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, 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

Vaclav Kotesovec, Table of n, a(n) for n = 0..220

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

Adjacent sequences: A351916 A351917 A351918 * A351920 A351921 A351922

Keyword

nonn

Author

Paul D. Hanna, Feb 25 2022

Status

approved