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A260613 Triangle read by rows: T(n, k) = coefficient of x^(n-k) in Product_{m=1..n} (x+prime(m)); 0 <= k <= n, n >= 0.

%I A260613 #43 Dec 22 2024 11:12:03
%S A260613 1,1,2,1,5,6,1,10,31,30,1,17,101,247,210,1,28,288,1358,2927,2310,1,41,
%T A260613 652,5102,20581,40361,30030,1,58,1349,16186,107315,390238,716167,
%U A260613 510510,1,77,2451,41817,414849,2429223,8130689,14117683,9699690
%N A260613 Triangle read by rows: T(n, k) = coefficient of x^(n-k) in Product_{m=1..n} (x+prime(m)); 0 <= k <= n, n >= 0.
%C A260613 Up to signs and order of coefficients the same as A070918. Except for signs and the first column the same as A238146. - _M. F. Hasler_, Aug 13 2015
%H A260613 Alois P. Heinz, <a href="/A260613/b260613.txt">Rows n = 0..140, flattened</a> (first 20 rows from Matthew Campbell)
%F A260613 T(n, 1) = A007504(n) for n >= 1.
%F A260613 T(n, 2) = A024447(n) for n >= 2.
%e A260613 The triangle starts:
%e A260613 Row 0: 1;
%e A260613 Row 1: 1, 2;  Coefficients of x + 2.
%e A260613 Row 2: 1, 5, 6;  Coefficients of (x+2)(x+3) = x^2 + 5x + 6.
%e A260613 Row 3: 1, 10, 31, 30; Coeff's of (x+2)(x+3)(x+5) = x^3 + 10x^2 + 31x + 30.
%e A260613 Row 5: 1, 17, 101, 247, 210;
%e A260613 Row 6: 1, 28, 288, 1358, 2927, 2310;
%e A260613 ...
%p A260613 T:= n-> (p-> seq(coeff(p, x, n-i), i=0..n))(mul(x+ithprime(i), i=1..n)):
%p A260613 seq(T(n), n=0..10);  # _Alois P. Heinz_, Aug 18 2019
%t A260613 row[n_] := CoefficientList[Product[x + Prime[m], {m, 1, n}] + O[x]^(n+1), x] // Reverse;
%t A260613 row /@ Range[0, 8] // Flatten (* _Jean-François Alcover_, Sep 16 2019 *)
%o A260613 (PARI) tabl(nn) = {for (n=0, nn, polp = prod(k=1, n, x+prime(k)); forstep (k= n, 0, -1, print1(polcoeff(polp, k), ", ");); print(););} \\ _Michel Marcus_, Aug 10 2015
%Y A260613 Main diagonal gives A002110.
%Y A260613 Row sums give A054640.
%Y A260613 Cf. A000040.
%K A260613 nonn,tabl
%O A260613 0,3
%A A260613 _Matthew Campbell_, Aug 10 2015
%E A260613 Corrected and edited by _M. F. Hasler_, Aug 13 2015
%E A260613 a(20) in b-file corrected by _Andrew Howroyd_, Dec 31 2017