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 A304825 Sum of binomial(Y(2,p), 2) over the partitions p of n, where Y(2,p) is the number of part sizes with multiplicity 2 or greater in p. 0
 1, 1, 3, 4, 9, 12, 22, 30, 50, 68, 105, 142, 210, 281, 400, 531, 736, 967, 1311, 1707, 2274, 2935, 3851, 4930, 6389, 8116, 10402, 13121, 16658, 20872, 26275, 32719, 40880, 50613, 62807, 77343, 95389, 116874, 143331, 174789, 213251, 258903, 314367, 380079, 459462 (list; graph; refs; listen; history; text; internal format)
 OFFSET 6,3 LINKS FORMULA a(n) = (A301313(n) - A024788(n))/4. G.f.: q^6 /((1-q^2)*(1-q^4))*Product_{j>=1} 1/(1-q^j). EXAMPLE For a(8), we sum over the partitions of eight. For each partition p, we take binomial(Y(2,p),2): that is, the number of parts with multiplicity at least two choose 2. 8................B(0,2) = 0 7,1..............B(0,2) = 0 6,2..............B(0,2) = 0 6,1,1............B(1,2) = 0 5,3..............B(0,2) = 0 5,2,1............B(0,2) = 0 5,1,1,1..........B(1,2) = 0 4,4..............B(1,2) = 0 4,3,1............B(0,2) = 0 4,2,2............B(1,2) = 0 4,2,1,1..........B(1,2) = 0 4,1,1,1,1........B(1,2) = 0 3,3,2............B(1,2) = 0 3,3,1,1..........B(2,2) = 1 3,2,2,1..........B(1,2) = 0 3,2,1,1,1........B(1,2) = 0 3,1,1,1,1,1......B(1,2) = 0 2,2,2,2..........B(1,2) = 0 2,2,2,1,1........B(2,2) = 1 2,2,1,1,1,1......B(2,2) = 1 2,1,1,1,1,1,1....B(1,2) = 0 1,1,1,1,1,1,1,1..B(1,2) = 0 --------------------------- Total.....................3 MAPLE b:= proc(n, i, p) option remember; `if`(n=0 or i=1,       binomial(`if`(n>1, 1, 0)+p, 2), add(       b(n-i*j, i-1, `if`(j>1, 1, 0)+p), j=0..n/i))     end: a:= n-> b(n\$2, 0): seq(a(n), n=6..60);  # Alois P. Heinz, May 19 2018 MATHEMATICA Array[Total[Binomial[Count[Split@#, _?(Length@# >= 2 &)], 2] & /@IntegerPartitions[#]] &, 50] (* Second program: *) b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1,      Binomial[If[n > 1, 1, 0] + p, 2], Sum[      b[n-i*j, i-1, If[j>1, 1, 0]+p], {j, 0, n/i}]]; a[n_] := b[n, n, 0]; a /@ Range[6, 60] (* Jean-François Alcover, May 30 2021, after Alois P. Heinz *) CROSSREFS Cf. A024786, A302347. Sequence in context: A025613 A097063 A293569 * A026476 A335604 A002513 Adjacent sequences:  A304822 A304823 A304824 * A304826 A304827 A304828 KEYWORD nonn AUTHOR Emily Anible, May 19 2018 STATUS approved

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Last modified June 21 04:10 EDT 2021. Contains 345354 sequences. (Running on oeis4.)