OFFSET
0,4
COMMENTS
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..2080
FORMULA
G.f. A(x) also satisfies:
(1) A(x) = 1/(1+x) - Sum_{n>=1} A107877(n) * (x/(1+x))^(n*(n+1)/2+1) / (1+x)^n.
(2) [x^(n*(n-1)/2)] (1+x)^(n*(n+1)/2) * A(x) = A107877(n) for n >= 0.
(3) [x^(n*(n+1)/2 + 1)] (1+x)^(n*(n+1)/2) * A(x) = -A107877(n) for n >= 0.
(5) [x^k] (1+x)^(n+k) * A(x) = A127496(n,k) for k = 0..n*(n+1)/2, for n>=0.
(6) [x^n] (1+x)^n * A(x) = A305605(n) for n >= 0.
EXAMPLE
G.f. A(x) = 1 - x + 2*x^3 - 7*x^4 + 21*x^5 - 56*x^6 + 125*x^7 - 209*x^8 + 154*x^9 + 572*x^10 - 3404*x^11 + 11930*x^12 - 35394*x^13 + 99144*x^14 - 274550*x^15 + 757813*x^16 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1+x)^n * A(x) begins:
n= 0: [1, -1, 0, 2, -7, 21, -56, 125, -209, 154, 572, -3404, ...];
n= 1: [1, 0, -1, 2, -5, 14, -35, 69, -84, -55, 726, -2832, ...];
n= 2: [1, 1, -1, 1, -3, 9, -21, 34, -15, -139, 671, -2106, ...];
n= 3: [1, 2, 0, 0, -2, 6, -12, 13, 19, -154, 532, -1435, ...];
n= 4: [1, 3, 2, 0, -2, 4, -6, 1, 32, -135, 378, -903, ...];
n= 5: [1, 4, 5, 2, -2, 2, -2, -5, 33, -103, 243, -525, ...];
n= 6: [1, 5, 9, 7, 0, 0, 0, -7, 28, -70, 140, -282, ...];
n= 7: [1, 6, 14, 16, 7, 0, 0, -7, 21, -42, 70, -142, ...];
n= 8: [1, 7, 20, 30, 23, 7, 0, -7, 14, -21, 28, -72, ...];
n= 9: [1, 8, 27, 50, 53, 30, 7, -7, 7, -7, 7, -44, ...];
n=10: [1, 9, 35, 77, 103, 83, 37, 0, 0, 0, 0, -37, ...];
n=11: [1, 10, 44, 112, 180, 186, 120, 37, 0, 0, 0, -37, ...];
n=12: [1, 11, 54, 156, 292, 366, 306, 157, 37, 0, 0, -37, ...];
n=13: [1, 12, 65, 210, 448, 658, 672, 463, 194, 37, 0, -37, ...];
n=14: [1, 13, 77, 275, 658, 1106, 1330, 1135, 657, 231, 37, -37, ...];
n=15: [1, 14, 90, 352, 933, 1764, 2436, 2465, 1792, 888, 268, 0, 0, 0, 0, 0, -268, ...]; ...
which illustrates the occurrences of zeros in the table.
RELATED SEQUENCES.
Notice that [x^(n*(n-1)/2)] (1+x)^(n*(n+1)/2) * A(x) = A107877(n), which begins:
[1, 1, 2, 7, 37, 268, 2496, 28612, 391189, 6230646, 113521387, ...].
Also, note that the coefficient of x^(n*(n-1)/2) in (1+x)^(n*(n+1)/2) * A(x) yields -A107877(n).
Also, observe that [x^n] (1+x)^(2*n) * A(x) = A127497(n), which begins:
[1, 1, 2, 7, 23, 83, 306, 1135, 4257, 16095, 61222, 233956, ...].
The initial terms of the diagonals in the above table forms the rows of irregular triangle A127496:
1;
1, 1;
1, 2, 2, 2;
1, 3, 5, 7, 7, 7, 7;
1, 4, 9, 16, 23, 30, 37, 37, 37, 37, 37;
1, 5, 14, 30, 53, 83, 120, 157, 194, 231, 268, 268, 268, 268, 268, 268;
1, 6, 20, 50, 103, 186, 306, 463, 657, 888, 1156, 1424, 1692, 1960, 2228, 2496, 2496, 2496, 2496, 2496, 2496, 2496; ...
in which row n equals the partial sums of the prior row with the final term repeated n more times at the end.
PROG
(PARI) /* Informal code to generate terms */
{A=[1, -1]; for(i=1, 465, A=concat(A, 0); m=floor(sqrt(2*#A-2) + 1/2); A[#A] = -polcoeff( (1+x +x*O(x^#A))^(m*(m+1)/2)*Ser(A), #A-1) ; print1(#A, ", ")); A}
/* Show that the definition is satisfied: */
for(n=1, floor(sqrt(2*#A) + 1/2), print1(n": "); for(k=n*(n-1)/2+1, n*(n+1)/2, print1(polcoeff( (1+x +x*O(x^#A))^(n*(n+1)/2)*Ser(A) , k), ", ")); print(""))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jun 14 2018
STATUS
approved