A323675 G.f.: Sum_{n>=0} x^n * (x^n + i)^n / (1 + i*x^(n+1))^(n+1), where i^2 = -1.

1, 0, -1, 2, 2, 0, -7, -2, 8, 0, 8, 12, -9, -28, -16, 4, 2, 0, 46, 104, 39, -100, -144, 4, 16, -144, -66, 28, 72, 336, 533, 178, -496, -448, 242, 288, -298, -1032, -1212, -480, 142, 1008, 2701, 2586, -538, -2040, 198, 1868, 686, -544, -678, -2588, -6510, -7904, -4230, 2028, 9567, 14888, 7382, -10216, -14648, 4280, 21572, 15920, 3642, 0, -398, -6840, -26546, -44936, -37498, -16464, 5259, 46792, 70792, 13936, -67514, -65232, 5118, 53024, 77328, 106400, 98680, 48228, 4428, -31456, -101674, -204336, -239578, -137424, 23687, 164954, 247346, 147720, -175264, -430724, -301290

Offset

0,4

Comments

It is remarkable that the generating function results in a power series in x with only real coefficients.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..4970

Formula

G.f.: Sum_{n>=0} x^n * (x^n + i)^n / (1 + i*x^(n+1))^(n+1).

G.f.: Sum_{n>=0} x^n * (x^n - i)^n / (1 - i*x^(n+1))^(n+1).

G.f.: Sum_{n>=0} x^n * (x^n + i)^n * (1 - i*x^(n+1))^(n+1) / (1 + x^(2*n+2))^(n+1).

G.f.: Sum_{n>=0} x^n * (x^n - i)^n * (1 + i*x^(n+1))^(n+1) / (1 + x^(2*n+2))^(n+1).

Example

G.f.: A(x) = 1 - x^2 + 2*x^3 + 2*x^4 - 7*x^6 - 2*x^7 + 8*x^8 + 8*x^10 + 12*x^11 - 9*x^12 - 28*x^13 - 16*x^14 + 4*x^15 + 2*x^16 + 46*x^18 + 104*x^19 + 39*x^20 - 100*x^21 - 144*x^22 + 4*x^23 + 16*x^24 - 144*x^25 - 66*x^26 + 28*x^27 + 72*x^28 + 336*x^29 + 533*x^30 + 178*x^31 - 496*x^32 - 448*x^33 + 242*x^34 + 288*x^35 - 298*x^36 - 1032*x^37 - 1212*x^38 - 480*x^39 + 142*x^40 + 1008*x^41 + 2701*x^42 + ...
such that
A(x) = 1/(1 + i*x) + x*(x + i)/(1 + i*x^2)^2 + x^2*(x^2 + i)^2/(1 + i*x^3)^3 + x^3*(x^3 + i)^3/(1 + i*x^4)^4 + x^4*(x^4 + i)^4/(1 + i*x^5)^5 + x^5*(x^5 + i)^5/(1 + i*x^6)^6 + x^6*(x^6 + i)^6/(1 + i*x^7)^7 + x^7*(x^7 + i)^7/(1 + i*x^8)^8 + ...
also,
A(x) = 1/(1 - i*x) + x*(x - i)/(1 - i*x^2)^2 + x^2*(x^2 - i)^2/(1 - i*x^3)^3 + x^3*(x^3 - i)^3/(1 - i*x^4)^4 + x^4*(x^4 - i)^4/(1 - i*x^5)^5 + x^5*(x^5 - i)^5/(1 - i*x^6)^6 + x^6*(x^6 - i)^6/(1 - i*x^7)^7 + x^7*(x^7 - i)^7/(1 - i*x^8)^8 + ...
ODD TERMS.
It appears that odd terms occur only at n*(n+1); the odd terms begin:
[1, -1, -7, -9, 39, 533, 2701, 9567, 5259, 23687, -531597, -6683401, -27445177, -251078037, -1245962509, -5523256133, -24464598853, ..., A323676(n), ...].
TRIANGLE FORM.
This sequence may be written as a triangle that begins
1, 0;
-1, 2, 2, 0;
-7, -2, 8, 0, 8, 12;
-9, -28, -16, 4, 2, 0, 46, 104;
39, -100, -144, 4, 16, -144, -66, 28, 72, 336;
533, 178, -496, -448, 242, 288, -298, -1032, -1212, -480, 142, 1008;
2701, 2586, -538, -2040, 198, 1868, 686, -544, -678, -2588, -6510, -7904, -4230, 2028;
9567, 14888, 7382, -10216, -14648, 4280, 21572, 15920, 3642, 0, -398, -6840, -26546, -44936, -37498, -16464;
5259, 46792, 70792, 13936, -67514, -65232, 5118, 53024, 77328, 106400, 98680, 48228, 4428, -31456, -101674, -204336, -239578, -137424;
23687, 164954, 247346, 147720, -175264, -430724, -301290, 20768, 136254, 108780, 245332, 527320, 640544, 477828, 329162, 262032, -66676, -748024, -1355778, -1299024; ...
in which the leftmost column consists only of odd integers (A323676).

Prog

(PARI) {a(n) = my(A=sum(m=0, n, x^m * (x^m + I +x*O(x^n))^m/(1 + I*x^(m+1) +x*O(x^n))^(m+1) )); polcoeff(A, n)}
for(n=0, 120, print1(a(n), ", "))
(PARI) {a(n) = my(A=sum(m=0, n, x^m * (x^m - I +x*O(x^n))^m/(1 - I*x^(m+1) +x*O(x^n))^(m+1) )); polcoeff(A, n)}
for(n=0, 120, print1(a(n), ", "))

Crossrefs

Cf. A323676 (odd terms).

Cf. A323557, A324300, A323695.

Sequence in context: A174104 A296492 A135006 * A243492 A086118 A104986

Adjacent sequences: A323672 A323673 A323674 * A323676 A323677 A323678

Keyword

sign

Author

Paul D. Hanna, Feb 10 2019

Status

approved