Search: a138745 -id:a138745
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A125079
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Excess of number of divisors of 2n+1 of form 12k+1, 12k+5 over those of form 12k+7, 12k+11.
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+10
11
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1, 1, 2, 0, 1, 0, 2, 2, 2, 0, 0, 0, 3, 1, 2, 0, 0, 0, 2, 2, 2, 0, 2, 0, 1, 2, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2, 3, 0, 0, 1, 0, 4, 2, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 2, 2, 0, 2, 0, 1, 2, 4, 0, 0, 0, 0, 2, 2, 0, 0, 0, 4, 1, 2, 0, 2, 0, 2, 2, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 2, 4, 0, 0, 0, 2, 4, 2, 0, 0, 0, 4, 0, 0
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OFFSET
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0,3
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COMMENTS
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REFERENCES
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Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 82, Eq. (32.56).
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LINKS
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FORMULA
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Expansion of q^(-1/2) * eta(q^3)^3 *eta(q^4) * eta(q^12) / (eta(q) * eta(q^6)^2) in powers of q.
Expansion of phi(-q^3) * psi(-q^3) / (chi(-q) * chi(-q^2)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Euler transform of period 12 sequence [ 1, 1, -2, 0, 1, 0, 1, 0, -2, 1, 1, -2, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 4), b(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A138745.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Dec 28 2023
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EXAMPLE
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1 + x + 2*x^2 + x^4 + 2*x^6 + 2*x^7 + 2*x^8 + 3*x^12 + x^13 + 2*x^14 + ...
q + q^3 + 2*q^5 + q^9 + 2*q^13 + 2*q^15 + 2*q^17 + 3*q^25 + q^27 + 2*q^29 + ...
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, With[{m = 2 n + 1}, Sum[ KroneckerSymbol[ -36, d], { d, Divisors[ m]}]]]
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PROG
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(PARI) {a(n) = if( n<0, 0, n = 2*n+1; sumdiv( n, d, kronecker( -36, d)))}
(PARI) {a(n) = if( n<0, 0, n = 2*n+1; sumdiv( n, d, kronecker( 6, d) * (-1)^(d\12)))}
(PARI) {a(n) = if( n<0, 0, if( n%6==1, n\=3, 1); sumdiv( 2*n + 1, d, kronecker( -4, d)) )}
(PARI) {a(n) = local(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, if( p==3, 1, if( p%4==1, e+1, !(e%2)))))))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^4 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^6 + A)^2), n))}
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A125061
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Expansion of psi(q) * psi(q^2) * chi(q^3) * chi(-q^6) in powers of q where psi(), chi() are Ramanujan theta functions.
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+10
9
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1, 1, 1, 3, 1, 2, 3, 0, 1, 1, 2, 0, 3, 2, 0, 6, 1, 2, 1, 0, 2, 0, 0, 0, 3, 3, 2, 3, 0, 2, 6, 0, 1, 0, 2, 0, 1, 2, 0, 6, 2, 2, 0, 0, 0, 2, 0, 0, 3, 1, 3, 6, 2, 2, 3, 0, 0, 0, 2, 0, 6, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 1, 2, 2, 9, 0, 0, 6, 0, 2, 1, 2, 0, 0, 4, 0, 6, 0, 2, 2, 0, 0, 0, 0, 0, 3, 2, 1, 0, 3, 2, 6, 0, 2
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OFFSET
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0,4
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COMMENTS
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REFERENCES
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Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 82, Eq. (32.53).
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LINKS
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FORMULA
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Expansion of eta(q^2) * eta(q^4)^2 * eta(q^6)^3 / (eta(q) * eta(q^3) * eta(q^12)^2) in powers of q.
Expansion of (theta_3(q)^2 + 3*theta_3(q^3)^2) / 4 in powers of q.
Euler transform of period 12 sequence [ 1, 0, 2, -2, 1, -2, 1, -2, 2, 0, 1, -2, ...].
Moebius transform is period 12 sequence [ 1, 0, 2, 0, 1, 0, -1, 0, -2, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 2-(-1)^e, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) == (1-(-1)^e)/2 if p == 3 (mod 4).
