A255935 Triangle read by rows: a(n) = Pascal's triangle A007318(n) + A197870(n+1).

0, 1, 2, 1, 2, 0, 1, 3, 3, 2, 1, 4, 6, 4, 0, 1, 5, 10, 10, 5, 2, 1, 6, 15, 20, 15, 6, 0, 1, 7, 21, 35, 35, 21, 7, 2, 1, 8, 28, 56, 70, 56, 28, 8, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 2, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 0

Offset

0,3

Comments

Consider the difference table of a sequence with A000004(n)=0's as main diagonal. (Example: A000045(n).) We call this sequence an autosequence of the first kind.

Based on Pascal's triangle, a(n) =

0, T1

1, 2,

1, 2, 0,

1, 3, 3, 2,

etc.

transforms every sequence s(n) in an autosequence of the first kind via the multiplication by the triangle

s0, T2

s0, s1,

s0, s1, s2,

s0, s1, s2, s3,

etc.

Examples.

1) s(n) = A198631(n)/A006519(n+1), the second fractional Euler numbers (see A209308). This yields 0*1, 1*1+2*1/2=2, 1*1+2*1/2+0*0=2, 1*1+3*1/2++3*0+2*(-1/4)=2, ... .

The autosequence is 0 followed by 2's or 2*(0,1,1,1,1,1,1,1,... = b(n)).

b(n), the basic autosequence of the first kind, is not in the OEIS (see A140575 and A054977).

2) s(n) = A164555(n)/A027642(n), the second Bernoulli numbers, yields 0,2,2,3,4,5,6,7,... = A254667(n).

Row sums of T1: A062510(n) = 3*A001045(n).

Antidiagonal sums of T1: A111573(n).

With 0's instead of the spaces, every column, i.e.,

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

0, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... = A001477(n) with 0 instead of 1 = A254667(n)

0, 0, 0, 3, 6, 10, 15, 21, 28, 36, 45, ... = A161680(n) with 0 instead of 1

0, 0, 0, 2, 4, 10, 20, 35, 56, 84, 120, ...

etc., is an autosequence of the first kind.

With T(0,0) = 1, it is (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, May 24 2015

Links

Table of n, a(n) for n=0..65.

Formula

a(n) = Pascal's triangle A007318(n) with main diagonal A010673(n) (= period 2: repeat 0, 2) instead of 1's=A000012(n).

a(n) = reversal abs(A140575(n)).

a(n) = A007318(n) + A197870(n+1).

T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 0, T(1,0) = 1, T(1,1) = 2, T(n,k) = 0 if k>n or if k<0 . - Philippe Deléham, May 24 2015

G.f.: (-1-2*x*y+x^2*y+x^2*y^2)/((x*y+1)*(x*y+x-1)) - 1. - R. J. Mathar, Aug 12 2015

Example

Triangle starts:
0;
1, 2;
1, 2, 0;
1, 3, 3, 2;
1, 4, 6, 4, 0;
1, 5, 10, 10, 5, 2;
1, 6, 15, 20, 15, 6, 0;
...

Mathematica

a[n_, k_] := If[k == n, 2*Mod[n, 2], Binomial[n, k]]; Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 23 2015 *)

Crossrefs

Cf. A001045, A001477, A010673, A011973, A054977, A062510, A007318, A197870, A006519, A198631, A140575, A209308, A164555, A027642, A111573, A161680, A254667.

Sequence in context: A143335 A099505 A156837 * A374394 A129559 A129680

Adjacent sequences: A255932 A255933 A255934 * A255936 A255937 A255938

Keyword

nonn,tabl

Author

Paul Curtz, Mar 11 2015

Status

approved