A014968 Expansion of (1/theta_4 - 1)/2.

0, 1, 2, 4, 7, 12, 20, 32, 50, 77, 116, 172, 252, 364, 520, 736, 1031, 1432, 1974, 2700, 3668, 4952, 6644, 8864, 11764, 15533, 20412, 26704, 34784, 45124, 58312, 75072, 96306, 123128, 156904, 199320, 252443, 318796, 401468, 504224, 631636, 789264, 983848, 1223532, 1518164, 1879620, 2322184, 2863040

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Let p(n) = the number of partitions of n, p(i,n) = the number of parts of the i-th partition of n, d(i,n) = the number of different parts in the i-th partition of n. Then a(n) = Sum_{i=1..p(n)} Sum_{j=1..d(i,n)} binomial(d(i,n)-1, j-1). - Thomas Wieder, May 08 2005

a(n) is the sum of the number of partitions of n-1 with two kinds of part 1 + the number of partitions of n-6 with two kinds of parts 1 through 3 + the number of partitions of n-15 with two kinds of parts 1 through 5 + ... . - Gregory L. Simay, Aug 03 2019

Examples

			G.f.: x + 2*x^2 + 4*x^3 + 7*x^4 + 12*x^5 + 20*x^6 + 32*x^7 + 50*x^8 + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Seiichi Manyama)
• J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 103.
• A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, Contrib. Discr. Math. 3 (2) (2008), 76-114

Crossrefs

Cf. A015128, A265835.

Cf. Expansion of ((Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k) - 1)/k in powers of x: this sequence (k=2), A277968 (k=3), A277974 (k=5), A160549 (k=7), A277912 (k=11).

Programs

Maple

A014968 := proc(n::integer) local a,i,j,prttn,prttnlst,ZahlTeile,ZahlVerschiedenerTeile; with(combinat); a := 0; prttnlst:=partition(n); for i from 1 to nops(prttnlst) do prttn := prttnlst[i]; ZahlTeile := nops(prttn); ZahlVerschiedenerTeile:=nops(convert(prttn,multiset)); for j from 1 to ZahlVerschiedenerTeile do a := a + binomial(ZahlVerschiedenerTeile-1,j-1); od; od; print("n, a(n): ",n, a); end proc;  for n from 0 to 20 do A014968(n) end do # Thomas Wieder, May 08 2005; fixed by Vaclav Kotesovec, Dec 16 2015
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1, 0,
      b(n, i-1))+add(2*b(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
    end:
a:= n-> `if`(n=0, 0, b(n$2)/2):
seq(a(n), n=0..49);  # Alois P. Heinz, Feb 10 2021

Mathematica

a[ n_] := SeriesCoefficient[ (1 / EllipticTheta[ 4, 0, q] - 1) / 2, {q, 0, n}]; (* Michael Somos, Nov 03 2013 *)
(QPochhammer[x^2]/QPochhammer[x]^2-1)/2 + O[x]^40 // CoefficientList[#, x]& (* Jean-François Alcover, Nov 07 2016 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A)^2 - 1 ) / 2, n))}; /* Michael Somos, Nov 03 2013 */

PARI

{a(n) = if( n<0, 0, polcoeff( sum(k=1, n, x^k / (1 + x^k) * prod(j=1, k, (1 + x^j) / (1 - x^j), 1 + x * O(x^(n-k)))), n))}; /* Michael Somos, Nov 03 2013 */

PARI

my(x='x+O('x^66)); concat([0],Vec(eta(x^2)/eta(x)^2-1)/2) \\ Joerg Arndt, Nov 27 2016

Formula

G.f.: Sum_{k>0} (x^k / (1 + x^k)) * Product_{j=1..k} (1 + x^j) / (1 - x^j). - Michael Somos, Nov 03 2013

2 * a(n) = A015128(n) unless n=0.

a(n) ~ exp(Pi*sqrt(n)) / (4*n) * (1 - 1/(Pi*sqrt(n))). - Vaclav Kotesovec, Nov 10 2016

G.f.: (Product_{k>=1} 1/(1-x^k))*(Sum_{k>=0} x^((2*k+1)*(k+1))/((1-x)*(1-x^2)*...*(1-x^(2*k+1)))). - Gregory L. Simay, Aug 03 2019