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A323687 Odd coefficients in Sum_{n>=0} (x^n + i)^n / (1 + i*x^n)^(n+1), in which the constant term is taken to be 1. 3
1, 3, 13, 101, 31, 1785, 10359, 48837, 361601, 1518439, 8107655, 45047205, 22174877, 1409934297, 7923887063, 58640750927, 268411539971, 1425834725577, 8083997355233, 45849429916389, 253004366571229, 1487729015517467, 8443414161401399, 48141245001933381, 155779268193228419, 1569245091203776687, 8970232353224094279, 51314027859988631817, 292380695300170801437, 1682471873186160627609, 10085943474769129981125, 55294491352291112750853 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) = A323689(n^2) for n >= 0.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..100

FORMULA

a(n) = [x^(n^2)] Sum_{k>=0} (x^k + i)^k / (1 + i*x^k)^(k+1) for n > 0.

a(n) = [x^(n^2)] Sum_{k>=0} (x^k - i)^k / (1 - i*x^k)^(k+1) for n > 0.

a(n) = [x^(n^2)] Sum_{n>=0} (-i)^n * (1 + i*x^n)^(2*n+1) / (1 + x^(2*n))^(n+1) for n > 0.

a(n) = [x^(n^2)] Sum_{n>=0} i^n * (1 - i*x^n)^(2*n+1) / (1 + x^(2*n))^(n+1) for n > 0.

EXAMPLE

The generating function of A323689 is

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

in which the constant term is taken to be 1, so that

G(x) = 1 + 3*x - 14*x^3 + 13*x^4 + 22*x^5 - 30*x^7 - 82*x^8 + 101*x^9 - 46*x^11 + 170*x^12 + 54*x^13 - 524*x^15 + 31*x^16 + 70*x^17 - 78*x^19 + 442*x^20 + 1236*x^21 - 94*x^23 - 3204*x^24 + 1785*x^25 - 2428*x^27 + 842*x^28 + 118*x^29 - 126*x^31 + 6208*x^32 + 4228*x^33 - 14508*x^35 + 10359*x^36 + ...

This sequence gives the odd coefficients of x^n, which occur at n = k^2 for k >= 0.

PROG

(PARI) {A323689(n) = my(SUM = sum(m=0, n, (x^m + I +x*O(x^n))^m / (1 + I*x^m +x*O(x^n))^(m+1) ) ); polcoeff(1 + SUM - I^n/(1+I), n)}

{a(n) = A323689(n^2)}

for(n=0, 30, print1(a(n), ", "))

(PARI) {A323689(n) = my(SUM = sum(m=0, n, (-I)^m*(1 + I*x^m +x*O(x^n))^(2*m+1) / (1 + x^(2*m) +x*O(x^n))^(m+1) ) ); polcoeff(1 + SUM - I^n/(1-I), n)}

{a(n) = A323689(n^2)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A323689.

Sequence in context: A220897 A267196 A268215 * A338697 A168417 A240167

Adjacent sequences:  A323684 A323685 A323686 * A323688 A323689 A323690

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 14 2019

STATUS

approved

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Last modified July 31 20:24 EDT 2021. Contains 346377 sequences. (Running on oeis4.)