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%I #56 Aug 10 2019 00:42:58
%S 1,0,3,-2,2,0,9,-14,8,0,12,-12,15,-52,76,-36,2,0,50,-104,79,-140,324,
%T -276,128,-144,118,-28,72,-336,657,-802,1184,-1568,1086,-288,302,
%U -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
%N G.f.: Sum_{n>=0} x^n * (1 + x^n)^n / (1 + x^(n+1))^(n+1).
%C Odd terms occur only at positions n*(n+1) for n >= 0 (conjecture; verified for initial 32600 terms).
%C a(n*(n+1)) = A323679(n).
%H Paul D. Hanna, <a href="/A323557/b323557.txt">Table of n, a(n) for n = 0..10100</a>
%F G.f.: Sum_{n>=0} x^n * (1 + x^n)^n / (1 + x^(n+1))^(n+1).
%F G.f.: Sum_{n>=0} (-x)^n * (1 - x^n)^n / (1 - x^(n+1))^(n+1).
%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^n - x^k)^(n-k).
%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (-1)^k * (x^n + x^k)^(n-k).
%F 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)).
%e 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 + ...
%e such that
%e 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 + ...
%e also,
%e 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 + ...
%e ODD TERMS.
%e It appears that odd terms occur only at n*(n+1); the odd terms begin:
%e [1, 3, 9, 15, 79, 657, 2789, 9679, 50187, 122379, 911783, 7942511, 71320919, 292307479, 1254424307, 5649367163, 25471489371, ..., A323679(n), ...];
%e this holds true for at least the initial 32600 terms.
%e TRIANGLE FORM.
%e This sequence may be written as a triangle that begins
%e 1, 0;
%e 3, -2, 2, 0;
%e 9, -14, 8, 0, 12, -12;
%e 15, -52, 76, -36, 2, 0, 50, -104;
%e 79, -140, 324, -276, 128, -144, 118, -28, 72, -336;
%e 657, -802, 1184, -1568, 1086, -288, 302, -1032, 1212, -480, 142, -1008;
%e 2789, -3706, 4502, -8040, 9534, -5132, 1166, -544, 778, -2692, 6514, -7904, 5346, -4380;
%e 9679, -16904, 19986, -26744, 41552, -47144, 34636, -16048, 3642, 0, 1454, -9000, 27654, -44936, 38338, -27552;
%e 50187, -90632, 112056, -124816, 172726, -223056, 185458, -98944, 77328, -106400, 98684, -48228, 14956, -31456, 101674, -204336, 240902, -159600;
%e 122379, -319610, 666586, -874488, 927588, -1072924, 1142134, -802912, 313534, -108780, 254532, -558520, 675852, -491140, 336026, -358128, 473868, -853576, 1369462, -1379520; ...
%e in which the odd terms a(n*(n+1)) = A323679(n) form the left border.
%e RELATED SEQUENCES.
%e Terms a(n*(n+2)) = A323677(n) form a diagonal in the above triangle, starting with
%e [1, -2, 8, -36, 128, -288, 1166, -16048, 77328, -108780, 220440, -5900816, 44395366, -339891804, 898603106, -5623621248, 2160154604, ..., A323677(n), ...].
%e Terms a(n*(n+3)) = A323678(n) form a diagonal in the above triangle, starting with
%e [1, 2, 12, 50, 72, 142, 5346, 38338, 240902, 1369462, 8927272, 29594702, 78001922, 259042422, 2690290778, 26069217364, 144738683318, ..., A323678(n), ...].
%e RELATED SERIES.
%e Below we illustrate the following identity at specific values of x:
%e 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).
%e (1) At x = 1/2, the following sums are equal
%e S1 = Sum_{n>=0} 2^(n+1) * (2^n + 1)^n / (2^(n+1) + 1)^(n+1),
%e S1 = Sum_{n>=0} 2^(n+1) * (2^n - 1)^n / (2^(n+1) - 1)^(n+1) * (-1)^n,
%e where S1 = 1.694294601066597605831822294976249717707326205881024725908408...
%e (2) At x = 1/3, the following sums are equal
%e S2 = Sum_{n>=0} 3^(n+1) * (3^n + 1)^n / (3^(n+1) + 1)^(n+1),
%e S2 = Sum_{n>=0} 3^(n+1) * (3^n - 1)^n / (3^(n+1) - 1)^(n+1) * (-1)^n,
%e where S2 = 1.291258733393015321539496095851028631331196714523786660740336...
%e (3) At x = 2/3, the following sums are equal
%e S3 = Sum_{n>=0} 2^n * 3^(n+1) * (3^n + 2^n)^n / (3^(n+1) + 2^(n+1))^(n+1),
%e S3 = Sum_{n>=0} 2^n * 3^(n+1) * (3^n - 2^n)^n / (3^(n+1) - 2^(n+1))^(n+1) * (-1)^n,
%e where S3 = 2.523590984213154172284965025135287234251707014722123198796878...
%o (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)}
%o for(n=0, 120, print1(a(n), ", "))
%o (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)}
%o for(n=0, 120, print1(a(n), ", "))
%Y Cf. A323679 (odd terms), A323677 (a(n*(n+2))), A323678 (a(n*(n+3))).
%Y Cf. A323675 (variant), A325046 (variant), A326602 (variant).
%Y Cf. A326285.
%K sign
%O 0,3
%A _Paul D. Hanna_, Feb 03 2019
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