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]).

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, 1, 34, 126, 224, 224, 126, 34, 1, 1, 43, 188, 406, 518, 406, 188, 43, 1, 1, 53, 267, 678, 1050, 1050, 678, 267, 53, 1

Offset

0,5

Comments

Beta[n+1,m+1] = Integrate[(-1 + t)^n/t^(m + n + 2), {t, 1, Infinity}].

Row sums are {1, 2, 6, 18, 50, 130, 322, 770, 1794, 4098, ...}.

The triangle modulo 2 is Sierpinski:

ListDensityPlot[Table[Table[Mod[ t[n, m], 2], {m, 0, 64}], {n, 0, 64}], Frame -> False, Mesh -> False].

Links

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

Formula

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]);

out_n,m = antidiagonal(t(n,m)) = A003506(n,m) - A051601(n,m).

Example

{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},
{1, 34, 126, 224, 224, 126, 34, 1},
{1, 43, 188, 406, 518, 406, 188, 43, 1},
{1, 53, 267, 678, 1050, 1050, 678, 267, 53, 1}

Mathematica

Clear[t, n]
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]);
a = Table[Table[t[n, m], {n, 0, 10}], {m, 0, 10}];
Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];
Flatten[%]

Crossrefs

Cf. A051601, A003506.

Sequence in context: A141541 A334552 A347676 * A132789 A319251 A100754

Adjacent sequences: A177944 A177945 A177946 * A177948 A177949 A177950

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, May 15 2010

Status

approved