|
%I #5 Mar 16 2022 16:32:37
%S 1,3,-10,50,-345,2681,-22416,196700,-1786715,16656155,-158443468,
%T 1531830328,-15007700345,148672680185,-1486712621330,14987306377954,
%U -152144993493979,1554005064929735,-15958686622754240,164676857033422880
%N G.f. A(x) satisfies: A(x)^5 = (1-x) * (A(x) + x)^4.
%F G.f. A(x) satisfies:
%F (1) A(x) = ( A(x)^5/(1-x) )^(1/4) - x.
%F (2) A(x)^5 = (1-x) * (A(x) + x)^4.
%F (3) A( x*(1+x)^4/(1 + x*(1+x)^4) ) = (1+x)^4/(1 + x*(1+x)^4).
%F (4) A(x) = x / Series_Reversion( x*(1+x)^4/(1 + x*(1+x)^4) ).
%F (5) Sum_{k=0..n} [x^k] A(x)^n = (-1)^(n-1) * 4, for n >= 1.
%e G.f.: A(x) = 1 + 3*x - 10*x^2 + 50*x^3 - 345*x^4 + 2681*x^5 - 22416*x^6 + 196700*x^7 - 1786715*x^8 + ...
%e where A(x)^5 equals (1-x)*(A(x) + x)^4, as can be seen from the following power series expansions:
%e A(x)^5 = 1 + 15*x + 40*x^2 - 80*x^3 - 20*x^4 + 48*x^5 - 420*x^6 + 8160*x^7 - 109230*x^8 + ...
%e (A(x) + x)^4 = 1 + 16*x + 56*x^2 - 24*x^3 - 44*x^4 + 4*x^5 - 416*x^6 + 7744*x^7 - 101486*x^8 + ...
%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, 3, -10, 50, -345, 2681, -22416, 196700, ...];
%e n=2: [1, 6, -11, 40, -290, 2292, -19346, 170784, ...];
%e n=3: [1, 9, -3, -3, -105, 1083, -10105, 94239, ...];
%e n=4: [1, 12, 14, -52, 21, 224, -3208, 35792, ...];
%e n=5: [1, 15, 40, -80, -20, 48, -420, 8160, ...];
%e n=6: [1, 18, 75, -60, -255, 294, -77, 720, ...];
%e n=7: [1, 21, 119, 35, -630, 350, 322, -214, ...]; ...
%e in which the partial sum of row n up to column n equals (-1)^(n-1)*4, as illustrated by:
%e n=1: 4 = 1 + 3;
%e n=2: -4 = 1 + 6 + -11;
%e n=3: 4 = 1 + 9 + -3 + -3;
%e n=4: -4 = 1 + 12 + 14 + -52 + 21;
%e n=5: 4 = 1 + 15 + 40 + -80 + -20 + 48;
%e n=6: -4 = 1 + 18 + 75 + -60 + -255 + 294 + -77;
%e n=7: 4 = 1 + 21 + 119 + 35 + -630 + 350 + 322 + -214;
%e ...
%o (PARI) {a(n) = polcoeff( x/serreverse( x*(1+x)^4/(1 + x*(1+x)^4 +x^2*O(x^n)) ),n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A352385, A352413, A352415.
%K sign
%O 0,2
%A _Paul D. Hanna_, Mar 15 2022
|
