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Expansion of Product_{i>=1, j>=0} (1 + x^(i * 7^j)).
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6
1, 1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 14, 18, 22, 28, 34, 41, 50, 60, 72, 86, 105, 124, 146, 174, 204, 240, 282, 332, 386, 450, 524, 606, 703, 812, 940, 1082, 1243, 1428, 1636, 1873, 2140, 2448, 2788, 3172, 3610, 4096, 4646, 5264, 5963, 6737, 7607, 8584, 9668, 10887, 12244, 13765, 15451, 17328
COMMENTS
This sequence is different from A093950. The first difference occurs at a(50) = 6737, whereas A093950(50) = 6736.
FORMULA
G.f.: Product_{k>=1} (1 + x^k)^ A373217(k). Let A(x) be the g.f. of this sequence, and B(x) be the g.f. of A000009, then B(x) = A(x)/A(x^7).
PROG
(PARI) my(N=60, x='x+O('x^N)); Vec(prod(k=1, N, (1+x^k)^(valuation(k, 7)+1)))
McKay-Thompson series of class 42C for the Monster group.
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5
1, -1, 0, -1, 1, -1, 1, -2, 3, -2, 3, -3, 4, -4, 4, -6, 7, -7, 7, -9, 10, -12, 13, -14, 17, -18, 19, -22, 26, -28, 29, -34, 38, -41, 44, -50, 57, -60, 65, -72, 81, -86, 94, -105, 114, -124, 133, -146, 161, -174, 187, -204, 224, -240, 258, -282, 309, -332, 354, -386, 419, -450, 481, -524, 569, -606, 651, -703
COMMENTS
Given g.f. A(x), the second term of the left side of Cayley's identity is -A(q). - Michael Somos, Dec 03 2013
REFERENCES
A. Cayley, An elliptic-transcendant identity, Messenger of Math., 2 (1873), p. 179.
FORMULA
Expansion of chi(-x) * chi(-x^7) in powers of x where chi() is a Ramanujan theta function.
Expansion of q^(1/3) * eta(q) * eta(q^7) / (eta(q^2) * eta(q^14)) in powers of q.
Euler transform of period 14 sequence [ -1, 0, -1, 0, -1, 0, -2, 0, -1, 0, -1, 0, -1, 0, ...].
Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^2 - u^2*v - 2*u.
G.f. is a period 1 Fourier series which satisfies f(-1 / (126 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A093950. G.f.: 1 / (Product_{k>0} (1 + x^k) * (1 + x^(7*k))).
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/21)) / (2 * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017
EXAMPLE
G.f. = 1 - x - x^3 + x^4 - x^5 + x^6 - 2*x^7 + 3*x^8 - 2*x^9 + 3*x^10 - 3*x^11 + ...
T42C = 1/q - q^2 - q^8 + q^11 - q^14 + q^17 - 2*q^20 + 3*q^23 - 2*q^26 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ x^7, x^14], {x, 0, n}]; (* Michael Somos, Aug 06 2011 *) a[ n_] := SeriesCoefficient[ 1 / ( Product[ 1 + x^k, {k, n}] Product[ 1 + x^k, {k, 7, n, 7}] ), {x, 0, n}]; (* Michael Somos, Aug 06 2011 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^7 + A) / (eta(x^2 + A) * eta(x^14 + A)), n))};
McKay-Thompson series of class 84C for the Monster group.
