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A351919
E.g.f. A(x) satisfies: Sum_{k=0..n} [x^k/k!] 1/A(x)^(n+1-k) = 0 for n > 0.
1
1, 1, 1, 2, 6, 22, 96, 486, 2816, 18362, 133092, 1060918, 9226068, 86913822, 881783456, 9584972462, 111135773688, 1369122271498, 17858966209908, 245895213956190, 3563864413568516, 54235164104218478, 864658341720196176, 14411626441272698566
OFFSET
0,4
LINKS
FORMULA
a(n) ~ c * d^n * n!, where d = 0.72467075187356681806169214268514... and c = 0.9182939975437585609088613585... - Vaclav Kotesovec, Feb 28 2022
EXAMPLE
G.f.: A(x) = 1 + x + x^2/2! + 2*x^3/3! + 6*x^4/4! + 22*x^5/5! + 96*x^6/6! + 486*x^7/7! + 2816*x^8/8! + 18362*x^9/9! + 133092*x^10/10! + ...
Related series.
1/A(x) = 1 - x + x^2/2! - 2*x^3/3! + 4*x^4/4! - 12*x^5/5! + 38*x^6/6! - 150*x^7/7! + 648*x^8/8! - 3218*x^9/9! + ... + A213058(n)*x^n + ...
log(A(x)) = x + x^3/3! + x^4/4! + 6*x^5/5! + 14*x^6/6! + 86*x^7/7! + 342*x^8/8! + 2394*x^9/9! + 13648*x^10/10! + ...
Illustration of definition.
The table of coefficients of x^k in A(x)^(-n) for n > 0 begins:
n=1: [1, -1, 1, -2, 4, -12, 38, -150, ...];
n=2: [1, -2, 4, -10, 30, -104, 420, -1896, ...];
n=3: [1, -3, 9, -30, 114, -486, 2316, -12210, ...];
n=4: [1, -4, 16, -68, 316, -1608, 8936, -54024, ...];
n=5: [1, -5, 25, -130, 720, -4280, 27330, -187230, ...];
n=6: [1, -6, 36, -222, 1434, -9792, 70908, -544800, ...];
n=7: [1, -7, 49, -350, 2590, -20034, 162680, -1389066, ...];
n=8: [1, -8, 64, -520, 4344, -37616, 339216, -3193200, ...]; ...
in which the antidiagonals add to zero (after the initial term):
0 = 1 + (-1) ;
0 = 1 + (-2) + 1 ;
0 = 1 + (-3) + 4 + (-2) ;
0 = 1 + (-4) + 9 + (-10) + 4 ;
0 = 1 + (-5) + 16 + (-30) + 30 + (-12) ;
0 = 1 + (-6) + 25 + (-68) + 114 + (-104) + 38 ;
...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = sum(k=0, #A-1, k!*polcoeff( 1/Ser(A)^(#A-k) , k)) / (#A-1)! ); n!*A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A213058.
Sequence in context: A268699 A253948 A248836 * A328500 A180389 A177389
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 25 2022
STATUS
approved