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A328786
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a(n) = (Sum_{k=0..n-1}(-1)^k*(39480*k+7321)*29700^(n-1-k)*T_k(14,1)*T_k(11,-11)^2)/n, where T_k(b,c) denotes the coefficient of x^k in the expansion of (x^2+b*x+c)^k.
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3
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7321, 69076403, 1423525024746, 31676475535509475, 752633551945067097470, 18627509719518121995003486, 474204125641606160260805604468, 12323377272130975561953028453412931, 325337163371714764552775702345400136950, 8696262375383068237957170325229215635055690
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OFFSET
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1,1
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COMMENTS
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Conjecture: (i) a(n) is a positive integer for each n > 0, and a(n) is odd if and only if n is a power of two. Moreover, we have the identity Sum_{k>=0}(39480*k+7321)(-29700)^(-k)*T_k(14,1)*T_k(11,-11)^2 = 6795*sqrt(5)/Pi.
(ii) For any prime p > 5, we have the congruence Sum_{k=0..p-1}(39480*k+7321)(-29700)^(-k)*T_k(14,1)*T_k(11,-11)^2 == p*(1513 + 70*Leg(3/p) + 5738*Leg(-5/p)) (mod p^2), where Leg(a/p) denotes the Legendre symbol.
(iii) Let p > 5 be a prime different from 11, and set S(p) = Sum_{k=0..p-1} T_k(14,1)*T_k(11,-11)^2/(-29700)^k. If Leg(-165/p) = -1, then S(p) == 0 (mod p^2). If Leg(-1/p) = Leg(p/3) = Leg(p/5) = Leg(p/11) = 1 and p = x^2 + 165*y^2 (with x and y integers), then S(p) == 4*x^2-2p (mod p^2). If Leg(-1/p) = Leg(p/3) = Leg(p/5) = Leg(p/11) = -1 and 2p = x^2 + 165*y^2, then S(p) == 2*x^2-2p (mod p^2). If Leg(-1/p) = Leg(p/5) = -1, Leg(p/3) = Leg(p/11) = 1, and p = 3*x^2 + 55*y^2, then S(p) == 12*x^2-2p (mod p^2). If Leg(-1/p) = Leg(p/5) = 1, Leg(p/3) = Leg(p/11) = -1, and 2p = 3*x^2 + 55*y^2, then S(p) == 6*x^2-2p (mod p^2). If Leg(-1/p) = Leg(p/11) = 1, Leg(p/3) = Leg(p/5) = -1, and p = 5*x^2 + 33*y^2, then S(p) == 2p-20*x^2 (mod p^2). If Leg(-1/p) = Leg(p/11) = -1, Leg(p/3) = Leg(p/5) = 1, and 2p = 5*x^2 + 33*y^2, then S(p) == 2p-10*x^2 (mod p^2). If Leg(-1/p) = Leg(p/3) = -1, Leg(p/5) = Leg(p/11) = 1, and p = 11*x^2 + 15*y^2, then S(p) == 44*x^2-2p (mod p^2). If Leg(-1/p) = Leg(p/3) = 1, Leg(p/5) = Leg(p/11) = -1, and 2p = 11*x^2 + 15*y^2, then S(p) == 22*x^2-2p (mod p^2).
In 2011 the author formulated many conjectural series for 1/Pi involving T_k(b,c). The conjectural series for 1/Pi in A329073, A329105, A329107 and the present one are of a new type which might be called type VIII to distinguish from the 7 known types (types I-VII).
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LINKS
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Zhi-Wei Sun, On sums related to central binomial and trinomial coefficients, in: M. B. Nathanson (ed.), Combinatorial and Additive Number Theory: CANT 2011 and 2012, Springer Proc. in Math. & Stat., Vol. 101, Springer, New York, 2014, pp. 257-312. Also available from arXiv:1101.0600 [math.NT], 2011-2014.
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EXAMPLE
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a(1) = 7321 since ((-1)^0*(39480*0+7321)*29700^(1-1-0)*T_0(14,1)*T_0(11,-11)^2)/1 = 7321.
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MATHEMATICA
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T[b_, c_, 0]=1; T[b_, c_, 1]=b;
T[b_, c_, n_]:=T[b, c, n]=(b(2n-1)T[b, c, n-1]-(b^2-4c)(n-1)T[b, c, n-2])/n;
a[n_]:=a[n]=Sum[(39480k+7321)T[14, 1, k]T[11, -11, k]^2*(-1)^k*29700^(n-1-k), {k, 0, n-1}]/n;
Table[a[n], {n, 1, 10}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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