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A305605 G.f. A(x) satisfies: [x^k] A(x) / (1-x)^n = 0 for k = n*(n+1)/2 + 1 through k = (n+1)*(n+2)/2 for n >= 0.

%I A305605 #21 Jun 15 2018 00:28:38
%S A305605 1,0,-1,0,-2,2,0,-7,14,-7,0,-37,111,-111,37,0,-268,1072,-1608,1072,
%T A305605 -268,0,-2496,12480,-24960,24960,-12480,2496,0,-28612,171672,-429180,
%U A305605 572240,-429180,171672,-28612,0,-391189,2738323,-8214969,13691615,-13691615,8214969,-2738323,391189,0,-6230646,49845168,-174458088,348916176,-436145220,348916176,-174458088,49845168,-6230646,0
%N A305605 G.f. A(x) satisfies: [x^k] A(x) / (1-x)^n = 0 for k = n*(n+1)/2 + 1 through k = (n+1)*(n+2)/2 for n >= 0.
%H A305605 Paul D. Hanna, <a href="/A305605/b305605.txt">Table of n, a(n) for n = 0..2556</a>
%F A305605 G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies:
%F A305605 (1) A(x) = 1 - x*Sum_{n>=1} A107877(n) * x^(n*(n+1)/2) * (1-x)^(n-1).
%F A305605 (2) [x^k] A(x) / (1-x)^n = 0 for k = n*(n+1)/2 + 1 through (n+1)*(n+2)/2, n >= 0.
%F A305605 (3) [x^k] A(x) / (1-x)^n = A107877(n) for k = n*(n-1)/2 through n*(n+1)/2, n >= 0.
%F A305605 (4) [x^k] A(x) / (1-x)^n = A127496(n,k) for k = 0..n*(n+1)/2 for n >= 0.
%F A305605 (5) [x^n] A(x) / (1-x)^n = A127497(n) for n >= 0.
%F A305605 FORMULAS INVOLVING TERMS.
%F A305605 a(n*(n+1)/2) = 0 for n >= 1.
%F A305605 a(n*(n-1)/2) = (-1)^n * A107877(n) for n >= 0.
%F A305605 a(n*(n+1)/2 + 1) = -A107877(n) for n >= 0.
%F A305605 a(n) = [x^n] (1+x)^n * G(x) where G(x) is the g.f. of A305601, which is the inverse binomial transform of this sequence.
%e A305605 G.f.: A(x) = 1 - x^2 - 2*x^4 + 2*x^5 - 7*x^7 + 14*x^8 - 7*x^9 - 37*x^11 + 111*x^12 - 111*x^13 + 37*x^14 - 268*x^16 + 1072*x^17 - 1608*x^18 + 1072*x^19 - 268*x^20 - 2496*x^22 + 12480*x^23 - 24960*x^24 + 24960*x^25 - 12480*x^26 + 2496*x^27 - 28612*x^29 + 171672*x^30 + ...
%e A305605 The table of coefficients of x^k in A(x) / (1-x)^n, for n >= 0, begins:
%e A305605 [1, 0, -1,  0,  -2,   2,   0,   -7,  14,    -7,    0, -37, 111,-111,   37, ...];
%e A305605 [1, 1,  0,  0,  -2,   0,   0,   -7,   7,     0,    0, -37,  74, -37,    0, ...];
%e A305605 [1, 2,  2,  2,   0,   0,   0,   -7,   0,     0,    0, -37,  37,   0,    0, ...];
%e A305605 [1, 3,  5,  7,   7,   7,   7,    0,   0,     0,    0, -37,   0,   0,    0, ...];
%e A305605 [1, 4,  9, 16,  23,  30,  37,   37,  37,    37,   37,   0,   0,   0,    0, ...];
%e A305605 [1, 5, 14, 30,  53,  83, 120,  157,  194,  231,  268, 268, 268, 268,  268, ...];
%e A305605 [1, 6, 20, 50, 103, 186, 306,  463,  657,  888, 1156,1424,1692,1960, 2228, ...];
%e A305605 [1, 7, 27, 77, 180, 366, 672, 1135, 1792, 2680, 3836,5260,6952,8912,11140, ...]; ...
%e A305605 illustrating the occurrence of zeros.
%e A305605 Note that the initial terms of the rows in the above table forms the rows of irregular triangle A127496.
%e A305605 TRIANGULAR FORM.
%e A305605 This sequence may be arranged into a triangle like so:
%e A305605 1,
%e A305605 0, -1,
%e A305605 0, -2, 2,
%e A305605 0, -7, 14, -7,
%e A305605 0, -37, 111, -111, 37,
%e A305605 0, -268, 1072, -1608, 1072, -268,
%e A305605 0, -2496, 12480, -24960, 24960, -12480, 2496,
%e A305605 0, -28612, 171672, -429180, 572240, -429180, 171672, -28612,
%e A305605 ...
%e A305605 in which the g.f. of the rows equal -x * A107877(n) * (1-x)^(n-1) for n > 0.
%o A305605 (PARI) /* Informal code to generate terms */
%o A305605 {A=[1, 0]; for(i=1, 465, A=concat(A, 0); m=floor(sqrt(2*#A-2) + 1/2); A[#A] = -polcoeff( Ser(A)/(1-x +x*O(x^#A))^(m-1), #A-1) ; print1(#A, ", ")); A}
%o A305605 /* Show that the definition is satisfied: */
%o A305605 for(n=0, sqrtint(2*#A)-1, print1(n": "); for(k=n*(n+1)/2+1, (n+1)*(n+2)/2, print1(polcoeff( Ser(A)/(1-x +x*O(x^#A))^n , k), ", ")); print(""))
%Y A305605 Cf. A305601, A107877, A127496.
%K A305605 sign
%O A305605 0,5
%A A305605 _Paul D. Hanna_, Jun 14 2018