A254933 Triangle used for the integral of even powers of the sine and cosine functions.

-1, -8, 1, -90, 18, -2, -1344, 336, -64, 6, -25200, 7200, -1800, 300, -24, -570240, 178200, -52800, 11880, -1728, 120, -15135120, 5045040, -1681680, 458640, -91728, 11760, -720, -461260800, 161441280, -58705920, 18345600, -4515840, 806400, -92160, 5040, -15878903040, 5774146560, -2245501440, 777288960, -222082560, 49351680, -7931520, 816480, -40320

Offset

1,2

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 this triangle to cover all even powers.

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

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

See the Gradstein - Ryshik reference, 2.513 1., p. 168, (after changing the summation variable k -> n - k).

The unsigned triangle T(n, k) appears in the formula 2^(2*n)*n!*int(cos^{2*n}x dx) = (2*n)!/n!*x + Sum {k = 1..n} |T(n, k)|*sin(2*k*x), n >= 1.

See the Gradstein - Ryshik reference, 2.513 3., p. 168.

For the integral of odd powers of sine and cosine see A254932.

Let f_n(x) := Sum_{k=1..n} -T(n, k) * sin(k*x). Then f_n(x) = x + O(x^(2*n+1)). - Michael Somos, Jun 07 2019

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=1..45.

Wolfdieter Lang, The first ten rows.

Formula

T(n, k) = (-1)^k*n!/k*binomial(2*n,n - k), 1 <= k <= n.

Example

The triangle T(n, k) begins:
n\k      1      2      3     4     5   6 ...
1:      -1
2:      -8      1
3:     -90     18     -2
4:   -1344    336    -64     6
5:  -25200   7200  -1800   300   -24
6: -570240 178200 -52800 11880 -1728 120
...
For more rows see the link.
n=3: 2^6*3!*int((sin x)^6 dx)  = 120*x - 90*sin(2*x) + 18*sin(4*x) - 2*sin(*x),
     that is: int((sin x)^6 dx) = (5/16)*x -(15/64)*sin(2*x) + (3/64)*sin(4*x) - (1/192)*sin(6*x).
     2^6*3!*int((cos x)^6 dx)  = 120*x  + 90*sin(2*x) + 18*sin(4*x) + 2*sin(6*x),
     that is: int((cos x)^6 dx) =  (5/16)*x + (15/64)*sin(2*x) + (3/64)*sin(4*x) + (1/192)*sin(6*x).
As defined in my comment: f_1(x) = sin(x) = x - x^3/6 + O(x^5), f_2(x) = (8*sin(x) - sin(2x)/6 = x - x^5/30 + O(x^7), f_3(x) = (90*sin(x) - 18*sin(2x) + 2*sin(3))/60 = x - x^7/140 + O(x^9). - Michael Somos, Jun 07 2019

Mathematica

T[n_, k_] := (-1)^k*n!/k*Binomial[2*n, n-k]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 18 2015 *)

Prog

(PARI) {T(n, k) = if( n<1 || n>k, 0, (-1)^k*n!/k*binomial(2*n, n-k))}; /* Michael Somos, Jun 07 2019 */

Crossrefs

Cf. A254932 (odd powers).

Sequence in context: A143499 A114152 A347111 * A174503 A048786 A240955

Adjacent sequences: A254930 A254931 A254932 * A254934 A254935 A254936

Keyword

sign,tabl,easy

Author

Wolfdieter Lang, Feb 16 2015

Status

approved