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A303562 G.f. A(x) satisfies: 0 = [x^(n-1)] 1 / A(x)^(n^2+n-1) for n>2.

%I A303562 #18 Apr 28 2018 00:07:16
%S A303562 1,1,6,50,490,5187,59080,675012,8723880,84841130,2106192682,
%T A303562 -26974249302,2765793096248,-163142299607490,11813146551718560,
%U A303562 -906751607066476056,75382006693375808940,-6718584345560312459292,639573513055226901933760,-64760465046707144137421880,6950351671309757070230871462
%N A303562 G.f. A(x) satisfies: 0 = [x^(n-1)] 1 / A(x)^(n^2+n-1) for n>2.
%C A303562 a(n) is odd iff n = (4^k - 1)/3 for k >= 0 (conjecture).
%H A303562 Paul D. Hanna, <a href="/A303562/b303562.txt">Table of n, a(n) for n = 0..400</a>
%F A303562 G.f. A(x) satisfies:
%F A303562 (1) 0 = [x^(n-1)] 1 / A(x)^(n^2+n-1) for n>2.
%F A303562 (2) 0 = [x^(n-1)] (x*A(x))' / A(x)^(n*(n+1)) for n>2.
%F A303562 (3) 0 = [x^(n-1)] (x*A(x)^n)' / A(x)^(n^2+2*n-1) for n>2.
%F A303562 (4) 0 = [x^(n-1)] (x*A(x)^(n+2))' / A(x)^((n+1)^2) for n>2.
%F A303562 (5) 0 = [x^(n-1)] (x*A(x)^(n+1))' / A(x)^(n*(n+1)) for n>1.
%e A303562 G.f. A(x) = 1 + x + 6*x^2 + 50*x^3 + 490*x^4 + 5187*x^5 + 59080*x^6 + 675012*x^7 + 8723880*x^8 + 84841130*x^9 + ...
%e A303562 such that the coefficient of x^(n-1) in 1/A(x)^(n^2+n-1) equals zero for n>2.
%e A303562 RELATED SERIES.
%e A303562 (x*A(x))' = 1 + 2*x + 18*x^2 + 200*x^3 + 2450*x^4 + 31122*x^5 + 413560*x^6 + 5400096*x^7 + 78514920*x^8 + ...
%e A303562 A'(x)/A(x) = 1 + 11*x + 133*x^2 + 1711*x^3 + 22386*x^4 + 304601*x^5 + 4019310*x^6 + 59971671*x^7 + 620401840*x^8 + ...
%e A303562 ILLUSTRATION OF DEFINITION.
%e A303562 The table of coefficients of x^k in 1/A(x)^(n^2+n-1) begins:
%e A303562 n=1: [1, -1, -5, -39, -371, -3842, -43425, -485860, ...];
%e A303562 n=2: [1, -5, -15, -105, -970, -9711, -110550, -1167485, ...];
%e A303562 n=3: [1, -11, 0, -44, -561, -5544, -74778, -601920, ...];
%e A303562 n=4: [1, -19, 76, 0, -95, 418, -27474, 277628, ...];
%e A303562 n=5: [1, -29, 261, -725, 0, 2871, -40716, 915501, ...];
%e A303562 n=6: [1, -41, 615, -4059, 10619, 0, -109347, 2014330, ...];
%e A303562 n=7: [1, -55, 1210, -13530, 80080, -225071, 0, 4884440, ...];
%e A303562 n=8: [1, -71, 2130, -35074, 343924, -2020731, 6422944, 0, ...]; ...
%e A303562 in which the main diagonal is all zeros after the initial terms, illustrating that 0 = [x^(n-1)] 1/A(x)^(n^2+n-1) for n>2.
%e A303562 RELATED TABLES.
%e A303562 The table of coefficients of x^k in (x*A(x))' / A(x)^(n*(n+1)) begins:
%e A303562 n=1: [1, 0, 5, 78, 1113, 15368, 217125, 2915160, ...];
%e A303562 n=2: [1, -4, -9, -42, -194, 0, 22110, 466994, 10357803, ...];
%e A303562 n=3: [1, -10, 0, -32, -357, -3024, -33990, -218880, ...];
%e A303562 n=4: [1, -18, 68, 0, -75, 308, -18798, 175344, ...];
%e A303562 n=5: [1, -28, 243, -650, 0, 2376, -32292, 694518, ...];
%e A303562 n=6: [1, -40, 585, -3762, 9583, 0, -93345, 1670420, ...];
%e A303562 n=7: [1, -54, 1166, -12792, 74256, -204610, 0, 4262784, ...];
%e A303562 n=8: [1, -70, 2070, -33592, 324548, -1878426, 5880160, 0, ...]; ...
%e A303562 in which the main diagonal is all zeros after the initial terms, illustrating that 0 = [x^(n-1)] (x*A(x))' / A(x)^(n*(n+1)) for n>2.
%e A303562 The table of coefficients of x^k in (x*A(x)^(n+1))' / A(x)^(n*(n+1)) begins:
%e A303562 n=1: [1, 2, 22, 266, 3422, 44772, 609202, 8038620, ...];
%e A303562 n=2: [1, 0, 12, 176, 2457, 33288, 469690, 6150600, ...];
%e A303562 n=3: [1, -4, 0, 44, 854, 12672, 201160, 2446320, ...];
%e A303562 n=4: [1, -10, 10, 0, 70, 1222, 43320, 135920, ...];
%e A303562 n=5: [1, -18, 78, -50, 0, -408, 13950, -460224, ...];
%e A303562 n=6: [1, -28, 252, -784, 497, 0, 13258, -547944, ...];
%e A303562 n=7: [1, -40, 592, -3944, 11172, -8176, 0, -526608, ...];
%e A303562 n=8: [1, -54, 1170, -12936, 76194, -220374, 194424, 0, ...]; ...
%e A303562 in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^(n-1)] (x*A(x)^(n+1))' / A(x)^(n*(n+1)) for n>1.
%o A303562 (PARI) {a(n) = my(A=[1,1]); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( 1/Ser(A)^(m*(m+1)-1) )[m]/(m*(m+1)-1) ); A[n+1]}
%o A303562 for(n=0,20,print1(a(n),", "))
%o A303562 (PARI) {a(n) = my(A=[1,1]); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( (x*Ser(A))'/Ser(A)^(m*(m+1)) )[m]/m^2 ); A[n+1]}
%o A303562 for(n=0,20,print1(a(n),", "))
%Y A303562 Cf. A292877, A303563.
%K A303562 sign
%O A303562 0,3
%A A303562 _Paul D. Hanna_, Apr 27 2018