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Search: id:A152474
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| A152474 |
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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. |
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+0
2
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| 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
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Sum_{k=0..binomial(n,2)} T(n,k)*exp(2*Pi*I*k/n)) = n!. [From Vladeta Jovovic (vladeta(AT)eunet.yu), Dec 05 2008]
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FORMULA
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Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), 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)
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EXAMPLE
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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; ...
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PROGRAM
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(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)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Dec 15 2008]
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CROSSREFS
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Cf. A137341(row sums), A000262(first column), A105219(second column).
Sequence in context: A118384 A133176 A089435 this_sequence A088814 A088729 A142888
Adjacent sequences: A152471 A152472 A152473 this_sequence A152475 A152476 A152477
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KEYWORD
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easy,nonn,tabf
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AUTHOR
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Vladeta Jovovic (vladeta(AT)eunet.yu), Dec 05 2008
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