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Search: id:A005798
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| A005798 |
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eta(z/2)^8*eta(2z)^16/eta(z)^24, where eta = Dedekind's function.
(Formerly M4528)
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+0
6
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| 0, 1, -8, 44, -192, 718, -2400, 7352, -20992, 56549, -145008, 356388, -844032, 1934534, -4306368, 9337704, -19771392, 40965362, -83207976, 165944732, -325393024, 628092832, -1194744096, 2241688744, -4152367104, 7599231223, -13749863984
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Expansion of elliptic lambda/16 = m/16 = (k/4)^2 in powers of the nome q.
Euler transform of period 4 sequence [ -8,16,-8,0,...].
G.f. A(x) satisfies 0=f(A(x),A(x^2)) where f(u,v)=u^2-v+16uv-32u^2v+256(uv)^2. - Michael Somos Mar 19 2004
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 121.
A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 23.
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
Eric Weisstein's World of Mathematics, Elliptic Lambda Function a section of The World of Mathematics.
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FORMULA
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G.f.: q* Product( (1+q^(2n))/(1+q^(2n-1)), n=1..inf )^8 = eta(q)^8*eta(q^4)^16/eta(q^2)^24; eta = Dedekind's function.
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MAPLE
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with (numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add (add (d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: aa:=etr (n-> [ -8, 16, -8, 0] [modp(n-1, 4)+1]): a:= n->aa(n-1): seq (a(n), n=0..26); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 08 2008]
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PROGRAM
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(PARI) a(n)=local(A, m); if(n<1, 0, m=1; A=x+O(x^2); while(m<n, m*=2; A=sqrt(subst(A, x, x^2)); A=A/(1+4*A)^2); polcoeff(A, n))
(PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( (eta(x+A)*eta(x^4+A)^2/eta(x^2+A)^3)^8, n))} /* Michael Somos Jul 16 2005 */
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CROSSREFS
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If initial 0 is omitted and sequence begins with a(0) = 1, then this is the convolution of A001938 with itself. G.f.s are related by A005798(x)=x*A001938(x)^2. Reversion of A005797. Cf. A007248, A029845.
Sequence in context: A000938 A165618 A059596 this_sequence A092877 A023007 A073380
Adjacent sequences: A005795 A005796 A005797 this_sequence A005799 A005800 A005801
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KEYWORD
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sign,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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