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A303563 G.f. A(x) satisfies: 0 = [x^(n-1)] 1 / A(x)^(n^2-n+1) for n>2.

%I A303563 #11 Apr 27 2018 23:50:27
%S A303563 1,1,4,21,132,840,6798,26187,982794,-23411010,1229958800,-63402693620,
%T A303563 3727765284702,-241049598495378,17055417754898346,
%U A303563 -1310070866036785677,108572840286328367574,-9656468474317765916970,917437511140569561151848,-92733586081750860360411954,9936829948115042380890921976,-1125196473407637775842431681496,134258639558360961220987962351588
%N A303563 G.f. A(x) satisfies: 0 = [x^(n-1)] 1 / A(x)^(n^2-n+1) for n>2.
%C A303563 a(n) is odd iff n = 2^k - 1 for k >= 0 (conjecture).
%H A303563 Paul D. Hanna, <a href="/A303563/b303563.txt">Table of n, a(n) for n = 0..400</a>
%F A303563 G.f. A(x) satisfies:
%F A303563 (1) 0 = [x^(n-1)] 1 / A(x)^(n^2-n+1) for n>2.
%F A303563 (2) 0 = [x^(n-1)] (x*A(x))' / A(x)^(n^2-n+2) for n>2.
%F A303563 (3) 0 = [x^(n-1)] (x*A(x)^n)' / A(x)^(n^2+1) for n>2.
%F A303563 (4) 0 = [x^(n-1)] (x*A(x)^(n-1))' / A(x)^(n^2) for n>2.
%F A303563 (5) 0 = [x^(n-1)] (x*A(x)^n)' / A(x)^(n^2) for n>1.
%e A303563 G.f.: A(x) = 1 + x + 4*x^2 + 21*x^3 + 132*x^4 + 840*x^5 + 6798*x^6 + 26187*x^7 + 982794*x^8 - 23411010*x^9 + 1229958800*x^10 + ...
%e A303563 such that the coefficient of x^(n-1) in 1/A(x)^(n^2-n+1) equals zero for n>2.
%e A303563 RELATED SERIES.
%e A303563 1/A(x) = 1 - x - 3*x^2 - 14*x^3 - 85*x^4 - 504*x^5 - 4424*x^6 - 6796*x^7 - 878157*x^8 + ... + (-1)^n*A230218(n)*x^n + ...
%e A303563 A'(x)/A(x) = 1 + 7*x + 52*x^2 + 427*x^3 + 3286*x^4 + 32938*x^5 + 108718*x^6 + 7379059*x^7 - 221516750*x^8 + 12494464362*x^9 - 710385713212*x^10 + ...
%e A303563 ILLUSTRATION OF DEFINITION.
%e A303563 The table of coefficients of x^k in 1/A(x)^(n^2-n+1) begins:
%e A303563 n=1: [1, -1, -3, -14, -85, -504, -4424, -6796, -878157, ...];
%e A303563 n=2: [1, -3, -6, -25, -153, -819, -8664, 18360, ...];
%e A303563 n=3: [1, -7, 0, -7, -98, -210, -10122, 141525, ...];
%e A303563 n=4: [1, -13, 39, 0, -78, 819, -15483, 380952, ...];
%e A303563 n=5: [1, -21, 147, -364, 0, 2457, -35805, 821916, ...];
%e A303563 n=6: [1, -31, 372, -2139, 5580, 0, -91698, 1792947, ...];
%e A303563 n=7: [1, -43, 774, -7525, 42097, -125517, 0, 4097298, ...];
%e A303563 n=8: [1, -57, 1425, -20482, 185877, -1089270, 3791298, 0, ...]; ...
%e A303563 in which the main diagonal equals zeros after the initial terms, illustrating that 0 = [x^(n-1)] 1/A(x)^(n^2-n+1) for n>2.
%e A303563 RELATED TABLES.
%e A303563 The table of coefficients of x^k in (x*A(x)^n)' / A(x)^(n^2+1) begins:
%e A303563 n=1: [1, 0, 3, 28, 255, 2016, 22120, 40776, ...];
%e A303563 n=2: [1, -1, 2, 25, 255, 1911, 25992, -67320, ...];
%e A303563 n=3: [1, -4, 0, 2, 70, 240, 15906, -283050, ...];
%e A303563 n=4: [1, -9, 15, 0, 18, -441, 13101, -439560, ...];
%e A303563 n=5: [1, -16, 77, -104, 0, -468, 15345, -547944, ...];
%e A303563 n=6: [1, -25, 228, -897, 1260, 0, 14790, -636207, ...];
%e A303563 n=7: [1, -36, 522, -3850, 14685, -23352, 0, -571716, ...];
%e A303563 n=8: [1, -49, 1025, -11858, 81525, -324870, 598626, 0,  ...]; ...
%e A303563 in which the main diagonal equals zeros after the initial terms, illustrating that 0 = [x^(n-1)] (x*A(x)^n)' / A(x)^(n^2+1) for n>2.
%e A303563 The table of coefficients of x^k in (x*A(x)^n)' / A(x)^(n^2) begins:
%e A303563 n=1: [1, 1, 7, 52, 427, 3286, 32938, 108718, ...];
%e A303563 n=2: [1, 0, 5, 44, 399, 3016, 35670, -3960, ...];
%e A303563 n=3: [1, -3, 0, 7, 120, 630, 19906, -265455, ...];
%e A303563 n=4: [1, -8, 10, 0, 21, -456, 13950, -450240, ...];
%e A303563 n=5: [1, -15, 65, -70, 0, -539, 16215, -566100, ...];
%e A303563 n=6: [1, -24, 207, -748, 882, 0, 16887, -665604, ...];
%e A303563 n=7: [1, -35, 490, -3451, 12299, -17017, 0, -645000, ...];
%e A303563 n=8: [1, -48, 980, -11008, 72870, -274880, 451776, 0, ...]; ...
%e A303563 in which the main diagonal equals zeros after the initial term, illustrating that 0 = [x^(n-1)] (x*A(x)^n)' / A(x)^(n^2) for n>1.
%o A303563 (PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( (x*Ser(A)^m)'/Ser(A)^(m^2+1) )[m] ); A[n+1]}
%o A303563 for(n=0,30,print1(a(n),", "))
%Y A303563 Cf. A303562, A230218, A292877.
%K A303563 sign
%O A303563 0,3
%A A303563 _Paul D. Hanna_, Apr 27 2018