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.

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

Author

Vladeta Jovovic, Dec 05 2008

Keywords

• nonn
• tabf
• easy

Examples

			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;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..36, flattened

Crossrefs

Cf. A000262 (first column), A105219(second column), A137341 (row sums), A152534.

T(n,n) gives A346981.

Programs

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

Formula

Sum_{k=0..binomial(n,2)} T(n,k)*exp(2*Pi*I*k/n) = n!. - Vladeta Jovovic, Dec 05 2008

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)

Extensions

T(0,0)=1 prepended by Alois P. Heinz, Feb 04 2018