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A322235 Triangle, read by rows, each row n being defined by g.f. Product_{k=1..n} (k + x + k*x^2), for n >= 0.

%I A322235 #19 Dec 30 2018 00:48:37
%S A322235 1,1,1,1,2,3,5,3,2,6,11,24,23,24,11,6,24,50,131,160,215,160,131,50,24,
%T A322235 120,274,825,1181,1890,1815,1890,1181,825,274,120,720,1764,5944,9555,
%U A322235 17471,19866,24495,19866,17471,9555,5944,1764,720,5040,13068,48412,85177,173460,223418,313628,302619,313628,223418,173460,85177,48412,13068,5040,40320,109584,440684,834372,1860153,2642220,4120122,4521924,5320667,4521924,4120122,2642220,1860153,834372,440684,109584,40320,362880,1026576,4438620,8936288,21541905,33149481,56464695,68597418,89489025,86715299,89489025,68597418,56464695,33149481,21541905,8936288,4438620,1026576,362880
%N A322235 Triangle, read by rows, each row n being defined by g.f. Product_{k=1..n} (k + x + k*x^2), for n >= 0.
%H A322235 Paul D. Hanna, <a href="/A322235/b322235.txt">Table of n, a(n) for n = 0..5040, as a flattened triangle of rows 0..70.</a>
%F A322235 Row sums equal (2*n+1)!/(n!*2^n), the odd double factorials.
%F A322235 Left and right borders equal n!.
%e A322235 This irregular triangle formed from coefficients of x^k in Product_{m=1..n} (m + x + m*x^2), for n >= 0, k = 0..2*n, begins
%e A322235 1;
%e A322235 1, 1, 1;
%e A322235 2, 3, 5, 3, 2;
%e A322235 6, 11, 24, 23, 24, 11, 6;
%e A322235 24, 50, 131, 160, 215, 160, 131, 50, 24;
%e A322235 120, 274, 825, 1181, 1890, 1815, 1890, 1181, 825, 274, 120;
%e A322235 720, 1764, 5944, 9555, 17471, 19866, 24495, 19866, 17471, 9555, 5944, 1764, 720;
%e A322235 5040, 13068, 48412, 85177, 173460, 223418, 313628, 302619, 313628, 223418, 173460, 85177, 48412, 13068, 5040;
%e A322235 40320, 109584, 440684, 834372, 1860153, 2642220, 4120122, 4521924, 5320667, 4521924, 4120122, 2642220, 1860153, 834372, 440684, 109584, 40320; ...
%e A322235 in which the central terms equal A322238.
%e A322235 RELATED SEQUENCES.
%e A322235 Note that the terms in the secondary diagonal A322237 in the above triangle
%e A322235 [1, 3, 24, 160, 1890, 19866, 313628, 4521924, 89489025, 1642616195, ...]
%e A322235 may be divided by triangular numbers to obtain A322236:
%e A322235 [1, 1, 4, 16, 126, 946, 11201, 125609, 1988645, 29865749, 592326527, ...].
%t A322235 row[n_] := CoefficientList[Product[k+x+k*x^2, {k, 1, n}] + O[x]^(2n+1), x];
%t A322235 Table[row[n], {n, 0, 9}] // Flatten (* _Jean-François Alcover_, Dec 26 2018 *)
%o A322235 (PARI) {T(n, k) = polcoeff( prod(m=1, n, m + x + m*x^2) +x*O(x^k), k)}
%o A322235 /* Print the irregular triangle */
%o A322235 for(n=0, 10, for(k=0, 2*n, print1( T(n, k), ", ")); print(""))
%Y A322235 Cf. A322236, A322237, A322238.
%Y A322235 Cf. A322225 (variant), A322891 (variant).
%K A322235 nonn,tabf
%O A322235 0,5
%A A322235 _Paul D. Hanna_, Dec 15 2018