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 A258323 Sum T(n,k) over all partitions lambda of n into k distinct parts of Product_{i:lambda} prime(i); triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows. 14
 1, 0, 2, 0, 3, 0, 5, 6, 0, 7, 10, 0, 11, 29, 0, 13, 43, 30, 0, 17, 94, 42, 0, 19, 128, 136, 0, 23, 231, 293, 0, 29, 279, 551, 210, 0, 31, 484, 892, 330, 0, 37, 584, 1765, 852, 0, 41, 903, 2570, 1826, 0, 43, 1051, 4273, 4207, 0, 47, 1552, 6747, 6595, 2310 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Alois P. Heinz, Rows n = 0..500, flattened EXAMPLE T(6,2) = 43 because the partitions of 6 into 2 distinct parts are {[5,1], [4,2]} and prime(5)*prime(1) + prime(4)*prime(2) = 11*2 + 7*3 = 22 + 21 = 43. Triangle T(n,k) begins:   1   0,  2;   0,  3;   0,  5,   6;   0,  7,  10;   0, 11,  29;   0, 13,  43,  30;   0, 17,  94,  42;   0, 19, 128, 136;   0, 23, 231, 293;   0, 29, 279, 551, 210; MAPLE g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, expand(       add(g(n-i*j, i-1)*(ithprime(i)*x)^j, j=0..min(1, n/i)))))     end: T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n\$2)): seq(T(n), n=0..20); MATHEMATICA g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, Expand[Sum[g[n-i*j, i-1] * (Prime[i]*x)^j, {j, 0, Min[1, n/i]}]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][g[n, n]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jan 06 2017, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A000007, A000040, A025129(n+1), A258358, A258359, A258360, A258361, A258362, A258363, A258364, A258365. Row sums give A147655. T(n*(n+1)/2,n) = A002110(n). T(n^2,n) = A321267(n). Cf. A000217, A003056, A145518, A246867. Sequence in context: A325836 A011013 A138325 * A117175 A228086 A090482 Adjacent sequences:  A258320 A258321 A258322 * A258324 A258325 A258326 KEYWORD nonn,tabf AUTHOR Alois P. Heinz, May 26 2015 STATUS approved

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Last modified October 20 20:04 EDT 2019. Contains 328270 sequences. (Running on oeis4.)