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.

1, 0, -1, 0, -2, 2, 0, -7, 14, -7, 0, -37, 111, -111, 37, 0, -268, 1072, -1608, 1072, -268, 0, -2496, 12480, -24960, 24960, -12480, 2496, 0, -28612, 171672, -429180, 572240, -429180, 171672, -28612, 0, -391189, 2738323, -8214969, 13691615, -13691615, 8214969, -2738323, 391189, 0, -6230646, 49845168, -174458088, 348916176, -436145220, 348916176, -174458088, 49845168, -6230646, 0

Offset: 0

Author

Paul D. Hanna, Jun 14 2018

Keywords

• sign

Examples

			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 + ...
The table of coefficients of x^k in A(x) / (1-x)^n, for n >= 0, begins:
[1, 0, -1,  0,  -2,   2,   0,   -7,  14,    -7,    0, -37, 111,-111,   37, ...];
[1, 1,  0,  0,  -2,   0,   0,   -7,   7,     0,    0, -37,  74, -37,    0, ...];
[1, 2,  2,  2,   0,   0,   0,   -7,   0,     0,    0, -37,  37,   0,    0, ...];
[1, 3,  5,  7,   7,   7,   7,    0,   0,     0,    0, -37,   0,   0,    0, ...];
[1, 4,  9, 16,  23,  30,  37,   37,  37,    37,   37,   0,   0,   0,    0, ...];
[1, 5, 14, 30,  53,  83, 120,  157,  194,  231,  268, 268, 268, 268,  268, ...];
[1, 6, 20, 50, 103, 186, 306,  463,  657,  888, 1156,1424,1692,1960, 2228, ...];
[1, 7, 27, 77, 180, 366, 672, 1135, 1792, 2680, 3836,5260,6952,8912,11140, ...]; ...
illustrating the occurrence of zeros.
Note that the initial terms of the rows in the above table forms the rows of irregular triangle A127496.
TRIANGULAR FORM.
This sequence may be arranged into a triangle like so:
1,
0, -1,
0, -2, 2,
0, -7, 14, -7,
0, -37, 111, -111, 37,
0, -268, 1072, -1608, 1072, -268,
0, -2496, 12480, -24960, 24960, -12480, 2496,
0, -28612, 171672, -429180, 572240, -429180, 171672, -28612,
...
in which the g.f. of the rows equal -x * A107877(n) * (1-x)^(n-1) for n > 0.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..2556

Crossrefs

Cf. A305601, A107877, A127496.

Programs

PARI

/* Informal code to generate terms */
{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}
/* Show that the definition is satisfied: */
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(""))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies:

(1) A(x) = 1 - x*Sum_{n>=1} A107877(n) * x^(n*(n+1)/2) * (1-x)^(n-1).

(2) [x^k] A(x) / (1-x)^n = 0 for k = n*(n+1)/2 + 1 through (n+1)*(n+2)/2, n >= 0.

(3) [x^k] A(x) / (1-x)^n = A107877(n) for k = n*(n-1)/2 through n*(n+1)/2, n >= 0.

(4) [x^k] A(x) / (1-x)^n = A127496(n,k) for k = 0..n*(n+1)/2 for n >= 0.

(5) [x^n] A(x) / (1-x)^n = A127497(n) for n >= 0.

FORMULAS INVOLVING TERMS.

a(n*(n+1)/2) = 0 for n >= 1.

a(n*(n-1)/2) = (-1)^n * A107877(n) for n >= 0.

a(n*(n+1)/2 + 1) = -A107877(n) for n >= 0.

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.