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A211207 G.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n / (A(x) + n*x)^n. 5
1, 1, 2, 5, 19, 104, 717, 5802, 53337, 546227, 6148507, 75331145, 997148390, 14176316764, 215415605318, 3484286692680, 59775418733049, 1084259223927576, 20735691656139651, 417032279964273318, 8799878770181560605, 194408503996438497630, 4487825374588467361095 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

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

FORMULA

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

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

EXAMPLE

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 +...

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, ...].

PROG

(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), ", "))

CROSSREFS

Cf. A222012.

Sequence in context: A076669 A093502 A009311 * A107882 A328977 A224691

Adjacent sequences:  A211204 A211205 A211206 * A211208 A211209 A211210

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 04 2013

STATUS

approved

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Last modified September 19 20:55 EDT 2020. Contains 337182 sequences. (Running on oeis4.)