OFFSET
0,5
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..2556
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.
EXAMPLE
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.
PROG
(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(""))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jun 14 2018
STATUS
approved