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 A073252 Coefficients of replicable function number "48g". 7
 1, 2, 1, 2, 4, 4, 5, 6, 9, 12, 13, 16, 21, 26, 29, 36, 46, 54, 62, 74, 90, 106, 122, 142, 171, 200, 227, 264, 311, 358, 408, 470, 545, 626, 709, 810, 933, 1062, 1198, 1362, 1555, 1760, 1980, 2238, 2536, 2858, 3205, 3602, 4063, 4560, 5092, 5704, 6400, 7150, 7966 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Old name was: McKay-Thompson series of class 48g for the Monster group. Number of partitions of n into distinct odd parts of two kinds. [Joerg Arndt, Jul 30 2011] Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). Combinatorial interpretation of sequence: [ X1,X2 ] = 2 strictly increasing sequences (possibly null) of odd positive integers; a(n)=#pairs with sum of entries = n. REFERENCES T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, q_2^2. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..1000 D. Foata and G.-N. Han, Jacobi and Watson Identities Combinatorially Revisited D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994). Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA G.f.: 1 / (Prod_{k>0} 1 + (-x)^k)^2 = (Prod_{k>0} 1 + x^(2*k - 1))^2. Expansion of q^(1/12) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^2 in powers of q. Expansion of chi(q)^2 = phi(q) / f(-q^2) = f(q) / psi(-q) = (phi(q) / f(q))^2 = (psi(q) / f(-q^4))^2 = (f(-q^2) / psi(-q))^2 = (phi(-q^2) / f(-q))^2 = (f(q) / f(-q^2))^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. Euler transform of period 4 sequence [2, -2, 2, 0, ...]. Convolution square of A000700. A022597(n) = (-1)^n * a(n). a(n) ~ exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015 G.f.: exp(2*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 07 2018 a(2*n) = A226622(n). a(2*n + 1) = 2 * A226635(n). - Michael Somos, Nov 03 2019 EXAMPLE a(4)=4:[ (1),(3) ],[ (3),(1) ],[ (),(1,3) ],[ (1,3),() ] G.f. = 1 + 2*x + x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 5*x^6 + 6*x^7 + 9*x^8 + 12*x^9 + ... G.f. = 1/q + 2*q^11 + q^23 + 2*q^35 + 4*q^47 + 4*q^59 + 5*q^71 + 6*q^83 +... MATHEMATICA nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^2, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *) QP = QPochhammer; s = (QP[q^2]^2 / (QP[q] * QP[q^4]))^2 + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *) a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^2, {x, 0, n}]; (* Michael Somos, Nov 03 2019 *) PROG (PARI) {a(n) = if( n<0, 0, polcoeff( prod( i=1, (1+n)\2, 1 + x^(2*i - 1), 1 + x * O(x^n))^2, n))}; (PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod( i=1, n, 1 + (-x)^i, 1 + x * O(x^n))^2, n))}; (PARI) {a(n) = my(A); if( n<0, 0, A = x*O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A) / eta(x^4 + A))^2, n))}; CROSSREFS Cf. A000700, A022597. Sequence in context: A023673 A132965 A022597 * A134005 A132320 A076369 Adjacent sequences:  A073249 A073250 A073251 * A073253 A073254 A073255 KEYWORD nonn,easy AUTHOR Michael Somos, Jul 22 2002 EXTENSIONS Comments from Len Smiley. New name from Michael Somos, Nov 03 2019 STATUS approved

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Last modified May 18 10:19 EDT 2021. Contains 343995 sequences. (Running on oeis4.)