A153272 - OEIS

%I #11 Sep 08 2022 08:45:39

%S 7,7,56,7,63,693,7,70,910,14560,7,77,1155,21945,504735,7,84,1428,

%T 31416,848232,27143424,7,91,1729,43225,1339975,49579075,2131900225,7,

%U 98,2058,57624,2016840,84707280,4150656720,232436776320,7,105,2415,74865,2919735,137227545,7547514975,475493443425,33760034483175

%N Triangle T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 4, read by rows.

%C Row sums are {7, 63, 763, 15547, 527919, 28024591, 2182864327, 236674216947, 34243215666247, 6391699984166119, 1497639790982770659, ...}.

%H G. C. Greubel, <a href="/A153272/b153272.txt">Rows n = 0..100 of triangle, flattened</a>

%F T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 4.

%e Triangle begins as:

%e   7;

%e   7, 56;

%e   7, 63,  693;

%e   7, 70,  910, 14560;

%e   7, 77, 1155, 21945,  504735;

%e   7, 84, 1428, 31416,  848232, 27143424;

%e   7, 91, 1729, 43225, 1339975, 49579075, 2131900225;

%p m:=4; seq(seq(`if`(k=0, ithprime(m), mul(j*n + ithprime(m), j=0..k)), k=0..n), n=0..10); # _G. C. Greubel_, Dec 03 2019

%t T[n_, k_, m_]:= If[k==0, Prime[m], Product[j*n + Prime[m], {j,0,k}]];

%t Table[T[n,k,4], {n,0,10}, {k,0,n}]//Flatten

%o (PARI) T(n,k) = my(m=4); if(k==0, prime(m), prod(j=0,k, j*n + prime(m)) ); \\ _G. C. Greubel_, Dec 03 2019

%o (Magma) m:=4;

%o function T(n,k)

%o   if k eq 0 then return NthPrime(m);

%o   else return (&*[j*n + NthPrime(m): j in [0..k]]);

%o   end if; return T; end function;

%o [T(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 03 2019

%o (Sage)

%o def T(n, k):

%o     m=4

%o     if (k==0): return nth_prime(m)

%o     else: return product(j*n + nth_prime(m) for j in (0..k))

%o [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 03 2019

%Y Cf. A153270 (m=2), A153271 (m=3), this sequence (m=4).

%Y Cf. A001730, A051579, A051604.

%K nonn,tabl

%O 0,1

%A _Roger L. Bagula_, Dec 22 2008

%E Edited by _G. C. Greubel_, Dec 03 2019