A323557 G.f.: Sum_{n>=0} x^n * (1 + x^n)^n / (1 + x^(n+1))^(n+1).

1, 0, 3, -2, 2, 0, 9, -14, 8, 0, 12, -12, 15, -52, 76, -36, 2, 0, 50, -104, 79, -140, 324, -276, 128, -144, 118, -28, 72, -336, 657, -802, 1184, -1568, 1086, -288, 302, -1032, 1212, -480, 142, -1008, 2789, -3706, 4502, -8040, 9534, -5132, 1166, -544, 778, -2692, 6514, -7904, 5346, -4380, 9679, -16904, 19986, -26744, 41552, -47144, 34636, -16048, 3642, 0, 1454, -9000, 27654, -44936, 38338, -27552, 50187, -90632, 112056, -124816, 172726

Offset

0,3

Comments

Odd terms occur only at positions n*(n+1) for n >= 0 (conjecture; verified for initial 32600 terms).

a(n*(n+1)) = A323679(n).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..10100

Formula

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

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

G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^n - x^k)^(n-k).

G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (-1)^k * (x^n + x^k)^(n-k).

G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (-1)^k * Sum_{j=0..n-k} binomial(n-k,j) * x^((n-k)*(n-j)).

Example

G.f.: A(x) = 1 + 3*x^2 - 2*x^3 + 2*x^4 + 9*x^6 - 14*x^7 + 8*x^8 + 12*x^10 - 12*x^11 + 15*x^12 - 52*x^13 + 76*x^14 - 36*x^15 + 2*x^16 + 50*x^18 - 104*x^19 + 79*x^20 + 140*x^21 + 324*x^22 - 276*x^23 + 128*x^24 - 144*x^25 + 118*x^26 - 28*x^27 + 72*x^28 - 336*x^29 + 657*x^30 - 802*x^31 + 1184*x^32 + ...
such that
A(x) = 1/(1 + x) + x*(1 + x)/(1 + x^2)^2 + x^2*(1 + x^2)^2/(1 + x^3)^3 + x^3*(1 + x^3)^3/(1 + x^4)^4 + x^4*(1 + x^4)^4/(1 + x^5)^5 + x^5*(1 + x^5)^5/(1 + x^6)^6 + x^6*(1 + x^6)^6/(1 + x^7)^7 + x^7*(1 + x^7)^7/(1 + x^8)^8 + ...
also,
A(x) = 1/(1 - x) - x*(1 - x)/(1 - x^2)^2 + x^2*(1 - x^2)^2/(1 - x^3)^3 - x^3*(1 - x^3)^3/(1 - x^4)^4 + x^4*(1 - x^4)^4/(1 - x^5)^5 - x^5*(1 - x^5)^5/(1 - x^6)^6 + x^6*(1 - x^6)^6/(1 - x^7)^7 - x^7*(1 - x^7)^7/(1 - x^8)^8 + ...
ODD TERMS.
It appears that odd terms occur only at n*(n+1); the odd terms begin:
[1, 3, 9, 15, 79, 657, 2789, 9679, 50187, 122379, 911783, 7942511, 71320919, 292307479, 1254424307, 5649367163, 25471489371, ..., A323679(n), ...];
this holds true for at least the initial 32600 terms.
TRIANGLE FORM.
This sequence may be written as a triangle that begins
1, 0;
3, -2, 2, 0;
9, -14, 8, 0, 12, -12;
15, -52, 76, -36, 2, 0, 50, -104;
79, -140, 324, -276, 128, -144, 118, -28, 72, -336;
657, -802, 1184, -1568, 1086, -288, 302, -1032, 1212, -480, 142, -1008;
2789, -3706, 4502, -8040, 9534, -5132, 1166, -544, 778, -2692, 6514, -7904, 5346, -4380;
9679, -16904, 19986, -26744, 41552, -47144, 34636, -16048, 3642, 0, 1454, -9000, 27654, -44936, 38338, -27552;
50187, -90632, 112056, -124816, 172726, -223056, 185458, -98944, 77328, -106400, 98684, -48228, 14956, -31456, 101674, -204336, 240902, -159600;
122379, -319610, 666586, -874488, 927588, -1072924, 1142134, -802912, 313534, -108780, 254532, -558520, 675852, -491140, 336026, -358128, 473868, -853576, 1369462, -1379520; ...
in which the odd terms a(n*(n+1)) = A323679(n) form the left border.
RELATED SEQUENCES.
Terms a(n*(n+2)) = A323677(n) form a diagonal in the above triangle, starting with
[1, -2, 8, -36, 128, -288, 1166, -16048, 77328, -108780, 220440, -5900816, 44395366, -339891804, 898603106, -5623621248, 2160154604, ..., A323677(n), ...].
Terms a(n*(n+3)) = A323678(n) form a diagonal in the above triangle, starting with
[1, 2, 12, 50, 72, 142, 5346, 38338, 240902, 1369462, 8927272, 29594702, 78001922, 259042422, 2690290778, 26069217364, 144738683318, ..., A323678(n), ...].
RELATED SERIES.
Below we illustrate the following identity at specific values of x:
Sum_{n>=0} x^n * (1 + x^n)^n / (1 + x^(n+1))^(n+1)  =  Sum_{n>=0} (-x)^n * (1 - x^n)^n / (1 - x^(n+1))^(n+1).
(1) At x = 1/2, the following sums are equal
S1 = Sum_{n>=0} 2^(n+1) * (2^n + 1)^n / (2^(n+1) + 1)^(n+1),
S1 = Sum_{n>=0} 2^(n+1) * (2^n - 1)^n / (2^(n+1) - 1)^(n+1) * (-1)^n,
where S1 = 1.694294601066597605831822294976249717707326205881024725908408...
(2) At x = 1/3, the following sums are equal
S2 = Sum_{n>=0} 3^(n+1) * (3^n + 1)^n / (3^(n+1) + 1)^(n+1),
S2 = Sum_{n>=0} 3^(n+1) * (3^n - 1)^n / (3^(n+1) - 1)^(n+1) * (-1)^n,
where S2 = 1.291258733393015321539496095851028631331196714523786660740336...
(3) At x = 2/3, the following sums are equal
S3 = Sum_{n>=0} 2^n * 3^(n+1) * (3^n + 2^n)^n / (3^(n+1) + 2^(n+1))^(n+1),
S3 = Sum_{n>=0} 2^n * 3^(n+1) * (3^n - 2^n)^n / (3^(n+1) - 2^(n+1))^(n+1) * (-1)^n,
where S3 = 2.523590984213154172284965025135287234251707014722123198796878...

Prog

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

Crossrefs

Cf. A323679 (odd terms), A323677 (a(n*(n+2))), A323678 (a(n*(n+3))).

Cf. A323675 (variant), A325046 (variant), A326602 (variant).

Cf. A326285.

Sequence in context: A308181 A361862 A326816 * A155917 A291770 A343229

Adjacent sequences: A323554 A323555 A323556 * A323558 A323559 A323560

Keyword

sign

Author

Paul D. Hanna, Feb 03 2019

Status

approved