+10
4
1, 1, 0, 1, 1, 1, 1, 2, 3, 2, 3, 3, 4, 4, 4, 6, 7, 7, 7, 9, 10, 12, 13, 14, 17, 18, 19, 22, 26, 28, 29, 34, 38, 41, 44, 50, 57, 60, 65, 72, 81, 86, 94, 105, 114, 124, 133, 146, 161, 174, 187, 204, 224, 240, 258, 282, 309, 332, 354, 386, 419, 450, 481, 524, 569, 606
COMMENTS
Given g.f. A(x), the first term of the left side of Cayley's identity is A(q). - Michael Somos, Dec 03 2013
REFERENCES
A. Cayley, An elliptic-transcendant identity, Messenger of Math., 2 (1873), p. 179.
FORMULA
Expansion of q^(1/3) * eta(q^2)^2 * eta(q^14)^2 / (eta(q) * eta(q^4) * eta(q^7) * eta(q^28)) in powers of q. - Michael Somos, Dec 03 2013 Euler transform of period 28 sequence [1, -1, 1, 0, 1, -1, 2, 0, 1, -1, 1, 0, 1, -2, 1, 0, 1, -1, 1, 0, 2, -1, 1, 0, 1, -1, 1, 0, ...]. - Michael Somos, Dec 03 2013 G.f. is a period 1 Fourier series which satisfies f(-1 / (28 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 03 2013 G.f.: Product_{k>0} (1 + x^(2*k - 1)) * (1 + x^(14*k - 7)). - Michael Somos, Dec 03 2013 a(n) ~ exp(2*Pi*sqrt(n/21)) / (2 * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 06 2015
EXAMPLE
G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + 2*x^7 + 3*x^8 + 2*x^9 + 3*x^10 + ...
T84C = 1/q + q^2 + q^8 + q^11 + q^14 + q^17 + 2*q^20 + 3*q^23 + 2*q^26 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^7, x^14], {x, 0, n}]; (* Michael Somos, Dec 03 2013 *) a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 1, n, 2}] Product[ 1 + x^k, {k, 7, n, 14}], {x, 0, n}]; (* Michael Somos, Dec 03 2013 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^14 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^7 + A) * eta(x^28 + A)), n))}; /* Michael Somos, Dec 03 2013 */
Expansion of 1 / (chi(x) * chi(x^7)) in powers of x where chi() is a Ramanujan theta function.
+10
2
1, -1, 1, -2, 2, -3, 4, -6, 7, -9, 12, -14, 18, -22, 28, -34, 41, -50, 60, -72, 86, -105, 124, -146, 174, -204, 240, -282, 332, -386, 450, -524, 606, -703, 812, -940, 1082, -1243, 1428, -1636, 1873, -2140, 2448, -2788, 3172, -3610, 4096, -4646, 5264, -5962
FORMULA
Expansion of q^(-1/3) * eta(q) * eta(q^4) * eta(q^7) * eta(q^28) / (eta(q^2) * eta(q^14))^2 in powers of q.
Euler transform of period 28 sequence [ -1, 1, -1, 0, -1, 1, -2, 0, -1, 1, -1, 0, -1, 2, -1, 0, -1, 1, -1, 0, -2, 1, -1, 0, -1, 1, -1, 0, ...].
Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u - v^2) * (v - u^2) - 2 * (u*v)^2 * (1 - u*v)^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (252 t)) = f(t) where q = exp(2 Pi i t).
G.f.: Product_{k>0} (1 + (-x)^k) * (1 + (-x)^(7*k)).
EXAMPLE
G.f. = 1 - x + x^2 - 2*x^3 + 2*x^4 - 3*x^5 + 4*x^6 - 6*x^7 + 7*x^8 - 9*x^9 + ...
G.f. = q - q^4 + q^7 - 2*q^10 + 2*q^13 - 3*q^16 + 4*q^19 - 6*q^22 + 7*q^25 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ Product[ 1 + (-x)^k, {k, n}] Product[ 1 + (-x)^k, {k, 7, n, 7}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x] QPochhammer[ x^7, -x^7], {x, 0, n}];
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-1/3)* eta[q]*eta[q^4]*eta[q^7]*eta[q^28]/(eta[q^2]*eta[q^14])^2, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jul 04 2018 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 + (-x)^k, 1 + x * O(x^n)) * prod( k=1, n\7, 1 + (-x)^(7*k), 1 + x * O(x^n)), n))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^7 + A) * eta(x^28 + A) / (eta(x^2 + A) * eta(x^14 + A))^2, n))};
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