A211207 G.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n / (A(x) + n*x)^n.

1, 1, 2, 5, 19, 104, 717, 5802, 53337, 546227, 6148507, 75331145, 997148390, 14176316764, 215415605318, 3484286692680, 59775418733049, 1084259223927576, 20735691656139651, 417032279964273318, 8799878770181560605, 194408503996438497630, 4487825374588467361095

Offset: 0

Author

Paul D. Hanna, Feb 04 2013

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 19*x^4 + 104*x^5 + 717*x^6 + 5802*x^7 +...
where, by definition,
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 +...
also, g.f. A(x) satisfies:
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 +...
RELATED TABLES.
Form an array of coefficients of x^k in A(x)^n, which begins:
n=1: [1, 1,  2,   5,   19,  104,   717,   5802,   53337, ...];
n=2: [1, 2,  5,  14,   52,  266,  1743,  13644,  122547, ...];
n=3: [1, 3,  9,  28,  105,  513,  3203,  24201,  211977, ...];
n=4: [1, 4, 14,  48,  185,  880,  5266,  38376,  327252, ...];
n=5: [1, 5, 20,  75,  300, 1411,  8155,  57365,  475650, ...];
n=6: [1, 6, 27, 110,  459, 2160, 12158,  82734,  666567, ...];
n=7: [1, 7, 35, 154,  672, 3192, 17640, 116509,  912086, ...];
n=8: [1, 8, 44, 208,  950, 4584, 25056, 161280, 1227665, ...];
n=9: [1, 9, 54, 273, 1305, 6426, 34965, 220320, 1632960, ...]; ...
then the main diagonal equals n*n!/2 for n > 1:
[1, 2, 9, 48, 300, 2160, 17640, 161280, 1632960, ...].
Form an array of coefficients of x^k in (A(x) + n*x)^n, which begins:
n=1: [1,  2,    2,     5,     19,    104,     717, ...];
n=2: [1,  6,   13,    22,     72,    342,    2159, ...];
n=3: [1, 12,   54,   127,    285,   1116,    6110, ...];
n=4: [1, 20,  158,   640,   1625,   4416,   19746, ...];
n=5: [1, 30,  370,  2425,   9375,  25536,   80155, ...];
n=6: [1, 42,  747,  7310,  43119, 163296,  474326, ...];
n=7: [1, 56, 1358, 18627, 158697, 875980, 3294172, ...]; ...
then the main diagonal equals n^n*(n+1)/2 for n >= 1:
[1, 6, 54, 640, 9375, 163296, 3294172, 75497472, ...].
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A222012.

Programs

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)}
for(n=0, 30, print1(a(n), ", "))

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)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) A(x) = Sum_{n>=0} n^n * x^n / (A(x) + n*x)^n.

(2) A(x) = 1 + Sum_{n>=1} (n+1)!/2 * x^n / A(x)^n.

(3) A(x) = x/Series_Reversion(x*B(x)), where B(x) = 1 + Sum_{n>=1} (n+1)!/2*x^n.

(4) [x^(n-1)] A(x)^n = n * n! / 2 for n > 1.

(5) [x^(n-1)] (A(x) + n*x)^n = n^n * (n+1) / 2 for n >= 1. - Paul D. Hanna, Aug 30 2023

a(n) ~ n! * n / (2*exp(1)). - Vaclav Kotesovec, Nov 23 2024