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A036015
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Number of partitions of n into parts not of form 4k+2, 8k, 8k+1 or 8k-1.
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4
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1, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 6, 7, 7, 8, 10, 12, 13, 14, 17, 21, 22, 24, 29, 33, 36, 40, 46, 53, 58, 63, 73, 83, 90, 99, 113, 127, 138, 152, 171, 191, 209, 228, 255, 285, 309, 338, 377, 416, 453, 495, 547, 603, 656, 714, 787, 865, 938, 1020, 1121, 1226
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OFFSET
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0,9
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COMMENTS
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Case k=2,i=1 of Gordon/Goellnitz/Andrews Theorem.
Also number of partitions in which no odd part is repeated, with no part of size less than or equal to 2 and where differences between adjacent parts are greater than 1 when the larger part is odd and greater than 2 when the larger part is even.
Euler transform of period 8 sequence [ 0, 0, 1, 1, 1, 0, 0, 0, ...]. - Michael Somos, Jun 28 2004
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REFERENCES
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G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 114.
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LINKS
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FORMULA
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Let qf(a, q) = Product(1-a*q^j, j=0..infinity); g.f. is 1/(qf(q^3, q^8)*qf(q^4, q^8)*qf(q^5, q^8)).
a(n) ~ sqrt(2-sqrt(2)) * exp(sqrt(n)*Pi/2) / (8*n^(3/4)). - Vaclav Kotesovec, Oct 04 2015
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EXAMPLE
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1 + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + 3*x^11 + 4*x^12 + ...
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MAPLE
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M:=100; qf:=(a, q)->mul(1-a*q^j, j=0..M); tS:=1/(qf(q^3, q^8)*qf(q^4, q^8)*qf(q^5, q^8)); series(%, q, M); seriestolist(%);
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ 1/ (QPochhammer[ q^3, q^8] QPochhammer[ q^4, q^8] QPochhammer[ q^5, q^8]), {q, 0, n}] (* Michael Somos, Jun 22 2012 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[q^(k^2 + 2 k) QPochhammer[ -q, q^2, k] / QPochhammer[ q^2, q^2, k], {k, 0, Sqrt[n + 1] - 1}], {q, 0, n}]] (* Michael Somos, Jun 22 2012 *)
nmax=60; CoefficientList[Series[Product[1/((1-x^(8*k-3))*(1-x^(8*k-4))*(1-x^(8*k-5))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 04 2015 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 - ([ 0, 0, 1, 1, 1, 0, 0, 0][(k-1)%8 + 1]) * x^k, 1 + x * O(x^n)), n))} /* Michael Somos, Jun 28 2004 */
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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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EXTENSIONS
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STATUS
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approved
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