OFFSET
0,3
COMMENTS
More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1,n) * (q^n + p)^n * r^n / (1 + p*q^n*r)^(n+k),
(2) Sum_{n>=0} binomial(n+k-1,n) * (q^n - p)^n * r^n / (1 - p*q^n*r)^(n+k),
for any fixed integer k; this sequence results when k=2, q = x, r = x, p = 1.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..10100
FORMULA
G.f.: Sum_{n>=0} (n+1) * x^n * (1 + x^n)^n / (1 + x^(n+1))^(n+2).
G.f.: Sum_{n>=0} (n+1) * (-x)^n * (1 - x^n)^n / (1 - x^(n+1))^(n+2).
G.f.: Sum_{n>=0} (n+1) * x^n * Sum_{k=0..n} binomial(n,k) * (x^n - x^k)^(n-k).
G.f.: Sum_{n>=0} (n+1) * x^n * Sum_{k=0..n} binomial(n,k) * (-1)^k * (x^n + x^k)^(n-k).
G.f.: Sum_{n>=0} (n+1) * 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 + 8*x^2 - 6*x^3 + 10*x^4 + 41*x^6 - 64*x^7 + 48*x^8 + 82*x^10 - 84*x^11 + 90*x^12 - 300*x^13 + 532*x^14 - 284*x^15 + 34*x^16 + 428*x^18 - 892*x^19 + 671*x^20 - 960*x^21 + 2620*x^22 - 2440*x^23 + 1184*x^24 - 1440*x^25 + 1408*x^26 - 420*x^27 + 618*x^28 - 3024*x^29 + 6788*x^30 - 8274*x^31 + 11022*x^32 + ...
such that
A(x) = 1/(1 + x)^2 + 2*x*(1 + x)/(1 + x^2)^3 + 3*x^2*(1 + x^2)^2/(1 + x^3)^4 + 4*x^3*(1 + x^3)^3/(1 + x^4)^5 + 5*x^4*(1 + x^4)^4/(1 + x^5)^6 + 6*x^5*(1 + x^5)^5/(1 + x^6)^7 + 7*x^6*(1 + x^6)^6/(1 + x^7)^8 + 8*x^7*(1 + x^7)^7/(1 + x^8)^9 + ...
also,
A(x) = 1/(1 - x)^2 - 2*x*(1 - x)/(1 - x^2)^3 + 3*x^2*(1 - x^2)^2/(1 - x^3)^4 - 4*x^3*(1 - x^3)^3/(1 - x^4)^5 + 5*x^4*(1 - x^4)^4/(1 - x^5)^6 - 6*x^5*(1 - x^5)^5/(1 - x^6)^7 + 7*x^6*(1 - x^6)^6/(1 - x^7)^8 - 8*x^7*(1 - x^7)^7/(1 - x^8)^9 + ...
TRIANGLE FORM.
This sequence may be written as a triangle that begins
1, 0;
8, -6, 10, 0;
41, -64, 48, 0, 82, -84;
90, -300, 532, -284, 34, 0, 428, -892;
671, -960, 2620, -2440, 1184, -1440, 1408, -420, 618, -3024;
6788, -8274, 11022, -15120, 11602, -3456, 3470, -12288, 15448, -6560, 1342, -10080;
31803, -44788, 49980, -89400, 115312, -65976, 16190, -9792, 11836, -33084, 85334, -112840, 83666, -62064;
119486, -216504, 258898, -329880, 534492, -660184, 524614, -259320, 62010, 0, 24092, -129628, 398778, -693624, 634072, -440304;
746315, -1354080, 1668322, -1795524, 2443316, -3314808, 2935124, -1626372, 1263716, -1827840, 1815130, -949356, 308128, -549528, 1637302, -3308360, 4092178, -2817360; ...
in which the leftmost border (A326286) begins:
[1, 8, 41, 90, 671, 6788, 31803, 119486, 746315, 1959108, 17687917, ...].
RELATED SERIES.
Below we illustrate the following identity at specific values of x:
Sum_{n>=0} (n+1) * x^n * (1 + x^n)^n / (1 + x^(n+1))^(n+2) = Sum_{n>=0} (n+1) * (-x)^n * (1 - x^n)^n / (1 - x^(n+1))^(n+2).
(1) At x = 1/2, the following sums are equal
S1 = Sum_{n>=0} (n+1) * 4^(n+1) * (2^n + 1)^n / (2^(n+1) + 1)^(n+2),
S1 = Sum_{n>=0} (n+1) * 4^(n+1) * (2^n - 1)^n / (2^(n+1) - 1)^(n+2) * (-1)^n,
where S1 = 3.25235487864227095443984862634129135387423736948777534415428...
(2) At x = 1/3, the following sums are equal
S2 = Sum_{n>=0} (n+1) * 9^(n+1) * (3^n + 1)^n / (3^(n+1) + 1)^(n+2)
S2 = Sum_{n>=0} (n+1) * 9^(n+1) * (3^n - 1)^n / (3^(n+1) - 1)^(n+2) * (-1)^n,
where S2 = 1.825405411464020940673050144823957018856761891814066044697067...
(3) At x = 2/3, the following sums are equal
S3 = Sum_{n>=0} (n+1) * 2^n * 9^(n+1) * (3^n + 2^n)^n / (3^(n+1) + 2^(n+1))^(n+2),
S3 = Sum_{n>=0} (n+1) * 2^n * 9^(n+1) * (3^n - 2^n)^n / (3^(n+1) - 2^(n+1))^(n+2) * (-1)^n,
where S3 = 7.382803343792781402458424946145387931796609475310335027992482...
PROG
(PARI) {a(n) = my(A=sum(m=0, n, (m+1) * x^m * (1 + x^m +x*O(x^n))^m/(1 + x^(m+1) +x*O(x^n))^(m+2) )); polcoeff(A, n)}
for(n=0, 120, print1(a(n), ", "))
(PARI) {a(n) = my(A=sum(m=0, n, (m+1) * (-x)^m * (1 - x^m +x*O(x^n))^m/(1 - x^(m+1) +x*O(x^n))^(m+2) )); polcoeff(A, n)}
for(n=0, 120, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jul 01 2019
STATUS
approved