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 A024940 Number of partitions of n into distinct triangular numbers C(k,2). 29
 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 0, 2, 3, 1, 1, 3, 2, 1, 4, 3, 0, 3, 3, 2, 4, 3, 3, 3, 2, 3, 3, 2, 4, 6, 4, 2, 5, 4, 2, 6, 5, 3, 7, 6, 3, 5, 5, 5, 6, 5, 4, 7, 7, 6, 8, 6, 5, 9, 7, 4, 9, 9, 6, 10, 9, 4, 9, 10, 8, 11, 11, 9, 10, 10, 9, 10, 10, 9, 14, 14, 7, 14, 14, 7, 15, 15, 8, 15, 17, 13 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,11 LINKS T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe) FORMULA For n>0: a(n) = b(n, 1) where b(n, k) = if n>k*(k+1)/2 then b(n-k*(k+1)/2, k+1) + b(n, k+1) else (if n=k*(k+1)/2 then 1 else 0). - Reinhard Zumkeller, Aug 26 2003 a(n) ~ exp(3*Pi^(1/3) * ((sqrt(2)-1)*Zeta(3/2))^(2/3) * n^(1/3) / 2^(4/3)) * ((sqrt(2)-1)*Zeta(3/2))^(1/3) / (2^(5/3) * sqrt(3) * Pi^(1/3) * n^(5/6)). - Vaclav Kotesovec, Jan 02 2017 EXAMPLE a(31) counts these partitions:  [28,3], [21,10], [21,6,3,1], [15,10,6] Clark Kimberling, Mar 09 2014 MATHEMATICA Drop[ CoefficientList[ Series[ Product[(1 + x^(k*(k + 1)/2)), {k, 1, 15}], {x, 0, 102}], x], 1] (* also *) t = Table[n (n + 1)/2, {n, 1, 200}] ; p[n_] := IntegerPartitions[n, All, t]; Table[p[n], {n, 0, 12}] (*shows unrestricted partitions*) d[n_] := Select[p[n], Max[Length /@ Split@#] == 1 &]; Table[d[n], {n, 1, 31}] (*shows strict partitions*) Table[Length[d[n]], {n, 1, 70}] (* Clark Kimberling, Mar 09 2014 *) nmax = 100; nn = Floor[Sqrt[8*nmax + 1]/2] + 1; poly = ConstantArray[0, nn*(nn+1)/2 + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += poly[[j - k*(k+1)/2 + 1]], {j, nn*(nn+1)/2, k*(k+1)/2, -1}]; , {k, 2, nn}]; Take[poly, nmax + 1] (* Vaclav Kotesovec, Dec 10 2016 *) PROG (Haskell) a024940 = p \$ tail a000217_list where    p _  0 = 1    p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m -- Reinhard Zumkeller, Jun 28 2013 CROSSREFS Cf. A000217, A033461, A007294, A280366. Sequence in context: A292518 A264997 A222759 * A205217 A054635 A003137 Adjacent sequences:  A024937 A024938 A024939 * A024941 A024942 A024943 KEYWORD nonn AUTHOR STATUS approved

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Last modified November 20 02:57 EST 2018. Contains 317371 sequences. (Running on oeis4.)