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%I #65 Jan 01 2018 04:15:10
%S 1,0,0,0,1,0,0,1,0,0,0,1,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,0,1,2,0,0,0,
%T 0,0,0,1,2,0,0,0,0,0,0,0,1,3,1,0,0,0,0,0,0,0,1,3,1,0,0,0,0,0,0,0,0,1,
%U 4,2,0,0,0,0,0,0,0,0,0,1,4,3,0,0,0,0,0,0,0,0,0,0,1,5,4,0,0,0,0,0,0,0,0,0,0
%N Table read downwards by antidiagonals: T(n,k) is the number of partitions of k into n distinct positive parts, k, n >= 0.
%C The array T is basically a transposed version of A008289, padded with zeros where needed.
%C Entry T(n,k) contains the count of the pairs such that <n, k> = <A000120(q), A029931(q)> for any q >= 0.
%C Index of the first nonzero term on the n-th row is k=A000217(n).
%H Sergey Shishmintzev and Andrew Howroyd, <a href="/A218218/b218218.txt">Table of n, a(n) for n = 0..299</a>
%H Sergey Shishmintzev, <a href="https://oeis.org/plot2a?name1=A029931&name2=A000120&tform1=untransformed&tform2=untransformed&shift=0&radiop1=xy&drawpoints=true">Scatter plot of A029931(q) vs. A000120(q) for 0 <= q < 2^10</a>
%H OEIS Wiki, <a href="http://oeis.org/wiki/Empty_sum">Empty sum</a>
%e Array begins:
%e 1,0,0,0,0,0,0,0,0,0,0,0,0,0, 0, 0, 0, 0, 0,...
%e 0,1,1,1,1,1,1,1,1,1,1,1,1,1, 1, 1, 1, 1, 1,...
%e 0,0,0,1,1,2,2,3,3,4,4,5,5,6, 6, 7, 7, 8, 8,...
%e 0,0,0,0,0,0,1,1,2,3,4,5,7,8,10,12,14,16,19,...
%e 0,0,0,0,0,0,0,0,0,0,1,1,2,3, 5, 6, 9,11,15,...
%e 0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0, 1, 1, 2, 3,...
%e ...
%e T(0, 0) = 1, because there is only zero, which has no parts (n=0) and the sum with no parts is zero (k=0). Or, there is only a single q such that <n, k> = <0, 0> = <A000120(q), A029931(q)>, namely q=0.
%e T(1, 10) = 1, because there is only a single partition of k=10 into n=1 positive part. Or, there is only a single pair such that <n, k> = <1, 10> = <A000120(q), A029931(q)> and this pair has q=512.
%e T(2, 3) = 1, because there is only a single partition of k=3 into n=2 positive parts: 3 = 2 + 1. Or, for q>=0, there is only a single pair such that <n, k> = <2, 3> = <A000120(q), A029931(q)> and this pair has q=3.
%e T(3, 3) = 0, because we cannot decompose k=3 into 3 positive parts. Or, there is no q >= 0 such that <n, k> = <3, 3> = <A000120(q), A029931(q)>.
%e T(4, 15) = 6, because there are 6 partitions of 15 into 4 distinct positive parts: 15 = 6 + 4 + 3 + 2; 15 = 6 + 5 + 3 + 1; 15 = 7 + 4 + 3 + 1; 15 = 7 + 5 + 2 + 1; 15 = 8 + 4 + 2 + 1; 15 = 9 + 3 + 2 + 1. Or, there are the corresponding pairs:
%e <4, 15> = <A000120(46), A029931(46)>;
%e <4, 15> = <A000120(53), A029931(53)>;
%e <4, 15> = <A000120(77), A029931(77)>;
%e <4, 15> = <A000120(83), A029931(83)>;
%e <4, 15> = <A000120(139), A029931(139)>;
%e <4, 15> = <A000120(263), A029931(263)>.
%t max = 14; coes = CoefficientList[ Product[1 + t*x^k, {k, 1, max}], {t, x}]; Flatten[ Table[coes[[n, k-n+1]], {k, 1, max}, {n, 1, k}]] (* _Jean-François Alcover_, Nov 08 2012, from given g.f. *)
%o (Python)import math; L = 10
%o def HW(x):
%o ..if x==0: return 0
%o ..return sum([ (x>>i)&1 for i in range(int(math.log(x, 2))+1) ])
%o def TW(x):
%o ..if x==0: return 0
%o ..return sum([ ((x>>i)&1)*(i+1) for i in range(int(math.log(x, 2))+1) ])
%o max_h = L+1; max_t = lambda h: L + (h-1)*h/2 + 1
%o T = [None] * max_h
%o for h in range(max_h): T[h] = [0] * max_t(h)
%o for i in range(2**L):
%o ..h = HW(i); t = TW(i)
%o ..if t < max_t(h): T[h][t] = T[h][t] + 1
%o for h in range(max_h):
%o ..for t in range(max_t(h)): print T[h][t], ';',
%o ..print
%Y Cf. A008289.
%K nonn,tabl
%O 0,31
%A _Sergey Shishmintzev_, Oct 23 2012
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