%I #28 Dec 30 2018 01:46:15
%S 1,1,1,2,2,3,9,6,8,6,11,42,45,84,44,48,24,50,227,310,717,620,908,400,
%T 384,120,274,1425,2277,6165,6917,12330,9108,11400,4384,3840,720,1764,
%U 10264,18375,56367,74991,154877,149982,225468,147000,164224,56448,46080,5040,13068,83692,163585,556640,838554,1948268,2254625,3896536,3354216,4453120,2617360,2678144,836352,645120,40320,109584,763244,1601460,5955777,9882432,25330938,33402132,64599201,66804264,101323752,79059456,95292432,51246720,48847616,14026752,10321920
%N Triangle, read by rows, each row n being defined by g.f. Product_{k=1..n} (k + x + 2*k*x^2), for n >= 0.
%C Row sums equal A007559(n+1), the triple factorial numbers.
%H Paul D. Hanna, <a href="/A322891/b322891.txt">Table of n, a(n) for n = 0..1680 terms of this irregular triangle, as read by rows 0..40.</a>
%F T(n,0) = n! for n >= 0.
%F T(n,2*n) = 2^n * n!, the even double factorials, for n >= 0.
%F Sum_{k=0..2*n} T(n,k) = Product_{k=0..n} (3*k + 1), the triple factorials, for n >= 0.
%e This irregular triangle of coefficients T(n,k) of x^k in Product_{m=1..n} (m + x + 2*m*x^2), for n >= 0, k = 0..2*n, begins
%e 1;
%e 1, 1, 2;
%e 2, 3, 9, 6, 8;
%e 6, 11, 42, 45, 84, 44, 48;
%e 24, 50, 227, 310, 717, 620, 908, 400, 384;
%e 120, 274, 1425, 2277, 6165, 6917, 12330, 9108, 11400, 4384, 3840;
%e 720, 1764, 10264, 18375, 56367, 74991, 154877, 149982, 225468, 147000, 164224, 56448, 46080;
%e 5040, 13068, 83692, 163585, 556640, 838554, 1948268, 2254625, 3896536, 3354216, 4453120, 2617360, 2678144, 836352, 645120;
%e 40320, 109584, 763244, 1601460, 5955777, 9882432, 25330938, 33402132, 64599201, 66804264, 101323752, 79059456, 95292432, 51246720, 48847616, 14026752, 10321920; ...
%e in which the central terms equal A322892:
%e [1, 1, 9, 45, 717, 6917, 154877, 2254625, 64599201, 1267075953, ...].
%e RELATED SEQUENCES.
%e Note that the terms in the secondary diagonal A322893 in this triangle,
%e [1, 3, 42, 310, 6165, 74991, 1948268, 33402132, 1070751825, ...],
%e may be divided by triangular numbers n*(n+1)/2 to obtain A322894:
%e [1, 1, 7, 31, 411, 3571, 69581, 927837, 23794485, 433057989, ...].
%o (PARI) {T(n, k) = polcoeff( prod(m=1, n, m + x + 2*m*x^2) +x*O(x^k), k)}
%o /* Print the irregular triangle */
%o for(n=0, 10, for(k=0, 2*n, print1( T(n, k), ", ")); print(""))
%Y Cf. A322892 (central terms), A322893 (diagonal), A322894.
%Y Cf. A322235 (variant), A322225 (variant), A000165, A007559.
%K nonn,tabf
%O 0,4
%A _Paul D. Hanna_, Dec 29 2018