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 A100534 Number of partitions of 2*n into parts of two kinds. 0
 1, 5, 20, 65, 185, 481, 1165, 2665, 5822, 12230, 24842, 49010, 94235, 177087, 326015, 589128, 1046705, 1831065, 3157789, 5374390, 9035539, 15018300, 24697480, 40210481, 64854575, 103679156, 164363280, 258508230, 403531208, 625425005 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS FORMULA Expansion of q^(1/24) * eta(q^4)^5 / (eta(q)^5 * eta(q^8)^2) in powers of q. - Michael Somos, Sep 24 2011 a(n) = A000712(2*n). EXAMPLE 1 + 5*x + 20*x^2 + 65*x^3 + 185*x^4 + 481*x^5 + 1165*x^6 + 2665*x^7 + ... q^-1 + 5*q^23 + 20*q^47 + 65*q^71 + 185*q^95 + 481*q^119 + 1165*q^143 + ... a(1)=5 because we have 2, 2', 11, 1'1 and 1'1'. MAPLE with(combinat): A000712:=n-> add(numbpart(k)*numbpart(n-k), k=0..n): seq(A000712(2*n), n=0..32); # Emeric Deutsch, Dec 16 2004 MATHEMATICA a[n_] := Sum[PartitionsP[k] PartitionsP[2 n - k], {k, 0, 2 n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 30 2015, adapted from Maple *) PROG (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^5 / (eta(x + A)^5 * eta(x^8 + A)^2), n))} /* Michael Somos, Sep 24 2011 */ (PARI) {a(n) = local(A); if( n<0, 0, n = 2*n; A = x * O(x^n); polcoeff( 1 / eta(x + A)^2, n))} /* Michael Somos, Sep 24 2011 */ CROSSREFS Cf. A000712. Sequence in context: A309919 A195861 A001939 * A285928 A160506 A277212 Adjacent sequences:  A100531 A100532 A100533 * A100535 A100536 A100537 KEYWORD nonn AUTHOR N. J. A. Sloane, Nov 27 2004 EXTENSIONS More terms from Emeric Deutsch, Dec 16 2004 STATUS approved

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Last modified June 24 02:56 EDT 2021. Contains 345415 sequences. (Running on oeis4.)