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%I #5 Dec 10 2016 17:35:16
%S 1,1,1,1,4,1,1,8,8,1,1,13,22,13,1,1,19,45,45,19,1,1,26,79,110,79,26,1,
%T 1,34,126,224,224,126,34,1,1,43,188,406,518,406,188,43,1,1,53,267,678,
%U 1050,1050,678,267,53,1
%N A symmetrical triangle sequence based on the beta function inverse and the spotlight tile A051601 as antidiagonal: t(n,m) = 1/Integrate[(-1 + t)^n/t^(m + n + 2), {t, 1, Infinity}] - (-2 Binomial[m + n, m] + Binomial[2 + m + n, 1 + m]).
%C Beta[n+1,m+1] = Integrate[(-1 + t)^n/t^(m + n + 2), {t, 1, Infinity}].
%C Row sums are {1, 2, 6, 18, 50, 130, 322, 770, 1794, 4098, ...}.
%C The triangle modulo 2 is Sierpinski:
%C ListDensityPlot[Table[Table[Mod[ t[n, m], 2], {m, 0, 64}], {n, 0, 64}], Frame -> False, Mesh -> False].
%F t(n,m) = 1/Integrate[(-1 + t)^n/t^(m + n + 2), {t, 1, Infinity}] - (-2 Binomial[m + n, m] + Binomial[2 + m + n, 1 + m]);
%F out_n,m = antidiagonal(t(n,m)) = A003506(n,m) - A051601(n,m).
%e {1},
%e {1, 1},
%e {1, 4, 1},
%e {1, 8, 8, 1},
%e {1, 13, 22, 13, 1},
%e {1, 19, 45, 45, 19, 1},
%e {1, 26, 79, 110, 79, 26, 1},
%e {1, 34, 126, 224, 224, 126, 34, 1},
%e {1, 43, 188, 406, 518, 406, 188, 43, 1},
%e {1, 53, 267, 678, 1050, 1050, 678, 267, 53, 1}
%t Clear[t, n]
%t t[n_, m_] = 1/Integrate[(-1 + t)^n/t^(m + n + 2), {t, 1, Infinity}] - (-2 Binomial[m + n, m] + Binomial[2 + m + n, 1 + m]);
%t a = Table[Table[t[n, m], {n, 0, 10}], {m, 0, 10}];
%t Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];
%t Flatten[%]
%Y Cf. A051601, A003506.
%K nonn,tabl,uned
%O 0,5
%A _Roger L. Bagula_, May 15 2010
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