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A254933
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Triangle used for the integral of even powers of the sine and cosine functions.
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2
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-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
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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I. S. Gradstein and I. M. Ryshik, Tables of series, products, and integrals, Volume 1, Verlag Harri Deutsch, 1981.
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LINKS
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FORMULA
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T(n, k) = (-1)^k*n!/k*binomial(2*n,n - k), 1 <= k <= n.
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(PARI) {T(n, k) = if( n<1 || n>k, 0, (-1)^k*n!/k*binomial(2*n, n-k))}; /* Michael Somos, Jun 07 2019 */
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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