cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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)) }.

Original entry on oeis.org

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
Offset: 0

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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

Examples

			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 - ...
		

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).

Crossrefs

Programs

  • 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 *)
  • 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 */

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

A005895 Weighted count of partitions with distinct parts.

Original entry on oeis.org

1, 2, 5, 7, 12, 18, 26, 35, 50, 67, 88, 116, 149, 191, 245, 306, 381, 477, 585, 718, 880, 1067, 1288, 1555, 1863, 2226, 2656, 3151, 3726, 4406, 5180, 6077, 7124, 8316, 9691, 11278, 13080, 15146, 17517, 20204, 23264, 26759, 30705, 35182, 40274, 46000, 52473, 59795, 68018, 77279, 87711, 99395, 112508
Offset: 1

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Comments

Also sum of largest parts of all partitions of n into distinct parts. - Vladeta Jovovic, Feb 15 2004

References

  • Andrews, George E.; Ramanujan's "lost" notebook. V. Euler's partition identity. Adv. in Math. 61 (1986), no. 2, 156-164.
  • S.-Y. Kang, Generalizations of Ramanujan's reciprocity theorem..., J. London Math. Soc., 75 (2007), 18-34. See Eq. (1.5) but beware errors.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

Programs

  • Maple
    M:=201; add( mul( (1+q^j),j=1..M) - mul( (1+q^j),j=1..n), n=0..M);
    # second Maple program:
    b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0, `if`(
          n=0, 1, b(n,i-1)+`if`(i>n, 0, b(n-i, min(n-i,i-1)))))
        end:
    a:= n-> add(j*b(n-j, min(n-j,j-1)), j=1..n):
    seq(a(n), n=1..80);  # Alois P. Heinz, Feb 03 2016
  • Mathematica
    m = 46; f[q_] :=  Sum[ Product[ (1+q^j), {j, 1, m}] - Product[ (1+q^j), {j, 1, n}], {n, 0, m}]; CoefficientList[ f[q], q][[2 ;; m+1]] (* Jean-François Alcover, Apr 13 2012, after Maple *)
  • PARI
    N=66;  x='x+O('x^N);
    S=prod(k=1,N, 1+x^k); gf=sum(n=0,N, S-prod(k=1,n, 1+x^k));
    /* alternative: Arndt's g.f.: */
    /* gf=sum(k=0,N, (k+1)*x^(k+1) * prod(j=1,k, 1+x^j) ); */
    Vec(gf)
    /* Joerg Arndt, Sep 17 2012 */

Formula

G.f.: sum(n>=0, S(q) - prod(k=1..n, 1+q^k) ), where S(q)=prod(k>=1, 1+q^k) (g.f. for A000009).
G.f. sum(k>=0, (k+1)*x^(k+1) * prod(j=1..k, 1+x^j) ). [Joerg Arndt, Sep 17 2012]

Extensions

More terms from James Sellers, Dec 24 1999
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