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 A116932 Number of partitions of n in which each part, with the possible exception of the largest, occurs at least three times. 13
 1, 2, 2, 3, 3, 6, 6, 9, 12, 14, 16, 24, 25, 32, 40, 49, 56, 73, 81, 102, 120, 142, 162, 202, 227, 270, 316, 367, 419, 506, 565, 663, 767, 879, 998, 1179, 1317, 1517, 1739, 1979, 2232, 2588, 2883, 3295, 3742, 4220, 4737, 5426, 6037, 6828, 7701, 8642, 9651, 10939 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Also, partitions of n in which any two distinct parts differ by at least 3. Example: a(5) = 3 because we have [5], [4,1] and [1,1,1,1,1]. LINKS Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz) FORMULA G.f.: sum(x^k*product(1+x^(3j)/(1-x^j), j=1..k-1)/(1-x^k), k=1..infinity). More generally, the g.f. of partitions of n in which each part, with the possible exception of the largest, occurs at least b times, is sum(x^k*product(1+x^(bj)/(1-x^j), j=1..k-1)/(1-x^k), k=1..infinity). It is also the g.f. of partitions of n in which any two distinct parts differ by at least b. log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-3*x)) dx = -0.77271248407593487127235205445116662610863126869... - Vaclav Kotesovec, Jan 28 2022 EXAMPLE a(5) = 3 because we have [5], [2,1,1,1] and [1,1,1,1,1]. MAPLE g:=sum(x^k*product(1+x^(3*j)/(1-x^j), j=1..k-1)/(1-x^k), k=1..70): gser:=series(g, x=0, 62): seq(coeff(gser, x^n), n=1..58); # second Maple program b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1) +add(b(n-i*j, i-3), j=1..n/i))) end: a:= n-> b(n, n): seq(a(n), n=1..70); # Alois P. Heinz, Nov 04 2012 MATHEMATICA b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-3], {j, 1, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, May 26 2015, after Alois P. Heinz *) CROSSREFS Cf. A116931, A100405. Column k=3 of A218698. - Alois P. Heinz, Nov 04 2012 Sequence in context: A133392 A101199 A032155 * A240579 A292225 A238786 Adjacent sequences: A116929 A116930 A116931 * A116933 A116934 A116935 KEYWORD nonn AUTHOR Emeric Deutsch, Feb 27 2006 STATUS approved

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