A254932 Triangle used for the integral of odd powers of the sine and cosine functions.

-1, -9, 1, -150, 25, -3, -3675, 735, -147, 15, -119070, 26460, -6804, 1215, -105, -4802490, 1143450, -343035, 81675, -12705, 945, -231891660, 57972915, -19324305, 5521230, -1171170, 159705, -10395, -13043905875, 3381753375, -1217431215, 395269875, -102477375, 19348875, -2338875, 135135, -837708621750, 223388965800, -85293968760, 30462131700, -9112603500, 2130219000, -360498600, 39054015, -2027025

Offset

0,2

Comments

This entry originated from a proposal by Ozgur Cem Birler for the integral of the fourth power of sin(x). Peter Bala suggested the use of triangles to cover all even and odd powers. See A254932.

The signed triangle T(n, k) appears in the formula

2^(2*n)*(2*n+1)!!*int(sin^{2*n+1}x dx) = Sum_{k=0..n} T(n, k)*cos((2*k+1)*x), n >= 0.

See the Gradstein - Ryshik reference, 2.513 2., p. 168. Here (2*n+1)!! = A001147(n+1), n >= 0.

The unsigned triangle T(n, k) appears in the formula

2^(2*n)* (2*n+1)!!*int(cos^{2*n+1}x dx) = Sum_{k=0..n} |T(n, k)|*sin((2*k+1)*x), n >= 0.

See the Gradstein & Ryshik reference, 2.513 4., p. 169.

References

I. S. Gradstein and I. M. Ryshik, Tables of series, products, and integrals, Volume 1, Verlag Harri Deutsch, 1981.

Links

Table of n, a(n) for n=0..44.

Wolfdieter Lang, The first 9 rows and more.

Formula

T(n, k) = (-1)^(k+1)*binomial(2*n+1,n-k)*(2*n+1)!!/(2*k+1), 0 <= k <= n, with the double factorials (2*n+1)!! = A001147(n+1).

Example

The triangle T(n, k) begins:
n\k       0       1       2     4      5   6...
0:       -1
1:       -9       1
2:     -150      25      -3
3:    -3675     735    -147    15
4:  -119070   26460   -6804  1215   -105
5: -4802490 1143450 -343035 81675 -12705 945
...
For more rows see the link.
n=2: 2^4*(1*3*5)*int((sin x)^5 dx)  = -150*cos(x) + 25*cos(3*x) - 3*cos(5*x),
  that is: int((sin x)^5 dx) = -(5/8)*cos(x) + (5/48)*cos(3*x) - (1/80)*cos(5*x).
     2^4*(1*3*5)*int((cos x)^5 dx)  = +150*sin(x) + 25*sin(3*x) + 3*sin(5*x),
  that is: int((cos x)^5 dx) = (5/8)*sin(x) + (5/48)*sin(3*x) + (1/80)*sin(5*x).

Mathematica

T[n_, k_] := (-1)^(k+1)*Binomial[2*n+1, n-k]*(2*n+1)!!/(2*k+1); Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 18 2015 *)

Crossrefs

Cf. A254933 (even powers).

Sequence in context: A136238 A113394 A243754 * A223511 A051231 A258437

Adjacent sequences: A254929 A254930 A254931 * A254933 A254934 A254935

Keyword

sign,easy,tabl

Author

Wolfdieter Lang, Feb 15 2015

Status

approved