727         m //= p
%o A306727     return arr[x]
%o A306727 def A(n, k):
%o A306727     p = 3
%o A306727     return aseq(p, p*n, k)
%o A306727 # A(n, k), 5 = A(3, 3) = aseq(3, 3*3, 3)
%o A306727 # _Serguei Zolotov_, Mar 13 2019
%Y A306727 Main diagonal gives A005704.
%Y A306727 A181322 gives array for base p=2.
%K A306727 nonn,tabl
%O A306727 0,5
%A A306727 _Serguei Zolotov_, Mar 06 2019