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A156732
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A triangular sequence: t(n,m)=((n + 1 - 2*m)^2/(n + 1 - m))*Binomial[n, m].
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0
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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; internal format)
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OFFSET
| 0,4
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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 (maxwell8(AT)yorku.ca), Sep 22 2010]
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REFERENCES
| J. Riordan, Combinatorial Identities, Wiley, 1968
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LINKS
| J.M. Campbell, Double Series Involving Binomial Coefficients and the Sine Integral, arXiv, Cornell University Library, 2010, p. 3-4. [From John M. Campbell (maxwell8(AT)yorku.ca), Sep 22 2010]
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FORMULA
| t(n,m)=((n + 1 - 2*m)^2/(n + 1 - m))*Binomial[n, m].
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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}
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MATHEMATICA
| Table[Table[((n + 1 - 2* m)^2/(n + 1 - m))*Binomial[n, m], {m, 1, n}], {n, 1, 10}];
Flatten[%]
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CROSSREFS
| Sequence in context: A055951 A165032 A088374 * A200341 A101980 A058536
Adjacent sequences: A156729 A156730 A156731 * A156733 A156734 A156735
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KEYWORD
| nonn,tabl,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 14 2009
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