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

0,4

Comments

Row sums are A025192(n).

Links

Indranil Ghosh, Rows 0..100, flattened

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).

Example

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;

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 *)

Crossrefs

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

Cf. Diagonals: A000012, A032766.

Sequence in context: A292770 A242107 A242108 * A247582 A359795 A224652

Adjacent sequences: A238938 A238939 A238940 * A238942 A238943 A238944

Keyword

nonn,tabl

Author

Philippe Deléham, Mar 07 2014

Extensions

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

Status

approved