|
| |
|
|
A292177
|
|
G.f.: Limit_{K->oo} Sum_{n=-oo..+oo} x^(n-K) * (1 - x^n + n*(n+1)/6 * x^(n+K))^n.
|
|
2
|
|
|
|
1, 0, 4, 0, 5, 0, 20, -24, 35, 0, 0, 0, 84, -160, 200, 0, -150, 0, 460, -560, 286, 0, 140, -200, 455, -1440, 2100, 0, -2180, 0, 3840, -3080, 969, -2240, 2730, 0, 1330, -5824, 5320, 0, -4235, 0, 16874, -21840, 2300, 0, 18440, -784, -20175, -16320, 37310, 0, -945, -42240, 49560, -25080, 4495, 0, 7560, 0, 5456, -50400, 102528, -120120, 40810, 0, 135660, -52624, -221690, 0, 278256, 0, 9139, -364000, 232750, -99792, 211120, 0, -106680, -100440, 12341, 0, 537992, -628320, 14190, -129920, 563420, 0, -195015, -480480, 591100, -168640, 18424, -1240320, 2138640, 0, -925120, -268224, -803250, 0
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
2,3
|
|
|
COMMENTS
|
Compare the g.f. to: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n = 0.
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: Limit_{K->oo} Sum_{n=-oo..+oo} -(-1)^n * x^(n^2-n-K) / (1 - x^n + n*(n-1)/6 * x^K)^n.
a(p) = 0 for odd prime p (conjecture).
|
|
|
EXAMPLE
|
G.f.: A(x) = x^2 + 4*x^4 + 5*x^6 + 20*x^8 - 24*x^9 + 35*x^10 + 84*x^14 - 160*x^15 + 200*x^16 - 150*x^18 + 460*x^20 - 560*x^21 + 286*x^22 + 140*x^24 - 200*x^25 + 455*x^26 - 1440*x^27 + 2100*x^28 - 2180*x^30 + 3840*x^32 - 3080*x^33 + 969*x^34 - 2240*x^35 + 2730*x^36 + 1330*x^38 - 5824*x^39 + 5320*x^40 +...
such that the g.f. equals the limit of the sum, as K tends to infinity,
S(K) = Sum_{n=-oo..+oo} x^(n-K) * (1 - x^n + n*(n+1)/6 * x^(n+K))^n.
Illustration of the limit.
S(1) = x^2 + 1/3*x^3 + 112/27*x^4 + 113/81*x^5 + 467/81*x^6 - 938/729*x^7 +...
S(2) = x^2 + 13/3*x^4 + 175/27*x^6 + 1550/81*x^8 - 24*x^9 + 2777/81*x^10 +...
S(3) = x^2 + 4*x^4 + 1/3*x^5 + 5*x^6 + 4/3*x^7 + 544/27*x^8 - 77/3*x^9 +...
S(4) = x^2 + 4*x^4 + 16/3*x^6 + 64/3*x^8 - 24*x^9 + 904/27*x^10 +...
S(5) = x^2 + 4*x^4 + 5*x^6 + 1/3*x^7 + 20*x^8 - 68/3*x^9 + 35*x^10 +...
S(6) = x^2 + 4*x^4 + 5*x^6 + 61/3*x^8 - 24*x^9 + 109/3*x^10 - 5/3*x^12 +...
S(7) = x^2 + 4*x^4 + 5*x^6 + 20*x^8 - 71/3*x^9 + 35*x^10 + 4/3*x^11 +...
S(8) = x^2 + 4*x^4 + 5*x^6 + 20*x^8 - 24*x^9 + 106/3*x^10 + 4/3*x^12 +...
S(9) = x^2 + 4*x^4 + 5*x^6 + 20*x^8 - 24*x^9 + 35*x^10 + 1/3*x^11 +...
...
At powers of 2, a(2^n) begins:
[1, 4, 20, 200, 3840, 102528, 8437440, 5275875200, 5635011683840, 2075681844543566848, 671078483184128826885120, ...].
|
|
|
PROG
|
(PARI) {a(n) = my(A=1, K=n); A = sum(m=-sqrtint(2*n+9), 2*n+1, x^(m-K) * (1 - x^m +m*(m+1)/6*x^(m+K) + O(x^(2*n+2)) )^m ); polcoeff(A, n)}
for(n=2, 80, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
sign
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