G.f.: 1 + Sum_{k>0} (x^k + x^(3*k)) / (1 - x^(2*k) + x^(4*k)).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3 (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A122857.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - Amiram Eldar, Nov 24 2023
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EXAMPLE
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G.f. = 1 + q + q^2 + 3*q^3 + q^4 + 2*q^5 + 3*q^6 + q^8 + q^9 + 2*q^10 + 3*q^12 + ...
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MATHEMATICA
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s = (EllipticTheta[3, 0, q]^2 + 3*EllipticTheta[3, 0, q^3]^2)/4 + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Dec 07 2015, from 2nd formula *)
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PROG
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(PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, ((d%2) * ((d%3==0)+1)) * (-1)^(d\6)))};
(PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); prod( k=1, matsize(A)[1],
[p, e] = A[k, ]; if( p==2, 1, p==3, 1+e%2*2, p%4==1, e+1, !(e%2) )))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^2 * eta(x^6 + A)^3 / (eta(x + A) * eta(x^3 + A) * eta(x^12 + A)^2), n))};
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CROSSREFS
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Cf. A002175, A008441, A019669, A053692, A113407, A121444, A122856, A122857, A122865, A125061, A138741, A138745, A138746, A138949, A138950, A138951, A138952, A163746.
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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A138746
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Expansion of 1 - eta(q) * eta(q^3) * eta(q^4)^3 / (eta(q^2)^2 * eta(q^12)) in powers of q.
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+10
4
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1, -1, 3, -1, 2, -3, 0, -1, 1, -2, 0, -3, 2, 0, 6, -1, 2, -1, 0, -2, 0, 0, 0, -3, 3, -2, 3, 0, 2, -6, 0, -1, 0, -2, 0, -1, 2, 0, 6, -2, 2, 0, 0, 0, 2, 0, 0, -3, 1, -3, 6, -2, 2, -3, 0, 0, 0, -2, 0, -6, 2, 0, 0, -1, 4, 0, 0, -2, 0, 0, 0, -1, 2, -2, 9, 0, 0, -6
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OFFSET
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1,3
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COMMENTS
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LINKS
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FORMULA
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Expansion of 1 - psi(-q) * psi(q^2) * chi(-q^3) * chi(-q^6) in powers of q where psi(), chi() are Ramanujan theta functions.
Moebius transform is period 24 sequence [ 1, -2, 2, 0, 1, -4, -1, 0, -2, -2, -1, 0, 1, 2, 2, 0, 1, 4, -1, 0, -2, 2, -1, 0, ...].
a(n) is multiplicative with a(2^e) = -1 if e>0, a(3^e) = 2 - (-1)^e, a(p^e) = e+1 if p == 1, 5 (mod 12), a(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12).
G.f.: Sum_{k>0} -(-1)^k * ( f(6*k - 1) + 2 * f(6*k - 3) + f(6*k - 5) ) where f(k) := x^k / (1 + x^k).
a(12*n + 7) = a(12*n + 11) = 0.
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EXAMPLE
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G.f = q - q^2 + 3*q^3 - q^4 + 2*q^5 - 3*q^6 - q^8 + q^9 - 2*q^10 - 3*q^12 + ...
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MATHEMATICA
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a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, (-1)^Quotient[#, 6] {1, 0, 2, 0, 1, 0}[[Mod[#, 6, 1]]] &]]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 3, -(-1)^#, # == 3, Mod[#2, 2] 2 + 1, Mod[#, 4] == 1, #2 + 1, True, 1 - Mod[#2, 2]] & @@@ FactorInteger @ n)]; (* Michael Somos, Sep 08 2015 *)
QP = QPochhammer; s = (1/q)*(1-QP[q]*QP[q^3]*(QP[q^4]^3/(QP[q^2]^2*QP[q^12] ))) + O[q]^80; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, ((d%2) * ((d%3==0) + 1)) * (-1)^(d\6)))};
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -1, p==3, 2 - (-1)^e, p%12<6, e+1, 1-e%2)))};
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CROSSREFS
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KEYWORD
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sign,mult
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AUTHOR
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STATUS
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approved
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