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 A098884 Number of partitions of n into distinct parts in which each part is congruent to 1 or 5 mod 6. 10
 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 2, 1, 0, 1, 2, 3, 3, 2, 1, 1, 3, 5, 5, 3, 1, 2, 5, 7, 7, 5, 3, 3, 7, 11, 11, 7, 4, 6, 11, 15, 15, 11, 7, 8, 15, 22, 22, 15, 10, 13, 22, 30, 30, 23, 16, 18, 30, 42, 42, 31, 22, 27, 43, 56, 56, 44, 33, 37, 57, 77, 77, 59, 45, 53, 79, 101, 101, 82, 64, 71 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). Convolution of A281244 and A280456. - Vaclav Kotesovec, Jan 18 2017 LINKS Reinhard Zumkeller and Vaclav Kotesovec, Table of n, a(n) for n = 0..20000 (terms 0..300 from Reinhard Zumkeller) Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004. Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of chi(x) / chi(x^3) in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Sep 20 2013 Expansion of f(x^1, x^5) / f(-x^6) in powers of x where f(,) is a Ramanujan theta function. - Michael Somos, Sep 20 2013 Expansion of G(x^6) * H(-x) + x * G(-x) * H(x^6) where G() (A003114), H() (A003106) are Rogers-Ramanujan functions. Expansion of q^(-1/12) * eta(q^2)^2 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6)^2) in powers of q. Euler transform of period 12 sequence [ 1, -1, 0, 0, 1, 0, 1, 0, 0, -1, 1, 0, ...]. - Michael Somos, Jun 26 2005 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227398. - Michael Somos, Sep 20 2013 G.f.: Product_{k>0} (1 - (-x)^k + x^(2*k)). G.f.: 1 / Product_{k>0} (1 - x^(2*k - 1) + x^(4*k - 2)). G.f.: 1 / Product_{k>0} ((1 + x^(6*k - 3)) / (1 + x^(2*k - 1))). G.f.: Product_{k>0} ((1 + x^(6*k - 1)) * (1 + x^(6*k - 5))). G.f.: 1 / Product_{k>0} (1  + (-x)^(3*k - 1)) * (1 + (-x)^(3*k - 2)). G.f.: (Sum_{k in Z} x^(k * (3*k - 2))) / (Sum_{k in Z} (-1)^k * x^(3*k * (3*k-1))). A109389(n) = (-1)^n * a(n). Convolution inverse of A227398. a(n) ~ exp(sqrt(n)*Pi/3)/ (2*sqrt(6)*n^(3/4)) * (1 + (Pi/72 - 9/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Aug 30 2015, extended Jan 18 2017 EXAMPLE E.g. a(25)=5 because 25=19+5+1=17+7+1=13+7+5=13+11+1. G.f. = 1 + x + x^5 + x^6 + x^7 + x^8 + x^11 + 2*x^12 + 2*x^13 + x^14 + x^16 + ... G.f. = q + q^13 + q^61 + q^73 + q^85 + q^97 + q^133 + 2*q^145 + 2*q^157 + q^169 + ... MAPLE series(product((1+x^(6*k-1))*(1+x^(6*k-5)), k=1..100), x=0, 100); MATHEMATICA a[ n_] := SeriesCoefficient[ Product[ 1 - (-x)^k + x^(2 k), {k, n}], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *) a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - x^k + x^(2 k), {k, 1, n, 2}], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *) a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 1, n, 2}] / Product[ 1 + x^k, {k, 3, n, 6}], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *) a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 1, n, 6}] Product[ 1 + x^k, {k, 5, n, 6}], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *) a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^6] QPochhammer[ -x^5, x^6], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *) a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] / QPochhammer[ -x^3, x^6], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *) PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2), n))}; /* Michael Somos, Jun 26 2005 */ (PARI) {a(n) = my(A, m); if( n<0, 0, A = x * O(x^n); m = sqrtint(3*n + 1); polcoeff( sum(k= -((m-1)\3), (m+1)\3, x^(k * (3*k - 2)), A) / eta(x^6 + A), n))}; /* Michael Somos, Sep 20 2013 */ (Haskell) a098884 = p a007310_list where    p _  0     = 1    p (k:ks) m = if k > m then 0 else p ks (m - k) + p ks m -- Reinhard Zumkeller, Feb 19 2013 CROSSREFS Cf. A109389, A227398. Cf. A007310, A056970, A096981, A097451. Sequence in context: A180835 A053188 A109389 * A297964 A296239 A039800 Adjacent sequences:  A098881 A098882 A098883 * A098885 A098886 A098887 KEYWORD nonn AUTHOR Noureddine Chair, Oct 14 2004 EXTENSIONS Typo in Maple program fixed by Vaclav Kotesovec, Nov 15 2016 STATUS approved

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Last modified August 22 02:44 EDT 2019. Contains 326169 sequences. (Running on oeis4.)