|
%I #5 Mar 16 2022 16:32:31
%S 1,4,-15,95,-815,7881,-81946,894100,-10097235,117019845,-1383816039,
%T 16631112009,-202545350505,2494192904025,-31003553499195,
%U 388498706407341,-4902323847971661,62240419152427905,-794494411812382465,10190561785036460125
%N G.f. A(x) satisfies: A(x)^6 = (1-x) * (A(x) + x)^5.
%F G.f. A(x) satisfies:
%F (1) A(x) = ( A(x)^6/(1-x) )^(1/5) - x.
%F (2) A(x)^6 = (1-x) * (A(x) + x)^5.
%F (3) A( x*(1+x)^5/(1 + x*(1+x)^5) ) = (1+x)^5/(1 + x*(1+x)^5).
%F (4) A(x) = x / Series_Reversion( x*(1+x)^5/(1 + x*(1+x)^5) ).
%F (5) Sum_{k=0..n} [x^k] A(x)^n = (-1)^(n-1) * 5, for n >= 1.
%e G.f.: A(x) = 1 + 4*x - 15*x^2 + 95*x^3 - 815*x^4 + 7881*x^5 - 81946*x^6 + 894100*x^7 - 10097235*x^8 + ...
%e where A(x)^6 equals (1-x)*(A(x) + x)^5, as can be seen from the following power series expansions:
%e A(x)^6 = 1 + 24*x + 150*x^2 + 50*x^3 - 675*x^4 + 480*x^5 - 35*x^6 + 1980*x^7 + ...
%e (A(x) + x)^5 = 1 + 25*x + 175*x^2 + 225*x^3 - 450*x^4 + 30*x^5 - 5*x^6 + 1975*x^7 + ...
%e Related table.
%e Another defining property of the g.f. A(x) is illustrated here.
%e The table of coefficients of x^k in A(x)^n begins:
%e n=1: [1, 4, -15, 95, -815, 7881, -81946, 894100, ...];
%e n=2: [1, 8, -14, 70, -645, 6392, -67369, 741352, ...];
%e n=3: [1, 12, 3, -11, -210, 2793, -32964, 385869, ...];
%e n=4: [1, 16, 36, -84, 26, 504, -9506, 135524, ...];
%e n=5: [1, 20, 85, -85, -145, 129, -1050, 27550, ...];
%e n=6: [1, 24, 150, 50, -675, 480, -35, 1980, ...];
%e n=7: [1, 28, 231, 385, -1260, -399, 1708, -689, ...]; ...
%e in which the partial sum of row n up to column n equals (-1)^(n-1)*5, as illustrated by:
%e n=1: 5 = 1 + 4;
%e n=2: -5 = 1 + 8 + -14;
%e n=3: 5 = 1 + 12 + 3 + -11;
%e n=4: -5 = 1 + 16 + 36 + -84 + 26;
%e n=5: 5 = 1 + 20 + 85 + -85 + -145 + 129;
%e n=6: -5 = 1 + 24 + 150 + 50 + -675 + 480 + -35;
%e n=7: 5 = 1 + 28 + 231 + 385 + -1260 + -399 + 1708 + -689;
%e ...
%o (PARI) {a(n) = polcoeff( x/serreverse( x*(1+x)^5/(1 + x*(1+x)^5 +x^2*O(x^n)) ),n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A352385, A352413, A352414.
%K sign
%O 0,2
%A _Paul D. Hanna_, Mar 15 2022
|
