cp's OEIS Frontend

A322894 a(n) = A322893(n) / (n*(n+1)/2), where A322893(n) = [x^(n-1)] Product_{k=1..n} (k + x + 2*k*x^2), for n >= 1.

%I A322894 #13 Dec 30 2018 00:59:50
%S A322894 1,1,7,31,411,3571,69581,927837,23794485,433057989,13747956267,
%T A322894 319028238387,12059110543767,341371258373471,14956914818390169,
%U A322894 500785356155724985,24937841088996528425,965337309260747987273,53822060004016654090607,2367108984768411034367975,146026942863362312725861811,7196976785684064477225272171,486563915009872154819986680357
%N A322894 a(n) = A322893(n) / (n*(n+1)/2), where A322893(n) = [x^(n-1)] Product_{k=1..n} (k + x + 2*k*x^2), for n >= 1.
%H A322894 Paul D. Hanna, <a href="/A322894/b322894.txt">Table of n, a(n) for n = 1..300</a>
%F A322894 a(n) = A322891(n, n-1) / (n*(n+1)/2).
%F A322894 a(n) = A322891(n, n+1) / (n*(n+1)).
%F A322894 a(n) appears to be odd for n >= 0.
%e A322894 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 A322894 1;
%e A322894 1, 1, 2;
%e A322894 2, 3, 9, 6, 8;
%e A322894 6, 11, 42, 45, 84, 44, 48;
%e A322894 24, 50, 227, 310, 717, 620, 908, 400, 384;
%e A322894 120, 274, 1425, 2277, 6165, 6917, 12330, 9108, 11400, 4384, 3840;
%e A322894 720, 1764, 10264, 18375, 56367, 74991, 154877, 149982, 225468, 147000, 164224, 56448, 46080; ...
%e A322894 Note that the terms in the secondary diagonal A322893 in the above triangle,
%e A322894 [1, 3, 42, 310, 6165, 74991, 1948268, 33402132, 1070751825, ...],
%e A322894 may be divided by triangular numbers n*(n+1)/2 to obtain this sequence:
%e A322894 [1, 1, 7, 31, 411, 3571, 69581, 927837, 23794485, 433057989, ...].
%o A322894 (PARI) {A322891(n, k) = polcoeff( prod(m=1, n, m + x + 2*m*x^2) +x*O(x^k), k)}
%o A322894 /* Print the irregular triangle */
%o A322894 for(n=0, 10, for(k=0, 2*n, print1( A322891(n, k), ", ")); print(""))
%o A322894 /* Print this sequence */
%o A322894 for(n=1, 30, print1( A322891(n, n-1)/(n*(n+1)/2), ", "))
%Y A322894 Cf. A322891, A322892, A322893.
%Y A322894 Cf. A322226 (variant), A322236 (variant).
%K A322894 nonn
%O A322894 1,3
%A A322894 _Paul D. Hanna_, Dec 29 2018