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Search: id:A104504
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| A104504 |
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Coefficients of the D-Dyson mod 27 identity. |
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+0
4
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| 1, 1, 2, 2, 4, 5, 8, 10, 15, 19, 27, 34, 47, 59, 79, 99, 130, 162, 209, 259, 330, 407, 512, 628, 782, 955, 1179, 1432, 1755, 2122, 2583, 3109, 3762, 4510, 5427, 6480, 7760, 9231, 11004, 13043, 15485, 18293, 21634, 25475, 30021, 35245, 41396, 48459, 56740
(list; graph; listen)
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OFFSET
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0,3
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LINKS
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Eric Weisstein's World of Mathematics, Dyson Mod 27 Identities
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FORMULA
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Expansion of f(-q^3,-q^24)/f(-q,-q^2) in powers of q where f() is Ramanujan's theta function.
Given A=A0+A1+A2+A3+A4 is the 5-section, then 0= 2*A0^2*A1^2 +A2^2*A4^2 -A2*A0^3 -A4*A1^3 -A0*A1*A2*A4.
G.f.: Product_{k>0} (1-x^(27k))(1-x^(27k-3))(1-x^(27k-24))/(1-x^k).
G.f.: Sum_{k>0} x^(k^2+3k) ( Product_{j=1..k} 1-x^(3j) )/ ( (Product_{j=1..2k+2} (1-x^j)) (Product_{j=1..k}(1-x^j)) ).
A104501(n) = A104503(n-1) + A104504(n-2) unless n=0. - Michael Somos Sep 29 2007
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EXAMPLE
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1 + q + 2*q^2 + 2*q^3 + 4*q^4 + 5*q^5 + 8*q^6 + 10*q^7 + 15*q^8 + 19*q^9 + ...
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PROGRAM
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(PARI) {a(n)=local(m); if(n<0, 0, m=sqrtint(24*n+49); polcoeff( sum(k= -((m-7)\18), (m+7)\18, (-1)^k*x^((9*k^2-7*k)*3/2), x*O(x^n))/ eta(x+x*O(x^n)), n))} /* Michael Somos Mar 15 2006 */
(PARI) {a(n)=if(n<1, n==0, polcoeff( sum(k=0, sqrtint(n+1)-1, x^(k^2+3*k)* prod(j=1, k, (1-x^(3*j))/(1-x^j)/(1-x^(2*j+1))/(1-x^(2*j+2)), 1+O(x^(n-k^2-2*k+1)))/(1-x)/(1-x^2) ), n))} /* Michael Somos Mar 15 2006 */
(PARI) {a(n) = local(A); if( n<0, 0, n+=2; A = eta(x + x*O(x^n)) ; polcoeff( - sum(k=0, n, (k%3==2) * polcoeff(A, k) * x^k) / A, n))} /* Michael Somos Sep 29 2007 */
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CROSSREFS
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Cf. A104501, A104502, A104503.
Sequence in context: A035981 A035991 A036002 this_sequence A027337 A027193 A126796
Adjacent sequences: A104501 A104502 A104503 this_sequence A104505 A104506 A104507
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KEYWORD
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nonn
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AUTHOR
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Eric Weisstein (eric(AT)weisstein.com), Mar 11, 2005
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EXTENSIONS
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Edited by Michael Somos, Mar 15 2006
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