A177944 Array T(n,m) = 1/Beta(n+1, m+1) - n - m read by antidiagonals.

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 26, 16, 1, 1, 25, 55, 55, 25, 1, 1, 36, 99, 134, 99, 36, 1, 1, 49, 161, 273, 273, 161, 49, 1, 1, 64, 244, 496, 622, 496, 244, 64, 1, 1, 81, 351, 831, 1251, 1251, 831, 351, 81, 1

Offset: 0

Author

Roger L. Bagula, May 15 2010

Keywords

• nonn
• easy
• tabl

Comments

Antidiagonal sums are 1, 2, 6, 20, 60, 162, 406, 968, 2232, 5030, ... = (d+1)*(2^d-d).

The values appear to be related to binomial(n,m)^2.

Examples

			The array starts in row n=0, column m=0 as:
1,....1,....1,....1,.....1,.....1,.....1,.....1,
1,....4,....9,...16,....25,....36,....49,....64, A000290
1,....9,...26,...55,....99,...161,...244,...351, A154560
1,...16,...55,..134,...273,...496,...831,..1310,
1,...25,...99,..273,...622,..1251,..2300,..3949,
1,...36,..161,..496,..1251,..2762,..5533,.10284,
1,...49,..244,..831,..2300,..5533,.12000,.24011,
1,...64,..351,.1310,..3949,.10284,.24011,.51466,
		

Programs

Mathematica

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

Formula

T(n,m) = Gamma(n+m+2)/(Gamma(n+1)*Gamma(m+1)) - n - m = T(m,n).

Extensions

Examples written in natural order, closed formula for antidiag. sum - The Assoc. Eds. of the OEIS, Nov 03 2010