OFFSET
0,4
COMMENTS
The general formula for integrals of the form int (x^(2*k))/((arcsin(x))^2) dx involves the triangular sequence t(2*k, n). For example, the solution to the integral Integral x^6/((arcsin(x))^2) dx involves the following sequence: -5*Si(arcsin(x)) + 27*Si(3*arcsin(x)) - 25*Si(5*arcsin(x)), where Si represents the sine integral. The sequence of integers 5, 27, 25 corresponds to the sixth row of this triangular sequence. The general formula for the integral Integral x^(2*k)/((arcsin(x))^2) dx is: (1/(2^(2*k)))*( -((2^(2*k)*sqrt(1-x^2)*(x^(2*k)))/arcsin(x) )+ (-1)^(k+1)*(1+(2*k))*Si((1+2*k)*arcsin(x)) + Sum_{n=1..k} (-1)^n*((1-2*n)^2/(k-n+1))*binomial(2*k, k+n)*Si((2*n-1)*arcsin(x)) ). - John M. Campbell, Sep 22 2010
REFERENCES
J. Riordan, Combinatorial Identities, Wiley, 1968.
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
John. M. Campbell, Double Series Involving Binomial Coefficients and the Sine Integral, arXiv:1009.0236 [math.NT], 2010, p. 3-4. [From John M. Campbell, Sep 22 2010]
FORMULA
T(n, k) = ((n-2*k)^2/(n-k+1))*binomial(n+1, k+1).
Sum_{k=0..floor(n/2)} T(n-1, k-1) = 2^n.
From G. C. Greubel, Feb 28 2021: (Start)
T(n, k) = T(n, n-k).
T(n, k) = ((n-2*k)^2/(n+2))*binomial(n+2, k+1).
Sum_{k=0..n} T(n, k) = 2*(2^(n+1) -n-2) = 4*A002662(n) + 2*n^2. (End)
EXAMPLE
Triangle begins as:
0;
1, 1;
4, 0, 4;
9, 2, 2, 9;
16, 10, 0, 10, 16;
25, 27, 5, 5, 27, 25;
36, 56, 28, 0, 28, 56, 36;
49, 100, 84, 14, 14, 84, 100, 49;
64, 162, 192, 84, 0, 84, 192, 162, 64;
81, 245, 375, 270, 42, 42, 270, 375, 245, 81;
MATHEMATICA
T[n_, k_]:= ((n-2*k)^2/(n-k+1))*Binomial[n+1, k+1];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 28 2021 *)
PROG
(Sage)
def A156732(n, k): return ((n-2*k)^2/(n+2))*binomial(n+2, k+1)
flatten([[A156732(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 28 2021
(Magma)
A156732:= func< n, k | ((n-2*k)^2/(n+2))*Binomial(n+2, k+1) >;
[A156732(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 28 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 14 2009
EXTENSIONS
Edited by G. C. Greubel, Feb 28 2021
STATUS
approved