Search: a324302 -id:a324302
|
| |
|
|
A324300
|
|
G.f.: Sum_{n>=0} x^n * (x^(n+1) + i)^n / (1 + i*x^(n+1))^(n+1).
|
|
+10
7
|
|
|
|
1, 0, -2, 3, 2, 0, -2, -14, 15, 0, -2, 22, 2, 0, -84, 33, 2, 0, -2, 38, 172, 0, -2, -508, 323, 0, -292, 54, 2, 0, -2, 1088, 444, 0, -2580, 1753, 2, 0, -628, -2396, 2, 0, -2, 86, 8142, 0, -2, -10366, 8991, 0, -1092, 102, 2, 0, -16724, -6716, 1372, 0, -2, 44844, 2, 0, -81846, 58495, 33284, 0, -2, 134, 2028, 0, -2, -127882, 2, 0, -62326, 150, 268492, 0, -2, -428606, 268541, 0, -2, 249196, 100100, 0, -3252, -26748, 2, 0, -738612, 182, 3724, 0, -157780, 1133312, 2, 0, -2517158, 1462761, 2, 0, -2, -44508, 2003576, 0, -2, 897068, 2, 0, -5332, -4625662, 2, 0, -344084, 230, 9352622, 0, -3769924, -15721720, 8097455
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
It is remarkable that the generating function results in a power series in x with only real coefficients.
Odd terms occur only at n = k*(k+2) for k >= 0 (conjecture);
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f. A(x) is defined by the following series.
(1) Sum_{n>=0} x^n * (x^(n+1) + i)^n / (1 + i*x^(n+1))^(n+1).
(2) Sum_{n>=0} x^n * (x^(n+1) - i)^n / (1 - i*x^(n+1))^(n+1).
(3) Sum_{n>=0} (i*x)^n * (1 - i*x^(n+1))^(2*n+1) / (1 + x^(2*n+2))^(n+1).
(4) Sum_{n>=0} (-i*x)^n * (1 + i*x^(n+1))^(2*n+1) / (1 + x^(2*n+2))^(n+1).
FORMULAS INVOLVING TERMS.
a(n*(n+2)) = 1 (mod 2) for n >= 0.
a(4*n+1) = 0 for n >= 0.
a(4*n+2) < 0 for n >= 0.
a(p-1) = (-1)^((p-1)/2) * 2 for odd primes p.
|
|
|
EXAMPLE
|
G.f.: A(x) = 1 - 2*x^2 + 3*x^3 + 2*x^4 - 2*x^6 - 14*x^7 + 15*x^8 - 2*x^10 + 22*x^11 + 2*x^12 - 84*x^14 + 33*x^15 + 2*x^16 - 2*x^18 + 38*x^19 + 172*x^20 - 2*x^22 - 508*x^23 + 323*x^24 - 292*x^26 + 54*x^27 + 2*x^28 - 2*x^30 + 1088*x^31 + 444*x^32 - 2580*x^34 + 1753*x^35 + ...
such that
A(x) = 1/(1+i*x) + x*(x^2+i)/(1+i*x^2)^2 + x^2*(x^3+i)^2/(1+i*x^3)^3 + x^3*(x^4+i)^3/(1+i*x^4)^4 + x^4*(x^5+i)^4/(1+i*x^5)^5 + x^5*(x^6+i)^5/(1+i*x^6)^6 + x^6*(x^7+i)^6/(1+i*x^7)^7 + x^7*(x^8+i)^7/(1+i*x^8)^8 + ...
also
A(x) = (1-i*x)/(1+x^2) + i*x*(1-i*x^2)^3/(1+x^4)^2 - x^2*(1-i*x^3)^5/(1+x^6)^3 - i*x^3*(1-i*x^4)^7/(1+x^8)^4 + x^4*(1-i*x^5)^9/(1+x^10)^5 + i*x^5*(1-i*x^6)^11/(1+x^12)^6 + i*x^6*(1-i*x^7)^11/(1+x^14)^7 + ...
Note that the imaginary components in the above sums vanish.
TRIANGLE FORM.
