cp's OEIS Frontend

A343751 A(n,k) is the sum of all compositions [c_1, c_2, ..., c_k] of n into k nonnegative parts encoded as Product_{i=1..k} prime(i)^(c_i); square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A343751 #29 Jan 31 2024 16:14:31
%S A343751 1,1,0,1,2,0,1,5,4,0,1,10,19,8,0,1,17,69,65,16,0,1,28,188,410,211,32,
%T A343751 0,1,41,496,1726,2261,665,64,0,1,58,1029,7182,14343,11970,2059,128,0,
%U A343751 1,77,2015,20559,93345,112371,61909,6305,256,0,1,100,3478,54814,360612,1139166,848506,315850,19171,512,0
%N A343751 A(n,k) is the sum of all compositions [c_1, c_2, ..., c_k] of n into k nonnegative parts encoded as Product_{i=1..k} prime(i)^(c_i); square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A343751 Alois P. Heinz, <a href="/A343751/b343751.txt">Antidiagonals n = 0..140, flattened</a>
%F A343751 A(n,k) = [x^n] Product_{i=1..k} 1/(1-prime(i)*x).
%F A343751 A(n,k) = A124960(n+k,k) for k >= 1.
%e A343751 A(1,3) = 10 = 5 + 3 + 2, sum of encoded compositions [0,0,1], [0,1,0], [1,0,0].
%e A343751 A(4,2) = 211 = 81 + 54 + 36 + 24 + 16, sum of encoded compositions [0,4], [1,3], [2,2], [3,1], [4,0].
%e A343751 Square array A(n,k) begins:
%e A343751   1,  1,    1,     1,      1,        1,        1, ...
%e A343751   0,  2,    5,    10,     17,       28,       41, ...
%e A343751   0,  4,   19,    69,    188,      496,     1029, ...
%e A343751   0,  8,   65,   410,   1726,     7182,    20559, ...
%e A343751   0, 16,  211,  2261,  14343,    93345,   360612, ...
%e A343751   0, 32,  665, 11970, 112371,  1139166,  5827122, ...
%e A343751   0, 64, 2059, 61909, 848506, 13379332, 89131918, ...
%p A343751 A:= proc(n, k) option remember; `if`(n=0, 1,
%p A343751      `if`(k=0, 0, add(ithprime(k)^i*A(n-i, k-1), i=0..n)))
%p A343751     end:
%p A343751 seq(seq(A(n, d-n), n=0..d), d=0..10);
%p A343751 # second Maple program:
%p A343751 A:= proc(n, k) option remember; `if`(n=0, 1,
%p A343751      `if`(k=0, 0, ithprime(k)*A(n-1, k)+A(n, k-1)))
%p A343751     end:
%p A343751 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A343751 A[n_, k_] := A[n, k] = If[n == 0, 1,
%t A343751      If[k == 0, 0, Prime[k] A[n-1, k] + A[n, k-1]]];
%t A343751 Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* _Jean-François Alcover_, Nov 06 2021, after 2nd Maple program *)
%Y A343751 Columns k=0-4 give: A000007, A000079, A001047(n+1), A016273, A025931.
%Y A343751 Rows n=0-2 give: A000012, A007504, A357251.
%Y A343751 Main diagonal gives A332967.
%Y A343751 Cf. A000040, A074140, A124960, A324939, A325054.
%K A343751 nonn,tabl
%O A343751 0,5
%A A343751 _Alois P. Heinz_, Apr 27 2021