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A156732 A triangular sequence: t(n,m)=((n + 1 - 2*m)^2/(n + 1 - m))*Binomial[n, m]. 0
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Row sums are:

{0, 2, 10, 36, 108, 290, 726, 1736, 4024, 9126, 20370,...}.

From the Riordan Identity:

2^n=Table[Sum[(( n + 1 - 2*m)^2/(n + 1 - m))*Binomial[n, m], {m, 0, Floor[n/2]}], {n, 0, 10}].

Because of the central zeros,

I call it an hollow sequence.

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 int (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 int (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))*((2*k)+1)Si(((2*k)+1)arcsin(x))+ sum^k_n=1 ((-1)^n)*(((1-(2*n))^2)/(k+1-n))*Binomial[2*k, k+n]*Si(((2*n)-1)arcsin(x))). [From John M. Campbell, Sep 22 2010]

REFERENCES

J. Riordan, Combinatorial Identities, Wiley, 1968

LINKS

Table of n, a(n) for n=0..54.

J.M. Campbell, Double Series Involving Binomial Coefficients and the Sine Integral, arXiv, Cornell University Library, 2010, p. 3-4. [From John M. Campbell, Sep 22 2010]

FORMULA

t(n,m)=((n + 1 - 2*m)^2/(n + 1 - m))*Binomial[n, m].

EXAMPLE

{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

Table[Table[((n + 1 - 2* m)^2/(n + 1 - m))*Binomial[n, m], {m, 1, n}], {n, 1, 10}];

Flatten[%]

CROSSREFS

Sequence in context: A055951 A165032 A088374 * A200341 A101980 A209134

Adjacent sequences:  A156729 A156730 A156731 * A156733 A156734 A156735

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, Feb 14 2009

STATUS

approved

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Last modified January 23 04:16 EST 2020. Contains 331168 sequences. (Running on oeis4.)