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A152474 Triangle T(n,k) read by rows: Sum_{k=0..binomial(n,2)} T(n,k)*q^k = n!*Sum_{pi} faq(n,q)/Product_{i=1..n} e(i)!*faq(i,q)^e(i), where pi runs over all nonnegative integer solutions to e(1) + 2*e(2) + ... + n*e(n) = n and faq(i,q) = Product_{j=1..i} (q^j-1)/(q-1), i = 1..n.

%I A152474 #23 Feb 02 2025 17:30:08
%S A152474 1,1,3,1,13,8,8,1,73,63,89,78,41,15,1,501,544,909,1095,1200,842,680,
%T A152474 315,129,24,1,4051,5225,9734,13799,18709,20441,20520,18101,14831,
%U A152474 10200,5891,3199,1109,314,35,1,37633,55656,112370,177457,270746,352969,442897
%N A152474 Triangle T(n,k) read by rows: Sum_{k=0..binomial(n,2)} T(n,k)*q^k = n!*Sum_{pi} faq(n,q)/Product_{i=1..n} e(i)!*faq(i,q)^e(i), where pi runs over all nonnegative integer solutions to e(1) + 2*e(2) + ... + n*e(n) = n and faq(i,q) = Product_{j=1..i} (q^j-1)/(q-1), i = 1..n.
%H A152474 Alois P. Heinz, <a href="/A152474/b152474.txt">Rows n = 0..36, flattened</a>
%F A152474 Sum_{k=0..binomial(n,2)} T(n,k)*exp(2*Pi*I*k/n) = n!. - _Vladeta Jovovic_, Dec 05 2008
%F A152474 From _Paul D. Hanna_, Dec 15 2008: (Start)
%F A152474 E.g.f.: A(x,q) = exp(e_q(x,q) - 1) = Sum_{n>=0} Sum_{k=0..n(n-1)/2} T(n,k)*q^k*x^n/(n!*faq(n,q)) where e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) and faq(n,q) = Product_{j=1..n} (q^j-1)/(q-1) with faq(0,q)=1.
%F A152474 Sum_{k=0..n(n-1)/2} T(n,k)*(-1)^k = n!*A000110((n+1)/2), where A000110 is the Bell numbers. (End)
%e A152474 Triangle T(n,k) begins:
%e A152474     1;
%e A152474     1;
%e A152474     3,   1;
%e A152474    13,   8,   8,    1;
%e A152474    73,  63,  89,   78,   41,  15,   1;
%e A152474   501, 544, 909, 1095, 1200, 842, 680, 315, 129, 24, 1;
%e A152474   ...
%o A152474 (PARI) {T(n,k)=local(e_q=sum(j=0,n,x^j/prod(i=1,j,(q^i-1)/(q-1)))+x*O(x^n)); n!*polcoeff(polcoeff(exp(e_q-1),n,x)*prod(j=1,n,(q^j-1)/(q-1)),k,q)} \\ _Paul D. Hanna_, Dec 15 2008
%Y A152474 Cf. A000262 (first column), A105219(second column), A137341 (row sums), A152534.
%Y A152474 T(n,n) gives A346981.
%K A152474 nonn,tabf,easy
%O A152474 0,3
%A A152474 _Vladeta Jovovic_, Dec 05 2008
%E A152474 T(0,0)=1 prepended by _Alois P. Heinz_, Feb 04 2018