A385577 Array read by ascending antidiagonals: A(n,m) = n*Pochhammer(n+1,m+1)/(m+2).

0, 1, 0, 3, 2, 0, 6, 8, 6, 0, 10, 20, 30, 24, 0, 15, 40, 90, 144, 120, 0, 21, 70, 210, 504, 840, 720, 0, 28, 112, 420, 1344, 3360, 5760, 5040, 0, 36, 168, 756, 3024, 10080, 25920, 45360, 40320, 0, 45, 240, 1260, 6048, 25200, 86400, 226800, 403200, 362880, 0

Offset

0,4

References

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 20.

Links

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

Paul W. Haggard and Bonnie L. Sadler, A Generalization of Triangular Numbers, International Journal of Mathematical Education in Science and Technology 25 (2): 195-202, (1994).

Formula

Sum_{m=0..n} A(n-m,m) = A006231(n+1).

A(n,1) = A007290(n+2).

A(1,n) = A000142(n+1).

A(2,n) = A001048(n+2).

A(3,n) = abs(A238474(n+1)).

A(n,n) = n!*A002740(n+2)

Example

Array begins as:
   0,  0,   0,    0,     0,     0,      0, ...
   1,  2,   6,   24,   120,   720,   5040, ...
   3,  8,  30,  144,   840,  5760,  45360, ...
   6, 20,  90,  504,  3360, 25920, 226800, ...
  10, 40, 210, 1344, 10080, 86400, 831600, ...
  ...

Mathematica

A[n_, m_]:=n*Pochhammer[n+1, m+1]/(m+2); Table[A[n-m, m], {n, 0, 9}, {m, 0, n}]//Flatten

Crossrefs

Cf. A000142, A001048, A002740, A006231, A007290, A068424, A238474.

Cf. A000217 (m=0), A033487 (m=2), A158874 (m=3).

Cf. A000004 (n=0).

Sequence in context: A159584 A257653 A378712 * A246834 A319730 A262294

Adjacent sequences: A385574 A385575 A385576 * A385578 A385579 A385580

Keyword

nonn,easy,tabl

Author

Stefano Spezia, Jul 03 2025

Status

approved