A156732 Triangle T(n, k) = ((n-2*k)^2/(n-k+1))*binomial(n+1, k+1), read by rows.

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

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

Cf. A000290, A002662.

Sequence in context: A165032 A345203 A088374 * A200341 A101980 A209134

Adjacent sequences: A156729 A156730 A156731 * A156733 A156734 A156735

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 14 2009

Extensions

Edited by G. C. Greubel, Feb 28 2021

Status

approved