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A322892 a(n) = [x^n] Product_{k=1..n} (k + x + 2*k*x^2), for n >= 0.

%I A322892 #9 Dec 30 2018 00:13:42
%S A322892 1,1,9,45,717,6917,154877,2254625,64599201,1267075953,44097148953,
%T A322892 1092097482333,44645622936189,1338624157833861,62791851488870493,
%U A322892 2213430779241737793,117082536584478235713,4748345510312622896993,279463602946698380026793,12824987274099379222626701,830920299335152521399853101,42586722790649923167650932101,3011022417317079016258969826109,170527854080899363788154404878305
%N A322892 a(n) = [x^n] Product_{k=1..n} (k + x + 2*k*x^2), for n >= 0.
%H A322892 Paul D. Hanna, <a href="/A322892/b322892.txt">Table of n, a(n) for n = 0..300</a>
%F A322892 a(n+1) = 4*(n+1) * A322893(n) + a(n), for n >= 1.
%F A322892 a(n+1) = 2*n*(n+1)^2 * A322894(n) + a(n), for n >= 1.
%e A322892 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 A322892 1;
%e A322892 1, 1, 2;
%e A322892 2, 3, 9, 6, 8;
%e A322892 6, 11, 42, 45, 84, 44, 48;
%e A322892 24, 50, 227, 310, 717, 620, 908, 400, 384;
%e A322892 120, 274, 1425, 2277, 6165, 6917, 12330, 9108, 11400, 4384, 3840;
%e A322892 720, 1764, 10264, 18375, 56367, 74991, 154877, 149982, 225468, 147000, 164224, 56448, 46080; ...
%e A322892 in which the main diagonal forms this sequence.
%e A322892 Note that the terms in the secondary diagonal A322893 in the above triangle
%e A322892 [1, 3, 42, 310, 6165, 74991, 1948268, 33402132, 1070751825, ...]
%e A322892 may be divided by triangular numbers n*(n+1)/2 to obtain A322894:
%e A322892 [1, 1, 7, 31, 411, 3571, 69581, 927837, 23794485, 433057989, ...].
%o A322892 (PARI) {A322891(n, k) = polcoeff( prod(m=1, n, m + x + 2*m*x^2) +x*O(x^k), k)}
%o A322892 /* Print the irregular triangle */
%o A322892 for(n=0, 10, for(k=0, 2*n, print1( A322891(n, k), ", ")); print(""))
%o A322892 /* Print this sequence */
%o A322892 for(n=0, 30, print1( A322891(n, n), ", "))
%Y A322892 Cf. A322891, A322893, A322894.
%Y A322892 Cf. A322238 (variant).
%K A322892 nonn
%O A322892 0,3
%A A322892 _Paul D. Hanna_, Dec 29 2018