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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.
3
1, 1, 3, 1, 13, 8, 8, 1, 73, 63, 89, 78, 41, 15, 1, 501, 544, 909, 1095, 1200, 842, 680, 315, 129, 24, 1, 4051, 5225, 9734, 13799, 18709, 20441, 20520, 18101, 14831, 10200, 5891, 3199, 1109, 314, 35, 1, 37633, 55656, 112370, 177457, 270746, 352969, 442897
OFFSET
0,3
COMMENTS
Sum_{k=0..binomial(n,2)} T(n,k)*exp(2*Pi*I*k/n)) = n!. - Vladeta Jovovic, Dec 05 2008
LINKS
FORMULA
From Paul D. Hanna, Dec 15 2008: (Start)
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.
Sum_{k=0..n(n-1)/2} T(n,k)*(-1)^k = n!*A000110((n+1)/2), where A000110 is the Bell numbers. (End)
EXAMPLE
Triangle T(n,k) begins:
1;
1;
3, 1;
13, 8, 8, 1;
73, 63, 89, 78, 41, 15, 1;
501, 544, 909, 1095, 1200, 842, 680, 315, 129, 24, 1;
...
PROG
(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
CROSSREFS
Cf. A000262 (first column), A105219(second column), A137341 (row sums), A152534.
T(n,n) gives A346981.
Sequence in context: A376863 A133176 A089435 * A088814 A088729 A270968
KEYWORD
nonn,tabf,easy
AUTHOR
Vladeta Jovovic, Dec 05 2008
EXTENSIONS
T(0,0)=1 prepended by Alois P. Heinz, Feb 04 2018
STATUS
approved