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%I #16 Aug 30 2023 02:09:27
%S 1,1,2,5,19,104,717,5802,53337,546227,6148507,75331145,997148390,
%T 14176316764,215415605318,3484286692680,59775418733049,
%U 1084259223927576,20735691656139651,417032279964273318,8799878770181560605,194408503996438497630,4487825374588467361095
%N G.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n / (A(x) + n*x)^n.
%H Paul D. Hanna, <a href="/A211207/b211207.txt">Table of n, a(n) for n = 0..500</a>
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F (1) A(x) = Sum_{n>=0} n^n * x^n / (A(x) + n*x)^n.
%F (2) A(x) = 1 + Sum_{n>=1} (n+1)!/2 * x^n / A(x)^n.
%F (3) A(x) = x/Series_Reversion(x*B(x)), where B(x) = 1 + Sum_{n>=1} (n+1)!/2*x^n.
%F (4) [x^(n-1)] A(x)^n = n * n! / 2 for n > 1.
%F (5) [x^(n-1)] (A(x) + n*x)^n = n^n * (n+1) / 2 for n >= 1. - _Paul D. Hanna_, Aug 30 2023
%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 19*x^4 + 104*x^5 + 717*x^6 + 5802*x^7 +...
%e where, by definition,
%e A(x) = 1 + x/(A(x) + x) + 2^2*x^2/(A(x) + 2*x)^2 + 3^3*x^3/(A(x) + 3*x)^3 + 4^4*x^4/(A(x) + 4*x)^4 +...+ n^n*x^n/(A(x) + n*x)^n +...
%e also, g.f. A(x) satisfies:
%e A(x) = 1 + x/A(x) + 3*x^2/A(x)^2 + 12*x^3/A(x)^3 + 60*x^4/A(x)^4 + 360*x^5/A(x)^5 + 2520*x^6/A(x)^6 +...+ (n+1)!/2*x^n/A(x)^n +...
%e RELATED TABLES.
%e Form an array of coefficients of x^k in A(x)^n, which begins:
%e n=1: [1, 1, 2, 5, 19, 104, 717, 5802, 53337, ...];
%e n=2: [1, 2, 5, 14, 52, 266, 1743, 13644, 122547, ...];
%e n=3: [1, 3, 9, 28, 105, 513, 3203, 24201, 211977, ...];
%e n=4: [1, 4, 14, 48, 185, 880, 5266, 38376, 327252, ...];
%e n=5: [1, 5, 20, 75, 300, 1411, 8155, 57365, 475650, ...];
%e n=6: [1, 6, 27, 110, 459, 2160, 12158, 82734, 666567, ...];
%e n=7: [1, 7, 35, 154, 672, 3192, 17640, 116509, 912086, ...];
%e n=8: [1, 8, 44, 208, 950, 4584, 25056, 161280, 1227665, ...];
%e n=9: [1, 9, 54, 273, 1305, 6426, 34965, 220320, 1632960, ...]; ...
%e then the main diagonal equals n*n!/2 for n > 1:
%e [1, 2, 9, 48, 300, 2160, 17640, 161280, 1632960, ...].
%e Form an array of coefficients of x^k in (A(x) + n*x)^n, which begins:
%e n=1: [1, 2, 2, 5, 19, 104, 717, ...];
%e n=2: [1, 6, 13, 22, 72, 342, 2159, ...];
%e n=3: [1, 12, 54, 127, 285, 1116, 6110, ...];
%e n=4: [1, 20, 158, 640, 1625, 4416, 19746, ...];
%e n=5: [1, 30, 370, 2425, 9375, 25536, 80155, ...];
%e n=6: [1, 42, 747, 7310, 43119, 163296, 474326, ...];
%e n=7: [1, 56, 1358, 18627, 158697, 875980, 3294172, ...]; ...
%e then the main diagonal equals n^n*(n+1)/2 for n >= 1:
%e [1, 6, 54, 640, 9375, 163296, 3294172, 75497472, ...].
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, m^m*x^m/(A+m*x+x*O(x^n))^m)); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n)=local(B=1+sum(m=1,n,(m+1)!/2*x^m)+x*O(x^n));polcoeff(x/serreverse(x*B),n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A222012.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 04 2013
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