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A322893 a(n) = [x^(n-1)] Product_{k=1..n} (k + x + 2*k*x^2), for n >= 1.

%I A322893 #7 Dec 30 2018 00:29:39
%S A322893 1,3,42,310,6165,74991,1948268,33402132,1070751825,23818189395,
%T A322893 907365113622,24884202594186,1097379059482797,35843982129214455,
%U A322893 1794829778206820280,68106808437178597960,3815489686616468849025,165072679883587905823683,10226191400763164277215330,497092886801366317217274750,33732223801436694239674078341,1820835126778068312737993859263
%N A322893 a(n) = [x^(n-1)] Product_{k=1..n} (k + x + 2*k*x^2), for n >= 1.
%H A322893 Paul D. Hanna, <a href="/A322893/b322893.txt">Table of n, a(n) for n = 1..300</a>
%F A322893 a(n) = A322891(n, n-1) for n >= 1.
%F A322893 a(n) = A322891(n, n+1)/2 for n >= 1.
%F A322893 a(n) = n*(n+1)/2 * A322894(n) for n >= 1.
%e A322893 The irregular triangle A322891 of coefficients of x^k in Product_{m=1..n} (m + x + 2*m*x^2), for n >= 0, k = 0..2*n, begins
%e A322893 1;
%e A322893 1, 1, 2;
%e A322893 2, 3, 9, 6, 8;
%e A322893 6, 11, 42, 45, 84, 44, 48;
%e A322893 24, 50, 227, 310, 717, 620, 908, 400, 384;
%e A322893 120, 274, 1425, 2277, 6165, 6917, 12330, 9108, 11400, 4384, 3840;
%e A322893 720, 1764, 10264, 18375, 56367, 74991, 154877, 149982, 225468, 147000, 164224, 56448, 46080; ...
%e A322893 Note that this sequence forms a secondary diagonal in the above triangle
%e A322893 [1, 3, 42, 310, 6165, 74991, 1948268, 33402132, 1070751825, ...]
%e A322893 and may be divided by triangular numbers n*(n+1)/2 to obtain A322894:
%e A322893 [1, 1, 7, 31, 411, 3571, 69581, 927837, 23794485, 433057989, ...].
%o A322893 (PARI) {A322891(n, k) = polcoeff( prod(m=1, n, m + x + 2*m*x^2) +x*O(x^k), k)}
%o A322893 /* Print the irregular triangle */
%o A322893 for(n=0, 10, for(k=0, 2*n, print1( A322891(n, k), ", ")); print(""))
%o A322893 /* Print this sequence */
%o A322893 for(n=1, 30, print1( A322891(n, n-1), ", "))
%Y A322893 Cf. A322891, A322892, A322894.
%Y A322893 Cf. A322237 (variant).
%K A322893 nonn
%O A322893 1,2
%A A322893 _Paul D. Hanna_, Dec 29 2018