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A035938
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Number of partitions in parts not of the form 7k, 7k+2 or 7k-2. Also number of partitions with 1 part of size 1 and differences between parts at distance 2 are greater than 1.
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0
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1, 1, 1, 2, 3, 3, 5, 6, 8, 10, 13, 16, 21, 25, 31, 38, 47, 56, 69, 82, 99, 118, 141, 166, 199, 233, 275, 322, 379, 440, 516, 598, 696, 805, 933, 1074, 1242, 1425, 1639, 1878, 2154, 2458, 2812, 3202, 3650, 4148, 4716, 5344, 6064, 6857, 7758, 8758, 9888
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Case k=3,i=2 of Gordon Theorem.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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REFERENCES
| G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of f(-x^7, -x^14)^2 / (f(-x, -x^6) * f(-x^3, -x^4)) in powers of x where f() is Rmanaujan's two variable theta function.
Euler transform of period 7 sequence [ 1, 0, 1, 1, 0, 1, 0, ...]. - Michael Somos, Feb 03 2012
G.f.: 1 / (Product_{k>0} (1 - x^(7*k - 6)) * (1 - x^(7*k - 4)) * (1 - x^(7*k - 3)) * (1 - x^(7*k - 1))). - Michael Somos, Feb 03 2012
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EXAMPLE
| 1 + x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 5*x^6 + 6*x^7 + 8*x^8 + 10*x^9 + ...
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MAPLE
| with (numtheory): a:= proc(n) option remember; `if`(n=0, 1, add (add (d*[0, 1, 0, 1, 1, 0, 1][1+irem(d, 7)], d=divisors(j)) *a(n-j), j=1..n)/n) end: seq (a(n), n=1..100); # Alois P. Heinz, Jan 22 2012
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PROG
| (Sage) # See A035937 for GordonsTheorem
def A035938_list(len) : return GordonsTheorem([1, 0, 1, 1, 0, 1, 0], len)
A035938_list(40) # Peter Luschny, Jan 22 2012
(PARI) {a(n) = local(A); if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 - [0, 1, 0, 1, 1, 0, 1][k%7 + 1] * x^k, 1 + x * O(x^n)), n))} /* Michael Somos, Feb 03 2012 */
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CROSSREFS
| Sequence in context: A008670 A193748 A039852 * A024503 A061790 A107236
Adjacent sequences: A035935 A035936 A035937 * A035939 A035940 A035941
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Olivier Gerard (olivier.gerard(AT)gmail.com)
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