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A275760 G.f. A(x,y) satisfies: A(x,y) = x*y + 1/A(x,x*y), with A(0,y) = 1.

%I A275760 #23 Aug 10 2016 13:07:50
%S A275760 1,0,1,0,-1,0,1,0,-1,1,0,1,-2,0,-1,2,-1,0,1,-2,3,0,-1,3,-4,1,0,1,-4,5,
%T A275760 -4,0,-1,4,-8,7,-1,0,1,-4,11,-10,5,0,-1,5,-13,16,-11,1,0,1,-6,16,-24,
%U A275760 18,-6,0,-1,6,-20,33,-30,16,-1,0,1,-6,24,-44,49,-30,7,0,-1,7,-28,57,-74,53,-22,1,0,1,-8,32,-74,105,-92,47,-8,0,-1,8,-37,94,-145,149,-89,29,-1,0,1,-8,43,-114,200,-226,163,-70,9,0,-1,9,-48,138,-268,332,-281,143,-37,1,0,1,-10,53,-168,346,-480,454,-276,100,-10,0,-1,10,-60,200,-442,675,-704,503,-221,46,-1,0,1,-10,67,-234,561,-922,1064,-860,450,-138,11,0,-1,11,-73,273,-701,1236,-1567,1402,-863,330,-56,1,0,1,-12,80,-318,861,-1634,2246,-2214,1554,-710,185,-12,0,-1,12,-88,367,-1047,2130,-3144,3403,-2657,1429,-478,67,-1,0,1,-12,96,-418,1268,-2732,4325,-5088,4378,-2700,1088,-242,13
%N A275760 G.f. A(x,y) satisfies: A(x,y) = x*y + 1/A(x,x*y), with A(0,y) = 1.
%C A275760 Row sums equals A275761.
%C A275760 Diagonal sums yield A275762.
%C A275760 G.f. A(x,y) evaluated at A(-x,-1) yields the g.f. of A143951.
%C A275760 G.f. A(x,y) evaluated at A(x,1/x) yields the g.f. of A275762.
%H A275760 Paul D. Hanna, <a href="/A275760/b275760.txt">Table of n, a(n) for n = 0..10201, for rows 0..200 of flattened form of triangle.</a>
%F A275760 G.f.: A(x,y) = 1/(1 - x*y/(1 + x*(1+y) - x^3*y/(1 + x^2*(1+y) - x^5*y/(1 + x^3*(1+y) - x^7*y/(1 + x^4*(1+y) - x^9*y/(1 - ...)))))), a continued fraction.
%F A275760 G.f.: A(x,y) = 1/(1 - x*y/(1+x + x*y/(1+x^2 - x^4*y/(1+x^3 + x^2*y/(1+x^4 - x^7*y/(1+x^5 + x^3*y/(1+x^6 - x^10*y/(1+x^7 + x^4*y/(1+x^8 - x^13*y/(1+x^9 + x^5*y/(1+x^10 - x^16*y/(1 + ...)))))))))))), a continued fraction.
%F A275760 Given g.f. A(x,y), then A(x,1/x) = 1 + 1/A(x,1).
%e A275760 G.f.: A(x,y) = 1 + y*x - y*x^2 + y*x^3 + (y^2 - y)*x^4 + (-2*y^2 + y)*x^5 + (-y^3 + 2*y^2 - y)*x^6 + (3*y^3 - 2*y^2 + y)*x^7 + (y^4 - 4*y^3 + 3*y^2 - y)*x^8 + (-4*y^4 + 5*y^3 - 4*y^2 + y)*x^9 + (-y^5 + 7*y^4 - 8*y^3 + 4*y^2 - y)*x^10 + (5*y^5 - 10*y^4 + 11*y^3 - 4*y^2 + y)*x^11 + (y^6 - 11*y^5 + 16*y^4 - 13*y^3 + 5*y^2 - y)*x^12 + (-6*y^6 + 18*y^5 - 24*y^4 + 16*y^3 - 6*y^2 + y)*x^13 + (-y^7 + 16*y^6 - 30*y^5 + 33*y^4 - 20*y^3 + 6*y^2 - y)*x^14 + (7*y^7 - 30*y^6 + 49*y^5 - 44*y^4 + 24*y^3 - 6*y^2 + y)*x^15 + (y^8 - 22*y^7 + 53*y^6 - 74*y^5 + 57*y^4 - 28*y^3 + 7*y^2 - y)*x^16 +...
%e A275760 such that the g.f. A(x,y) satisfies:
%e A275760 A(x,y) = x*y + 1/(x^2*y + 1/A(x,x^2*y)),
%e A275760 A(x,y) = x*y + 1/(x^2*y + 1/(x^3*y + 1/A(x,x^3*y))),
%e A275760 A(x,y) = x*y + 1/(x^2*y + 1/(x^3*y + 1/(x^4*y + 1/(x^5*y + 1/(x^6*y + 1/A(x^6*y)))))), ...
