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

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, 1, 15, 100, 390, 985, 1683, 1970, 1560, 800, 240, 32, 1, 18, 147, 720, 2355, 5418, 8989, 10836, 9420, 5760, 2352, 576, 64, 1, 21, 203, 1197, 4809, 13923, 29953, 48639, 59906, 55692, 38472, 19152, 6496, 1344, 128

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1295; Rows n = 0..35, flattened.

We-jin Woan, The Lagrange Inversion formula and divisibility properties, JIS 10 (2007) 07.7.8, Example 3.

Formula

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

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

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

Example

The triangle begins:
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;
1, 15, 100, 390, 985, 1683, 1970, 1560, 800, 240, 32;
1, 18, 147, 720, 2355, 5418, 8989, 10836, 9420, 5760, 2352, 576, 64;
1, 21, 203, 1197, 4809, 13923, 29953, 48639, 59906, 55692, 38472, 19152, 6496, 1344, 128;
1, 24, 268, 1848, 8806, 30744, 81340, 166344, 265729, 332688, 325360, 245952, 140896, 59136, 17152, 3072, 256; ...
where row n equals the coefficients in (1+x)^n*(1+2*x)^n.

Prog

(PARI) {T(n, k)=polcoeff((1+3*x+2*x^2+x*O(x^k))^n, k)}
{for(n=0, 10, for(k=0, 2*n, print1(T(n, k), ", ")); print(""))}

Crossrefs

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

Cf. A200537.

Sequence in context: A196843 A367023 A143778 * A164645 A115755 A300003

Adjacent sequences: A200533 A200534 A200535 * A200537 A200538 A200539

Keyword

nonn,tabf

Author

Paul D. Hanna, Nov 18 2011

Status

approved