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A200536 Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.

%I A200536 #13 Aug 28 2015 09:51:24
%S A200536 1,1,3,2,1,6,13,12,4,1,9,33,63,66,36,8,1,12,62,180,321,360,248,96,16,
%T A200536 1,15,100,390,985,1683,1970,1560,800,240,32,1,18,147,720,2355,5418,
%U A200536 8989,10836,9420,5760,2352,576,64,1,21,203,1197,4809,13923,29953,48639,59906,55692,38472,19152,6496,1344,128
%N A200536 Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.
%H A200536 Paul D. Hanna, <a href="/A200536/b200536.txt">Table of n, a(n) for n = 0..1295; Rows n = 0..35, flattened.</a>
%H A200536 We-jin Woan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Woan/woan655.html">The Lagrange Inversion formula and divisibility properties</a>, JIS 10 (2007) 07.7.8, Example 3.
%F A200536 Central terms in rows form the central Delannoy numbers: T(n,n) = A001850(n).
%F A200536 T(2*n,n) = A190726(n).
%F A200536 T(n,n+1) = n*A006318(n), where A006318 form the large Schroeder numbers.
%e A200536 The triangle begins:
%e A200536 1;
%e A200536 1, 3, 2;
%e A200536 1, 6, 13, 12, 4;
%e A200536 1, 9, 33, 63, 66, 36, 8;
%e A200536 1, 12, 62, 180, 321, 360, 248, 96, 16;
%e A200536 1, 15, 100, 390, 985, 1683, 1970, 1560, 800, 240, 32;
%e A200536 1, 18, 147, 720, 2355, 5418, 8989, 10836, 9420, 5760, 2352, 576, 64;
%e A200536 1, 21, 203, 1197, 4809, 13923, 29953, 48639, 59906, 55692, 38472, 19152, 6496, 1344, 128;
%e A200536 1, 24, 268, 1848, 8806, 30744, 81340, 166344, 265729, 332688, 325360, 245952, 140896, 59136, 17152, 3072, 256; ...
%e A200536 where row n equals the coefficients in (1+x)^n*(1+2*x)^n.
%o A200536 (PARI) {T(n,k)=polcoeff((1+3*x+2*x^2+x*O(x^k))^n,k)}
%o A200536 {for(n=0,10,for(k=0,2*n,print1(T(n,k),","));print(""))}
%Y A200536 Cf. A001850 (central Delannoy numbers), A006318, A190726; related triangle: A118384.
%Y A200536 Cf. A200537.
%K A200536 nonn,tabf
%O A200536 0,3
%A A200536 _Paul D. Hanna_, Nov 18 2011