This sequence may be written in the form of a triangle like so:
1;
0, -2, 3;
2, 0, -2, -14, 15;,
0, -2, 22, 2, 0, -84, 33;
2, 0, -2, 38, 172, 0, -2, -508, 323;
0, -292, 54, 2, 0, -2, 1088, 444, 0, -2580, 1753;
2, 0, -628, -2396, 2, 0, -2, 86, 8142, 0, -2, -10366, 8991;
0, -1092, 102, 2, 0, -16724, -6716, 1372, 0, -2, 44844, 2, 0, -81846, 58495;
33284, 0, -2, 134, 2028, 0, -2, -127882, 2, 0, -62326, 150, 268492, 0, -2, -428606, 268541;
0, -2, 249196, 100100, 0, -3252, -26748, 2, 0, -738612, 182, 3724, 0, -157780, 1133312, 2, 0, -2517158, 1462761;
2, 0, -2, -44508, 2003576, 0, -2, 897068, 2, 0, -5332, -4625662, 2, 0, -344084, 230, 9352622, 0, -3769924, -15721720, 8097455; ...
in which the right border consists of all the odd numbers in this sequence.
RELATED SEQUENCES.
The left border, a(n^2), in the above triangle begins:
[1, 0, 2, 0, 2, 0, 2, 0, 33284, 0, 2, 0, 883460, 0, 2, ..., A324301(n), ...].
The main diagonal, a(n*(n+1)), in the above triangle begins:
[1, -2, -2, 2, 172, -2, -2, 1372, 2, -738612, -5332, ..., A324302(n), ...].
The right border, a(n*(n+2)), in the above triangle begins:
[1, 3, 15, 33, 323, 1753, 8991, 58495, 268541, 1462761, ..., A324303(n), ...],
and appears to consist of all the odd terms in this sequence.
|
|
|
PROG
|
(PARI) {a(n) = my(SUM = sum(m=0, n, x^m*(x^(m+1) + I +x*O(x^n))^m / (1 + I*x^(m+1) +x*O(x^n))^(m+1) ) ); polcoeff(SUM, n)}
for(n=0, 120, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
sign
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A324303
|
|
a(n) = [x^(n*(n+2))] Sum_{m>=0} x^m * (x^(m+1) + i)^m / (1 + i*x^(m+1))^(m+1), for n >= 0.
|
|
+10
5
|
|
|
|
1, 3, 15, 33, 323, 1753, 8991, 58495, 268541, 1462761, 8097455, 13837475, 251595971, 1409934009, 9674895101, 46242140669, 252055236611, 1426069096367, 8079317057871, 45129781821155, 260949012214457, 1482376214228889, 8443414161166175, 28159474726319741, 274738209468977925, 1569245074591691433, 8970254867527278283, 51182160225347277091, 293733710358893793731, 1737504244377865756267
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
a(n) = A324300(n*(n+2)) for n >= 0.
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = [x^(n*(n+2))] Sum_{n>=0} x^n * (x^(n+1) + i)^n / (1 + i*x^(n+1))^(n+1).
a(n) = [x^(n*(n+2))] Sum_{n>=0} x^n * (x^(n+1) - i)^n / (1 - i*x^(n+1))^(n+1).
a(n) = [x^(n*(n+2))] Sum_{n>=0} (i*x)^n * (1 - i*x^(n+1))^(2*n+1) / (1 + x^(2*n+2))^(n+1).
a(n) = [x^(n*(n+2))] Sum_{n>=0} (-i*x)^n * (1 + i*x^(n+1))^(2*n+1) / (1 + x^(2*n+2))^(n+1).
|
|
|
EXAMPLE
|
G(x) = Sum_{n>=0} x^n * (x^(n+1) + i)^n / (1 + i*x^(n+1))^(n+1)
where
G(x) = 1 - 2*x^2 + 3*x^3 + 2*x^4 - 2*x^6 - 14*x^7 + 15*x^8 - 2*x^10 + 22*x^11 + 2*x^12 - 84*x^14 + 33*x^15 + 2*x^16 - 2*x^18 + 38*x^19 + 172*x^20 - 2*x^22 - 508*x^23 + 323*x^24 - 292*x^26 + 54*x^27 + 2*x^28 - 2*x^30 + 1088*x^31 + 444*x^32 - 2580*x^34 + 1753*x^35 + ...
This sequence gives the odd coefficients of x^n in G(x), which occur at n = k*(k+2) for k >= 0.
Explicitly,
G(x) = 1/(1+i*x) + x*(x^2+i)/(1+i*x^2)^2 + x^2*(x^3+i)^2/(1+i*x^3)^3 + x^3*(x^4+i)^3/(1+i*x^4)^4 + x^4*(x^5+i)^4/(1+i*x^5)^5 + x^5*(x^6+i)^5/(1+i*x^6)^6 + x^6*(x^7+i)^6/(1+i*x^7)^7 + x^7*(x^8+i)^7/(1+i*x^8)^8 + ...
also
G(x) = (1-i*x)/(1+x^2) + i*x*(1-i*x^2)^3/(1+x^4)^2 - x^2*(1-i*x^3)^5/(1+x^6)^3 - i*x^3*(1-i*x^4)^7/(1+x^8)^4 + x^4*(1-i*x^5)^9/(1+x^10)^5 + i*x^5*(1-i*x^6)^11/(1+x^12)^6 + i*x^6*(1-i*x^7)^11/(1+x^14)^7 + ...
