OFFSET
0,3
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
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
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^((2k+1)(k+1))/((1-x)...(1-x^(2k+1))). - Gregory L. Simay, Aug 03 2019
EXAMPLE
G.f.: x + 2*x^2 + 4*x^3 + 7*x^4 + 12*x^5 + 20*x^6 + 32*x^7 + 50*x^8 + ...
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 *)
PROG
(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) x='x+O('x^66); concat([0], Vec(eta(x^2)/eta(x)^2-1)/2) \\ Joerg Arndt, Nov 27 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved