A200536 - OEIS

%I #13 Aug 28 2015 09:51:24

%S 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 1,15,100,390,985,1683,1970,1560,800,240,32,1,18,147,720,2355,5418,

%U 8989,10836,9420,5760,2352,576,64,1,21,203,1197,4809,13923,29953,48639,59906,55692,38472,19152,6496,1344,128

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

%H Paul D. Hanna, <a href="/A200536/b200536.txt">Table of n, a(n) for n = 0..1295; Rows n = 0..35, flattened.</a>

%H 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 Central terms in rows form the central Delannoy numbers: T(n,n) = A001850(n).

%F T(2*n,n) = A190726(n).

%F T(n,n+1) = n*A006318(n), where A006318 form the large Schroeder numbers.

%e The triangle begins:

%e 1;

%e 1, 3, 2;

%e 1, 6, 13, 12, 4;

%e 1, 9, 33, 63, 66, 36, 8;

%e 1, 12, 62, 180, 321, 360, 248, 96, 16;

%e 1, 15, 100, 390, 985, 1683, 1970, 1560, 800, 240, 32;

%e 1, 18, 147, 720, 2355, 5418, 8989, 10836, 9420, 5760, 2352, 576, 64;

%e 1, 21, 203, 1197, 4809, 13923, 29953, 48639, 59906, 55692, 38472, 19152, 6496, 1344, 128;

%e 1, 24, 268, 1848, 8806, 30744, 81340, 166344, 265729, 332688, 325360, 245952, 140896, 59136, 17152, 3072, 256; ...

%e where row n equals the coefficients in (1+x)^n*(1+2*x)^n.

%o (PARI) {T(n,k)=polcoeff((1+3*x+2*x^2+x*O(x^k))^n,k)}

%o {for(n=0,10,for(k=0,2*n,print1(T(n,k),","));print(""))}

%Y Cf. A001850 (central Delannoy numbers), A006318, A190726; related triangle: A118384.

%Y Cf. A200537.

%K nonn,tabf

%O 0,3

%A _Paul D. Hanna_, Nov 18 2011