Note that the imaginary components in the above sums vanish.
|
|
|
PROG
|
(PARI) {A324300(n) = my(SUM = sum(m=0, n, x^m*(x^(m+1) + I +x*O(x^n))^m / (1 + I*x^(m+1) +x*O(x^n))^(m+1) ) ); polcoeff(SUM, n)}
for(n=0, 60, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A324301
|
|
a(n) = [x^(n^2)] Sum_{m>=0} x^m * (x^(m+1) + i)^m / (1 + i*x^(m+1))^(m+1), for n >= 0.
|
|
+10
4
|
|
|
|
1, 0, 2, 0, 2, 0, 2, 0, 33284, 0, 2, 0, 883460, 0, 2, 0, 2, 0, 979339580806, 0, 2, 0, 115667460, 0, 2, 0, 2, 0, 799941380, 0, 23305156883711135492, 0, 38029776687864458395142, 0, 4012026683676222852, 0, 2, 0, 44190267844175072072966, 0, 2, 0, 20586734084, 0, 1978148160774064661252, 0, 29798631414807274867656075256452, 0, 59960389124, 0, 1826037115915435009712832114231669892, 0, 113795930884, 0, 2, 0, 2, 0, 272716398084, 0, 3420684050214265377966852
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = [x^(n^2)] Sum_{n>=0} x^n * (x^(n+1) + i)^n / (1 + i*x^(n+1))^(n+1).
a(n) = [x^(n^2)] Sum_{n>=0} x^n * (x^(n+1) - i)^n / (1 - i*x^(n+1))^(n+1).
a(n) = [x^(n^2)] Sum_{n>=0} (i*x)^n * (1 - i*x^(n+1))^(2*n+1) / (1 + x^(2*n+2))^(n+1).
a(n) = [x^(n^2)] Sum_{n>=0} (-i*x)^n * (1 + i*x^(n+1))^(2*n+1) / (1 + x^(2*n+2))^(n+1).
a(2*n+1) = 0 for n >= 0.
|
|
|
EXAMPLE
|
G(x) = Sum_{n>=0} x^n * (x^(n+1) + i)^n / (1 + i*x^(n+1))^(n+1)
where
G(x) = 1 - 2*x^2 + 3*x^3 + 2*x^4 - 2*x^6 - 14*x^7 + 15*x^8 - 2*x^10 + 22*x^11 + 2*x^12 - 84*x^14 + 33*x^15 + 2*x^16 - 2*x^18 + 38*x^19 + 172*x^20 - 2*x^22 - 508*x^23 + 323*x^24 - 292*x^26 + 54*x^27 + 2*x^28 - 2*x^30 + 1088*x^31 + 444*x^32 - 2580*x^34 + 1753*x^35 + ...
such that
G(x) = 1/(1+i*x) + x*(x^2+i)/(1+i*x^2)^2 + x^2*(x^3+i)^2/(1+i*x^3)^3 + x^3*(x^4+i)^3/(1+i*x^4)^4 + x^4*(x^5+i)^4/(1+i*x^5)^5 + x^5*(x^6+i)^5/(1+i*x^6)^6 + x^6*(x^7+i)^6/(1+i*x^7)^7 + x^7*(x^8+i)^7/(1+i*x^8)^8 + ...
also
G(x) = (1-i*x)/(1+x^2) + i*x*(1-i*x^2)^3/(1+x^4)^2 - x^2*(1-i*x^3)^5/(1+x^6)^3 - i*x^3*(1-i*x^4)^7/(1+x^8)^4 + x^4*(1-i*x^5)^9/(1+x^10)^5 + i*x^5*(1-i*x^6)^11/(1+x^12)^6 + i*x^6*(1-i*x^7)^11/(1+x^14)^7 + ...
Note that the imaginary components in the above sums vanish.
|
|
|
PROG
|
(PARI) {A324300(n) = my(SUM = sum(m=0, n, x^m*(x^(m+1) + I +x*O(x^n))^m / (1 + I*x^(m+1) +x*O(x^n))^(m+1) ) ); polcoeff(SUM, n)}
for(n=0, 60, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
Search completed in 0.009 seconds
|
