A238941 Triangle T(n,k), read by rows given by (1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

1, 1, 1, 2, 3, 1, 5, 8, 4, 1, 13, 21, 13, 6, 1, 34, 55, 40, 25, 7, 1, 89, 144, 120, 90, 33, 9, 1, 233, 377, 354, 300, 132, 51, 10, 1, 610, 987, 1031, 954, 483, 234, 62, 12, 1, 1597, 2584, 2972, 2939, 1671, 951, 308, 86, 13, 1, 4181, 6765, 8495, 8850, 5561, 3573, 1345, 480, 100, 15, 1

Offset: 0

Author

Philippe Deléham, Mar 07 2014

Keywords

• nonn
• tabl

Comments

Row sums are A025192(n).

Examples

			Triangle begins:
1;
1, 1;
2, 3, 1;
5, 8, 4, 1;
13, 21, 13, 6, 1;
34, 55, 40, 25, 7, 1;
89, 144, 120, 90, 33, 9, 1;
233, 377, 354, 300, 132, 51, 10, 1;
		

Links

• Indranil Ghosh, Rows 0..100, flattened

Crossrefs

Cf. Columns: A001519, A001906, A238846, A001871.

Cf. Diagonals: A000012, A032766.

Programs

Mathematica

nmax=10; Column[CoefficientList[Series[CoefficientList[Series[(1 - 2*x + x*y)/(1 - 3*x + x^2 - x^2*y^2), {x, 0, nmax }], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 14 2017 *)

Formula

G.f. for the column k: x^k*(1-2*x)^A059841(k)/(1-3*x+x^2)^A008619(k).

G.f.: (1-2*x+x*y)/(1-3*x+x^2-x^2*y^2).

T(n,k) = 3*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.

Sum_{k = 0..n} T(n,k)*x^k = A000007(n), A001519(n), A025192(n), A030195(n+1) for x = -1, 0, 1, 2 respectively.

Sum_{k = 0..n} T(n,k)*3^k = A015525(n) + A015525(n+1).

Extensions

Data section corrected and extended by Indranil Ghosh, Mar 14 2017