A258358 -id:A258358

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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.

+10
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

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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:

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.

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, May 26 2015

STATUS

approved

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