A080995 Characteristic function of generalized pentagonal numbers A001318.

1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0

Offset

0,1

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Repeatedly [1,[0,]^2k,1,[0,]^k], k>=0; characteristic function of generalized pentagonal numbers: a(A001318(n))=1, a(A090864(n))=0. - Reinhard Zumkeller, Apr 22 2006

Starting with offset 1 with 1's signed (++--++,...), i.e., (1, 1, 0, 0, -1, 0, -1, 0, ...); is the INVERTi transform of A000041 starting (1, 2, 3, 5, 7, 11, ...). - Gary W. Adamson, May 17 2013

Number 9 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

References

Percy A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p. 81, Article 331.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1001 from T. D. Noe)

S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math. 274 (2004), no. 1-3, 9-24. See P(q).

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Jacobi Theta Functions.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Don Zagier, Elliptic modular forms and their applications, in "The 1-2-3 of modular forms", Springer-Verlag, 2008.

Index entries for characteristic functions.

Formula

Expansion of phi(-x^3) / chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions. - Michael Somos, Sep 14 2007

Expansion of psi(x) - x * psi(x^9) in powers of x^3 where psi() is a Ramanujan theta function. - Michael Somos, Sep 14 2007

Expansion of f(x, x^2) in powers of x where f() is Ramanujan's two-variable theta function.

Expansion of q^(-1/24) * eta(q^2) * eta(q^3)^2 / (eta(q) * eta(q^6)) in powers of q.

a(n) = b(24*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p>3. - Michael Somos, Jun 06 2005

Euler transform of period 6 sequence [ 1, 0, -1, 0, 1, -1, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 2^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A089810.

G.f.: Product_{k>0} (1 - x^(3*k)) / (1 - x^k + x^(2*k)). - Michael Somos, Jan 26 2008

G.f.: Sum x^(n*(3n+1)/2), n=-inf..inf [the exponents are the pentagonal numbers, A000326].

a(n) = |A010815(n)| = A089806(2*n) = A033683(24*n + 1).

For n > 0, a(n) = b(n) - b(n-1) + c(n) - c(n-1), where b(n) = floor(sqrt(2n/3+1/36)+1/6) (= A180447(n)) and c(n) = floor(sqrt(2n/3+1/36)-1/6) (= A085141(n)). - Mikael Aaltonen, Mar 08 2015

a(n) = (-1)^n * A133985(n). - Michael Somos, Jul 12 2015

a(n) = A000009(n) (mod 2). - John M. Campbell, Jun 29 2016

Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 2*sqrt(2/3) = 1.632993... . - Amiram Eldar, Jan 13 2024

Example

G.f. = 1 + x + x^2 + x^5 + x^7 + x^12 + x^15 + x^22 + x^26 + x^35 + x^40 + x^51 + ...
G.f. = q + q^25 + q^49 + q^121 + q^169 + q^289 + q^361 + q^529 + q^625 + ...

Mathematica

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ (Series[ EllipticTheta[ 3, Log[y] / (2 I), x^(3/2)], {x, 0, n + Floor@Sqrt[n]}] // Normal // TrigToExp) /. {y -> x^(1/2)}, {x, 0, n}]]; (* Michael Somos, Nov 18 2011 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] / QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Jun 08 2013 *)
a[ n_] := If[ n < 0, 0, Boole[ IntegerQ[ Sqrt[ 24 n + 1]]]]; (* Michael Somos, Jun 08 2013 *)

Prog

(PARI) {a(n) = if( n<0, 0, abs( polcoeff( eta(x + x * O(x^n)), n)))};
(PARI) {a(n) = issquare( 24*n + 1)}; /* Michael Somos, Apr 13 2005 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 / (eta(x + A) * eta(x^6 + A)), n))};
(Haskell)
a080995 = a033683 . (+ 1) . (* 24)  -- Reinhard Zumkeller, Nov 14 2015

Crossrefs

Cf. A001318 (support), A010815 (absolute values), A033683, A089806.

Cf. A000326, A180447, A085141, A133985.

Cf. A000009, A000041, A000122, A000700, A010054, A090864, A121373.

Sequence in context: A010815 A206958 A206959 * A121373 A199918 A229894

Adjacent sequences: A080992 A080993 A080994 * A080996 A080997 A080998

Keyword

nonn,easy

Author

Michael Somos, Feb 27 2003

Extensions

Minor edits by N. J. A. Sloane, Feb 03 2012

Status

approved