A080418 Generalized Pascal triangle.

1, 1, 3, 1, 2, 4, 1, 5, 5, 5, 1, 4, 11, 9, 6, 1, 7, 14, 21, 14, 7, 1, 6, 22, 34, 36, 20, 8, 1, 9, 27, 57, 69, 57, 27, 9, 1, 8, 37, 83, 127, 125, 85, 35, 10, 1, 11, 44, 121, 209, 253, 209, 121, 44, 11, 1, 10, 56, 164, 331, 461, 463, 329, 166, 54, 12

Offset

1,3

Links

Table of n, a(n) for n=1..66.

Formula

T(n, 1)=1, T(n, k)=0 for k>n, T(n, 2) = T(n-1, 1) + T(n-1, 2) + 2*(-1)^n, T(n, k) = T(n-1, k-1) + T(n-1, k) + (-1)^(n+k) for k>2. [corrected by Frank M Jackson, Mar 27 2012]

Example

First rows are:
{1},
{1,3},
{1,2,4},
{1,5,5,5},
{1,4,11,9,6},
{1,7,14,21,14,7},
...
For example, 2 = 1 + 3 - 2, 5 = 1 + 2 + 2; 11 = 5 + 5 + 1, 14 = 4 + 11 - 1.

Mathematica

t[n_, k_] := t[n, k]=Which[k==1, 1, n<k, 0, k==2, t[n-1, 1]+t[n-1, 2]+2(-1)^n, k>2, t[n-1, k-1] + t[n-1, k] + (-1)^(n+k)]; Flatten[Table[t[n, k], {n, 1, 20}, {k, 1, n}]] (* Frank M Jackson, Mar 27 2012 *)

Crossrefs

Columns include A000012, A004442, A000217+(-1)^n, A000292+(-1)^n and in general, binomial(n+k, k)+(-1)^n. Diagonals include A000096, A063258.

Sequence in context: A130419 A083110 A059016 * A073892 A280348 A332397

Adjacent sequences: A080415 A080416 A080417 * A080419 A080420 A080421

Keyword

easy,nonn,tabl

Author

Paul Barry, Feb 18 2003

Extensions

Terms corrected and extended by Frank M Jackson, Mar 27 2012

Status

approved