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A177947 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]).

%I A177947 #5 Dec 10 2016 17:35:16
%S A177947 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 A177947 1,34,126,224,224,126,34,1,1,43,188,406,518,406,188,43,1,1,53,267,678,
%U A177947 1050,1050,678,267,53,1
%N A177947 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 A177947 Beta[n+1,m+1] = Integrate[(-1 + t)^n/t^(m + n + 2), {t, 1, Infinity}].
%C A177947 Row sums are {1, 2, 6, 18, 50, 130, 322, 770, 1794, 4098, ...}.
%C A177947 The triangle modulo 2 is Sierpinski:
%C A177947 ListDensityPlot[Table[Table[Mod[ t[n, m], 2], {m, 0, 64}], {n, 0, 64}], Frame -> False, Mesh -> False].
%F A177947 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 A177947 out_n,m = antidiagonal(t(n,m)) = A003506(n,m) - A051601(n,m).
%e A177947 {1},
%e A177947 {1, 1},
%e A177947 {1, 4, 1},
%e A177947 {1, 8, 8, 1},
%e A177947 {1, 13, 22, 13, 1},
%e A177947 {1, 19, 45, 45, 19, 1},
%e A177947 {1, 26, 79, 110, 79, 26, 1},
%e A177947 {1, 34, 126, 224, 224, 126, 34, 1},
%e A177947 {1, 43, 188, 406, 518, 406, 188, 43, 1},
%e A177947 {1, 53, 267, 678, 1050, 1050, 678, 267, 53, 1}
%t A177947 Clear[t, n]
%t A177947 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 A177947 a = Table[Table[t[n, m], {n, 0, 10}], {m, 0, 10}];
%t A177947 Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];
%t A177947 Flatten[%]
%Y A177947 Cf. A051601, A003506.
%K A177947 nonn,tabl,uned
%O A177947 0,5
%A A177947 _Roger L. Bagula_, May 15 2010