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G.f.: A(x) = Sum_{n>=0} (1 + x)^(n(n+1)/2) / 2^(n+1).
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%I #9 Jan 15 2020 22:47:44

%S 1,2,12,124,1800,33648,769336,20796960,648841680,22945907520,

%T 907036108432,39631833652320,1896696894062880,98669609894805600,

%U 5543804125505195040,334563594743197602272,21583554094995765302592

%N G.f.: A(x) = Sum_{n>=0} (1 + x)^(n(n+1)/2) / 2^(n+1).

%C a(n) is the number of nonnegative integer matrices with n distinct columns and any number of nonzero rows with 2 ones in every column and columns in decreasing lexicographic order. - _Andrew Howroyd_, Jan 15 2020

%H Andrew Howroyd, <a href="/A173219/b173219.txt">Table of n, a(n) for n = 0..100</a>

%F a(n) = A265937(n)/2. - _Vaclav Kotesovec_, Oct 08 2019

%F a(n) ~ 2^n * n^n / (2^(log(2)/4) * log(2)^(2*n+1) * exp(n)). - _Vaclav Kotesovec_, Oct 08 2019

%F a(n) = 2*A121251(n) for n > 0. - _Andrew Howroyd_, Jan 15 2020

%t Table[Sum[StirlingS1[n, j] * Sum[Binomial[j, s]*HurwitzLerchPhi[1/2, -j - s, 0], {s, 0, j}] / 2^(j+1), {j, 0, n}] / n!, {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 08 2019 *)

%o (PARI) {a(n)=local(A=sum(m=0,n^2+100,(1+x +O(x^(n+2)))^(m*(m+1)/2)/2^(m+1)));round(polcoeff(A,n))}

%Y Row n=2 of A331278.

%Y Cf. A121251, A173217, A173218.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 05 2010