%e A275760 with the initial condition A(0,y) = 1.
%e A275760 RELATED SERIES.
%e A275760 The g.f. evaluated at A(-x,-1) yields the g.f. of A143951:
%e A275760 A(-x,-1) = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 6*x^7 + 9*x^8 + 14*x^9 + 21*x^10 + 31*x^11 + 47*x^12 +...+ A143951(n)*x^n +...
%e A275760 which enumerates Dyck paths such that the area between the x-axis and the path is n.
%e A275760 The g.f. evaluated at A(x,1/x) yields the g.f. of A275762:
%e A275760 A(x,1/x) = 2 - x + 2*x^2 - 4*x^3 + 7*x^4 - 12*x^5 + 22*x^6 - 41*x^7 + 74*x^8 - 133*x^9 + 243*x^10 - 444*x^11 +...+ A275762(n)*x^n +...
%e A275760 Compare A(x,1/x) to 1/A(x,1), which begins:
%e A275760 1/A(x,1) = 1 - x + 2*x^2 - 4*x^3 + 7*x^4 - 12*x^5 + 22*x^6 - 41*x^7 + 74*x^8 - 133*x^9 + 243*x^10 - 444*x^11 +...+ A275762(n)*x^n +...
%e A275760 This triangle of coefficients in A(x,y) begins:
%e A275760 1;
%e A275760 0, 1;
%e A275760 0, -1;
%e A275760 0, 1;
%e A275760 0, -1, 1;
%e A275760 0, 1, -2;
%e A275760 0, -1, 2, -1;
%e A275760 0, 1, -2, 3;
%e A275760 0, -1, 3, -4, 1;
%e A275760 0, 1, -4, 5, -4;
%e A275760 0, -1, 4, -8, 7, -1;
%e A275760 0, 1, -4, 11, -10, 5;
%e A275760 0, -1, 5, -13, 16, -11, 1;
%e A275760 0, 1, -6, 16, -24, 18, -6;
%e A275760 0, -1, 6, -20, 33, -30, 16, -1;
%e A275760 0, 1, -6, 24, -44, 49, -30, 7;
%e A275760 0, -1, 7, -28, 57, -74, 53, -22, 1;
%e A275760 0, 1, -8, 32, -74, 105, -92, 47, -8;
%e A275760 0, -1, 8, -37, 94, -145, 149, -89, 29, -1;
%e A275760 0, 1, -8, 43, -114, 200, -226, 163, -70, 9;
%e A275760 0, -1, 9, -48, 138, -268, 332, -281, 143, -37, 1;
%e A275760 0, 1, -10, 53, -168, 346, -480, 454, -276, 100, -10;
%e A275760 0, -1, 10, -60, 200, -442, 675, -704, 503, -221, 46, -1;
%e A275760 0, 1, -10, 67, -234, 561, -922, 1064, -860, 450, -138, 11;
%e A275760 0, -1, 11, -73, 273, -701, 1236, -1567, 1402, -863, 330, -56, 1;
%e A275760 0, 1, -12, 80, -318, 861, -1634, 2246, -2214, 1554, -710, 185, -12;
%e A275760 0, -1, 12, -88, 367, -1047, 2130, -3144, 3403, -2657, 1429, -478, 67, -1;
%e A275760 0, 1, -12, 96, -418, 1268, -2732, 4325, -5088, 4378, -2700, 1088, -242, 13;
%e A275760 0, -1, 13, -104, 474, -1521, 3459, -5863, 7416, -7002, 4830, -2295, 674, -79, 1;
%e A275760 0, 1, -14, 112, -538, 1803, -4342, 7819, -10598, 10884, -8290, 4537, -1624, 310, -14;
%e A275760 0, -1, 14, -121, 607, -2124, 5397, -10274, 14895, -16478, 13769, -8473, 3588, -928, 92, -1;
%e A275760 0, 1, -14, 131, -678, 2492, -6638, 13348, -20582, 24408, -22200, 15126, -7406, 2367, -390, 15;
%e A275760 0, -1, 15, -140, 755, -2905, 8095, -17160, 27998, -35485, 34829, -26052, 14411, -5476, 1251, -106, 1; ...
%o A275760 (PARI) /* Print first N rows of this triangle: */ N=32;
%o A275760 {a(n) = my(A=1 +x*O(x^n)); for(k=0, n, A = 1/A + y*x^(n+1-k)); polcoeff(A, n)}
%o A275760 {for(n=0, N, for(k=0, n, if(k==0, print1(polcoeff(a(n)+y*O(y^n), k, y)", "), if(polcoeff(a(n)+y*O(y^n), k, y)==0, break, print1(polcoeff(a(n)+y*O(y^n), k, y), ", ")))); print(""))}
%Y A275760 Cf. A275761, A275762, A143951.
%K A275760 sign,tabf
%O A275760 0,13
%A A275760 _Paul D. Hanna_, Aug 08 2016