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%I M1825 N0725 #32 Dec 20 2021 20:25:28
%S 1,1,2,8,10,24,53,74,153,280,436,793,1322,2085,3510,5648,8796,14042,
%T 21921,33490,51796,78843,118108,178029,265225,390852,576946,843694,
%U 1224329,1775450,2556360,3658111,5224159,7418887,10481780,14773012,20723154,28941023
%N Expansion of a modular function for Gamma_0(15).
%D Newman, Morris; Construction and application of a class of modular functions. II. Proc. London Math. Soc. (3) 9 1959 373-387.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A002510/b002510.txt">Table of n, a(n) for n = 6..1000</a>
%H Morris Newman, <a href="/A002507/a002507.pdf">Construction and application of a class of modular functions, II</a>, Proc. London Math. Soc. (3) 9 1959 373-387. [Annotated scanned copy, barely legible]
%F Expansion of eta(q^15)^13 / (eta(q) * eta(q^3)^5 * eta(q^5)^7) in powers of q.
%F Expansion of (c(q^5)^2 / (3 * c(q)))^2 / (b(q) * b(q^5)) in powers of q where b(), c() are cubic AGM theta functions. - _Michael Somos_, Jun 10 2012
%F Euler transform of period 15 sequence [1, 1, 6, 1, 8, 6, 1, 1, 6, 8, 1, 6, 1, 1, 0, ...]. - _Michael Somos_, Nov 10 2005
%F a(n) ~ exp(4*Pi*sqrt(2*n/15)) / (2^(1/4) * 3^(17/4) * 5^(13/4) * n^(3/4)). - _Vaclav Kotesovec_, Apr 09 2018
%e q^6 + q^7 + 2*q^8 + 8*q^9 + 10*q^10 + 24*q^11 + 53*q^12 + 74*q^13 + 153*q^14 + ...
%p 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-> [1,1,6,1,8, 6,1,1,6,8, 1,6,1,1,0] [modp(n-1,15)+1]): a:=n-> aa(n-6): seq(a(n), n=6..42); # _Alois P. Heinz_, Sep 08 2008
%t etr[p_] := Module[{b}, b[n_] := b[n] = Module[{d, j}, If[n == 0, 1, Sum [Sum [d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]]; b]; aa = etr[ Function[n, {1, 1, 6, 1, 8, 6, 1, 1, 6, 8, 1, 6, 1, 1, 0}[[Mod[n-1, 15] + 1]]]]; a[n_] := aa[n-6]; Table[a[n], {n, 6, 41}] (* _Jean-François Alcover_, Mar 03 2014, after _Alois P. Heinz_ *)
%o (PARI) {a(n) = local(A); if( n<6, 0, n -= 6; A = x * O(x^n); polcoeff( eta(x^15 + A)^13 / (eta(x + A) * eta(x^3 + A)^5 * eta(x^5 + A)^7), n))} /* _Michael Somos_, Nov 10 2005 */
%K nonn,easy
%O 6,3
%A _N. J. A. Sloane_
%E More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jan 14 2001
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