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A003406 Expansion of Ramanujan's function R(x) = 1 + Sum_{n >= 1} { x^(n*(n+1)/2) / ((1+x)(1+x^2)(1+x^3)...(1+x^n)) }.
(Formerly M0206)
11
1, 1, -1, 2, -2, 1, 0, 1, -2, 0, 2, 0, -1, -2, 2, 1, 0, -2, 2, -2, 0, 0, 3, 0, -2, -2, 1, 0, 2, 0, 0, 0, -2, 0, 0, 1, 0, 0, 0, 2, -1, 0, -2, -2, 0, 4, 0, 2, -2, 0, -2, -1, 2, 0, -2, 2, 0, 1, 0, 0, 0, 0, -2, 0, 0, 0, 0, -2, 4, 2, -1, 0, 0, -2, -2, -2, 2, 1, 2, 0, 0, 0, 0, -2, 2, 0, 0, -2, 2, -2, -2, 0, 3, 0, 0, 2, 0, 0, 0, -2, 1, -2, 0, -2, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

a(n) = A117192(n) - A117193(n) for n>0 (number of partitions into distinct parts with even rank minus those with odd rank); see also A000025. - Reinhard Zumkeller, Mar 03 2006

Ramanujan showed that R(x) = 2*Sum_{n>=0} (S(x) - P(n,x)) - 2*S(x)*D(x), where P(n,x) = Product_{k=1..n} (1+x^k), S(x) = g.f. A000009 = P(oo,x) and D(x) = -1/2 + Sum_{n>=1} x^n/(1-x^n) = -1/2 + g.f. A000005. - Michael Somos

REFERENCES

G. E. Andrews, Ramanujan's "lost" notebook V: Euler's partition identity, Adv. in Math. 61 (1986), no. 2, 156-164; Math. Rev. 87i:11137. [ The expansion in (2.8) is incorrect. ]

F. J. Dyson, A walk through Ramanujan's garden, pp. 7-28 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.

F. J. Dyson, Selected Papers, Am. Math. Soc., 1996, p. 200.

B. Gordon and D. Sinor, Multiplicative properties of eta-products, Number theory, Madras 1987, pp. 173-200, Lecture Notes in Math., 1395, Springer, Berlin, 1989. see page 182. MR1019331 (90k:11050)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2000 from T. D. Noe)

G. E. Andrews, Questions and conjectures in partition theory, Amer. Math. Monthly, 93 (1986), 708-711.

G. E. Andrews, Some debts I owe, Séminaire Lotharingien Combinatoire, Paper B42a, Issue 42, 2000.

G. E. Andrews, F. J. Dyson and D. Hickerson, Partitions and indefinite quadratic forms, Invent. Math. 91 (1988) 391-407.

S.-Y. Kang, Generalizations of Ramanujan's reciprocity theorem and their applications, J. London Math. Soc., 75 (2007), 18-34.

Alexander E. Patkowski, A note on the rank parity function, Discrete Math. 310 (2010), 961-965.

D. Zagier, Quantum modular forms, Example 1 in Quanta of Maths: Conference in honor of Alain Connes, Clay Mathematics Proceedings 11, AMS and Clay Mathematics Institute 2010, 659-675

FORMULA

G.f.: 1 - Sum_{n > 0} (-x)^n * (1 - x) * (1 - x^2) * ... * (1  -x^(n-1)).

G.f.: 1 + Sum_{n>=1}(x^(n(n+1)/2)/Product_{j=1..n}(1+x^j)). - Emeric Deutsch, Mar 30 2006

Define c(24*k + 1) = A003406(k), c(24*k - 1) = -2*A003475(k), c(n) = 0 otherwise. Then c(n) is multiplicative with c(2^e) = c(3^e) = 0^e, c(p^e) = (-1)^(e/2) * (1+(-1)^e)/2 if p == 7, 17 (mod 24), c(p^e) = (1+(-1)^e)/2 if p == 5, 11, 13, 19 (mod 24), c(p^e) = (e+1)*(-1)^(y*e) where p == 1, 23 (mod 24) and p = x^2 - 72*y^2 . - Michael Somos, Aug 17 2006

Also R(x) = -2 + Sum_{n>=0} (n+1)*x^(n(n-1)/2)/(Product_{k=1..n} (1+x^k)). - Paul D. Hanna, May 22 2010

EXAMPLE

1 + x - x^2 + 2*x^3 - 2*x^4 + x^5 + x^7 - 2*x^8 + 2*x^10 - x^12 - 2*x^13 + ...

q + q^25 - q^49 + 2*q^73 - 2*q^97 + q^121 + q^169 - 2*q^193 + 2*q^241 - ...

MAPLE

g:=1+sum(x^(n*(n+1)/2)/product(1+x^j, j=1..n), n=1..20): gser:=series(g, x=0, 110): seq(coeff(gser, x, n), n=0..104); # Emeric Deutsch, Mar 30 2006

t1:= add( (-1)^n*q^(n*(3*n+1)/2)*(1-q^(2*n+1))* add( (-1)^j*q^(-j^2), j=-n..n), n=0..20); t2:=series(t1, q, 40); # N. J. A. Sloane, Jun 27 2011

MATHEMATICA

max = 105; f[x_] := 1 + Sum[ x^(n*(n+1)/2) / Product[ 1+x^j, {j, 1, n}], {n, 1, max}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* Jean-François Alcover, Dec 02 2011 *)

max = 105; s = 1 + Sum[2*q^(n*(n+1)/2)/QPochhammer[-1, q, n+1], {n, 1, Ceiling[Sqrt[2 max]]}] + O[q]^max; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)

PROG

(PARI) {a(n) = local(t); if( n<0, 0, t = 1 + O(x^n); polcoeff( sum( k=1, n, t *= if( k>1, x^k - x, x) + O(x^(n-k+2)), 1), n))} /* Michael Somos, Mar 07 2006 */

(PARI) {a(n) = local(t); if( n<0, 0, t = 1 + O(x^n); polcoeff( sum( k=1, (sqrtint(8*n + 1)-1)\2, t *= x^k / (1 + x^k) + x * O(x^(n - (k^2-k)/2)), 1), n))} /* Michael Somos, Aug 17 2006 */

(PARI) {a(n) = local(A, p, e, x, y); if( n<0, 0, n = 24*n+1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p<5, 0, if( p%24>1 && p%24<23, if(e%2, 0, if( p%24==7 || p%24==17, (-1)^(e/2), 1)), x=y=0; if( p%24==1, forstep(i=1, sqrtint(p), 2, if( issquare( (i^2+p)/2, &y), x=i; break)), for( i=1, sqrtint(p\2), if( issquare(2*i^2 + p, &x), y=i; break))); (e+1)*(-1)^( (x + if((x-y)%6, y, -y))/6*e))))))} /* Michael Somos, Aug 17 2006 */

CROSSREFS

Cf. A003475, A005895, A005896, A158690.

Sequence in context: A180009 A180010 A287401 * A226289 A107063 A290453

Adjacent sequences:  A003403 A003404 A003405 * A003407 A003408 A003409

KEYWORD

sign,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 17 10:59 EST 2019. Contains 320219 sequences. (Running on oeis4.)