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%I #3 Nov 02 2014 14:49:05
%S 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,
%T 11,1,0,18332,13292,4441,897,128,13,1,0,167544,127224,44712,9864,1486,
%U 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
%N 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).
%C 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.
%e Triangle begins:
%e 1;
%e 1, 0;
%e 3, 1, 0;
%e 11, 5, 1, 0;
%e 56, 32, 7, 1, 0;
%e 324, 204, 57, 9, 1, 0;
%e 2324, 1604, 487, 89, 11, 1, 0;
%e 18332, 13292, 4441, 897, 128, 13, 1, 0;
%e 167544, 127224, 44712, 9864, 1486, 174, 15, 1, 0;
%e 1674264, 1311384, 485592, 111744, 18486, 2286, 227, 17, 1, 0;
%e 18615432, 14986632, 5735616, 1393872, 240318, 31734, 3329, 287, 19, 1, 0;
%e 223686792, 183769992, 72550296, 18223632, 3296958, 455742, 51009, 4647, 354, 21, 1, 0;
%e 2937715296, 2458713696, 993598248, 257587416, 48076704, 6958656, 801880, 77896, 6272, 428, 23, 1, 0;
%e 41233157952, 35006137152, 14438206776, 3835359192, 738870048, 110022696, 13300084, 1330300, 114164, 8236, 509, 25, 1, 0; ...
%e GENERATING FUNCTION.
%e G.f.: A(x,y) = 1 + (1)*x + (3 + y)*x^2/2! + (11 + 5*y + y^2)*x^3/3! +
%e (56 + 32*y + 7*y^2 + y^3)*x^4/4! +
%e (324 + 204*y + 57*y^2 + 9*y^3 + y^4)*x^5/5! +
%e (2324 + 1604*y + 487*y^2 + 89*y^3 + 11*y^4 + y^5)*x^6/6! +...
%e such that
%e A(x,y) = exp(y)*P(x) - Q(x,y)
%e where
%e P(x) = 1/Product_{n>=1} (1 - x^n/n) = Sum_{n>=0} A007841(n)*x^n/n!, and
%e Q(x,y) = Sum_{n>=1} y^n / Product_{k=1..n} (k - x^k).
%e More explicitly,
%e P(x) = 1/((1-x)*(1-x^2/2)*(1-x^3/3)*(1-x^4/4)*(1-x^5/5)*...)
%e 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)) +...
%e Column zero of this triangle forms the e.g.f. of A007841:
%e 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! +...
%o (PARI) {T(n,k)=local(A=1, P=((prod(j=1, n+1, 1/(1 - x^j/j +x^2*O(x^n))))),
%o Q=((sum(m=1, n+1, y^m * prod(j=1, m, 1/(j - x^j +x^2*O(x^n)))))) );
%o A=exp(y)*P - Q; n!*polcoeff(polcoeff(A,n,x),k,y)}
%o for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))
%Y Cf. A007841, A249078, A249474, A249475, A249476, A249477, A249478.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Nov 02 2014
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