A249480 E.g.f.: A(x,y) = exp(y)*P(x) - Q(x,y), where P(x) = 1/Product_{n>=1} (1 - x^n/n) and Q(x,y) = Sum_{n>=1} y^n / Product_{k=1..n} (k - x^k).

1, 1, 0, 3, 1, 0, 11, 5, 1, 0, 56, 32, 7, 1, 0, 324, 204, 57, 9, 1, 0, 2324, 1604, 487, 89, 11, 1, 0, 18332, 13292, 4441, 897, 128, 13, 1, 0, 167544, 127224, 44712, 9864, 1486, 174, 15, 1, 0, 1674264, 1311384, 485592, 111744, 18486, 2286, 227, 17, 1, 0, 18615432, 14986632, 5735616, 1393872, 240318, 31734, 3329, 287, 19, 1, 0

Offset: 0

Author

Paul D. Hanna, Nov 02 2014

Keywords

• nonn

Comments

The function P(x) = Product_{n>=1} 1/(1 - x^n/n) equals the e.g.f. of A007841, the number of factorizations of permutations of n letters into cycles in nondecreasing length order.

Examples

			 Triangle begins:
1;
1, 0;
3, 1, 0;
11, 5, 1, 0;
56, 32, 7, 1, 0;
324, 204, 57, 9, 1, 0;
2324, 1604, 487, 89, 11, 1, 0;
18332, 13292, 4441, 897, 128, 13, 1, 0;
167544, 127224, 44712, 9864, 1486, 174, 15, 1, 0;
1674264, 1311384, 485592, 111744, 18486, 2286, 227, 17, 1, 0;
18615432, 14986632, 5735616, 1393872, 240318, 31734, 3329, 287, 19, 1, 0;
223686792, 183769992, 72550296, 18223632, 3296958, 455742, 51009, 4647, 354, 21, 1, 0;
2937715296, 2458713696, 993598248, 257587416, 48076704, 6958656, 801880, 77896, 6272, 428, 23, 1, 0;
41233157952, 35006137152, 14438206776, 3835359192, 738870048, 110022696, 13300084, 1330300, 114164, 8236, 509, 25, 1, 0; ...
GENERATING FUNCTION.
G.f.: A(x,y) = 1 + (1)*x + (3 + y)*x^2/2! + (11 + 5*y + y^2)*x^3/3! +
(56 + 32*y + 7*y^2 + y^3)*x^4/4! +
(324 + 204*y + 57*y^2 + 9*y^3 + y^4)*x^5/5! +
(2324 + 1604*y + 487*y^2 + 89*y^3 + 11*y^4 + y^5)*x^6/6! +...
such that
A(x,y) = exp(y)*P(x) - Q(x,y)
where
P(x) = 1/Product_{n>=1} (1 - x^n/n) = Sum_{n>=0} A007841(n)*x^n/n!, and
Q(x,y) = Sum_{n>=1} y^n / Product_{k=1..n} (k - x^k).
More explicitly,
P(x) = 1/((1-x)*(1-x^2/2)*(1-x^3/3)*(1-x^4/4)*(1-x^5/5)*...)
Q(x,y) = y/(1-x) + y^2/((1-x)*(2-x^2)) + y^3/((1-x)*(2-x^2)*(3-x^3)) + y^4/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)) + y^5/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)*(5-x^5)) +...
Column zero of this triangle forms the e.g.f. of A007841:
P(x) = 1 + x + 3*x^2/2! + 11*x^3/3! + 56*x^4/4! + 324*x^5/5! + 2324*x^6/6! + 18332*x^7/7! + 167544*x^8/8! +...
		

Crossrefs

Cf. A007841, A249078, A249474, A249475, A249476, A249477, A249478.

Programs

PARI

{T(n,k)=local(A=1, P=((prod(j=1, n+1, 1/(1 - x^j/j +x^2*O(x^n))))),
Q=((sum(m=1, n+1, y^m * prod(j=1, m, 1/(j - x^j +x^2*O(x^n)))))) );
A=exp(y)*P - Q; n!*polcoeff(polcoeff(A,n,x),k,y)}
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))