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 A338085 a(n) is the cardinality of S(n), the subset of partitions of n such that there are enough smaller parts to add together to be greater than a larger part. 0
 0, 0, 0, 0, 1, 1, 5, 5, 12, 18, 30, 36, 65, 83, 120, 159, 225, 284, 395, 495, 665, 848, 1094, 1348, 1757, 2184, 2746, 3399, 4250, 5199, 6469, 7867, 9667, 11756, 14310, 17266, 20988, 25216, 30372, 36371, 43648, 52041, 62187, 73866, 87837, 104105, 123279, 145453 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS In George Andrews’s partition notation, exponents mean repeated addition, not repeated multiplication. So (p^K)(q^L) with p0 and the parts pk arranged in increasing order, suppose E1p1+E2p2+..Ekpk>p(k+1) for some 17. Also, (1^3)(3^2)(7^1)(20^4), a partition of 96, is in S(96) since 3*1+2*3=9>7. But (1^3)(4^5) is not in S(23) because 3*1 is not greater than 4. MATHEMATICA ispart[p_] := Module[{s = 0}, For[i = 1, i <= Length[p], i++, If[s > p[[i]] && p[[i]] > p[[i-1]], Return[1]]; s += p[[i]]]; 0]; a[n_] := a[n] = Module[{c = 0}, Do[ c += ispart[p], {p, Reverse /@ IntegerPartitions[n]}]; c]; Table[Print[n, " ", a[n]]; a[n], {n, 1, 50}] (* Jean-François Alcover, Nov 13 2020, after Andrew Howroyd *) PROG (PARI) ispart(p)={my(s=0); for(i=1, #p, if(s>p[i]&&p[i]>p[i-1], return(1)); s+=p[i]); 0} a(n)={my(c=0); forpart(p=n, c+=ispart(p)); c} \\ Andrew Howroyd, Oct 25 2020 (PARI) a(n)={local(Cache=Map()); my(F(r, k, b)=my(hk=[r, k, b], z); if(!mapisdefined(Cache, hk, &z), z = if(k<=1, b, sum(m=0, r\k, self()(r-m*k, k-1, b||(m&&r-m*k>k)))); mapput(Cache, hk, z)); z); F(n, n, 0)} \\ Andrew Howroyd, Nov 03 2020 CROSSREFS Sequence in context: A168336 A123133 A328367 * A302676 A339947 A206553 Adjacent sequences: A338082 A338083 A338084 * A338086 A338087 A338088 KEYWORD nonn AUTHOR Richard Peterson, Oct 08 2020 EXTENSIONS Terms a(15) and beyond from Andrew Howroyd, Nov 03 2020 STATUS approved

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Last modified September 28 13:02 EDT 2023. Contains 365735 sequences. (Running on oeis4.)