1,2

T(n,k)=C(n+1,k+1)*sin(pi*(k+1)/2); - Paul Barry (pbarry(AT)wit.ie), May 21 2006

The trigonometric expansion of sin(4x) is 4*cos(x)^3*sin(x) - 4*cos(x)*sin(x)^3, so the fourth row is 4, 0, -4, 0.

Triangle begins:

1

2 0

3 0 -1

4 0 -4 0

5 0 -10 0 1

6 0 -20 0 6 0

7 0 -35 0 21 0 -1

8 0 -56 0 56 0 -8 0

Flatten[ Table[ Plus @@ CoefficientList[ TrigExpand[ Sin[n*x]], {Sin[x], Cos[x]}], {n, 13}]]

First column is A000027 = C(n, 1), third column is A000292 = C(n, 3), fifth column is A000389 = C(n, 5), seventh column is A000580 = C(n, 7), ninth column is A000582 = C(n, 9).

A001288 = C(n, 11), A010966 = C(n, 13), A010968 = C(n, 15), A010970 = C(n, 17), A010972 = C(n, 19),

A010974 = C(n, 21), A010976 = C(n, 23), A010978 = C(n, 25), A010980 = C(n, 27), A010982 = C(n, 29),

A010984 = C(n, 31), A010986 = C(n, 33), A010988 = C(n, 35), A010990 = C(n, 37), A010992 = C(n, 39),

A010994 = C(n, 41), A010996 = C(n, 43), A010998 = C(n, 45), A011090 = C(n, 47), A017713 = C(n, 49)

Another version of the triangle in A034867. Cf. A096754.

A017715 = C(n, 51), A017717 = C(n, 53), A017719 = C(n, 55), A017721 = C(n, 57), etc.

Sequence in context: A088673 A035614 A133735 this_sequence A163496 A092241 A128144

Adjacent sequences: A095701 A095702 A095703 this_sequence A095705 A095706 A095707

sign,tabl

Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 06 2004