|
%I #5 Jan 07 2022 12:03:47
%S 1,2,-26,366,-6270,99922,-1630730,-33526706,1685562866,-390576999182,
%T -2936125610490,-3666605533359442,-376235732409401630,
%U -85462222978639050222,-16821593963787582554986,-3977651379054471070911090,-1019617793745769995713403822,-288252359877865826549093001294,-89096129151626329798167571168346
%N E.g.f. A(x) satisfies: 1 + 4*x = exp(-1) * Sum_{n>=0} A(x)^(n^2) / n!.
%e E.g.f. A(x) = 1 + 2*x - 26*x^2/2! + 366*x^3/3! - 6270*x^4/4! + 99922*x^5/5! - 1630730*x^6/6! - 33526706*x^7/7! + 1685562866*x^8/8! - 390576999182*x^9/9! + ...
%e where
%e 1 + 4*x = exp(-1) * (1 + A(x) + A(x)^4/2! + A(x)^9/3! + A(x)^16/4! + A(x)^25/5! + A(x)^36/6! + A(x)^49/7! + ... + A(x)^(n^2)/n! + ...).
%e RELATED TABLE.
%e The table of coefficients of x^k/k! in A(x)^(n^2) begins:
%e n=0: [1, 0, 0, 0, 0, 0, 0, ...];
%e n=1: [1, 2, -26, 366, -6270, 99922, -1630730, ...];
%e n=2: [1, 8, -56, -216, 19800, -706472, 14847688, ...];
%e n=3: [1, 18, 54, -3906, 34290, 1326978, -99273402, ...];
%e n=4: [1, 32, 544, -4704, -308640, 6962272, 154469920, ...];
%e n=5: [1, 50, 1750, 25950, -936750, -37790750, 1459186150, ...];
%e n=6: [1, 72, 4104, 159336, 1906200, -192221928, -7838021880, ...];
%e n=7: [1, 98, 8134, 535374, 23730210, 239390578, -52296366122, ...]; ...
%e in which infinite sums of terms along the columns may be illustrated by:
%e 1 = (1 + 1 + 1/2! + 1/3! + 1/4! + 1/5! + ...)/e;
%e 4 = (0 + 2 + 8/2! + 18/3! + 32/4! + 50/5! + ...)/e;
%e 0 = (0 + -26 + -56/2! + 54/3! + 544/4! + 1750/5! + ...);
%e 0 = (0 + 366 + -216/2! + -3906/3! + -4704/4! + 25950/5! + ...);
%e 0 = (0 + -6270 + 19800/2! + 34290/3! + -308640/4! + -936750/5! + ...);
%e 0 = (0 + 99922 + -706472/2! + 1326978/3! + 6962272/4! + -37790750/5! ...); ...
%e and can be used to determine all the terms of this sequence.
%K sign
%O 0,2
%A _Paul D. Hanna_, Jan 04 2022
|
