A134558 Array read by antidiagonals, a(n,k) = gamma(n+1,k)*e^k, where gamma(n,k) is the upper incomplete gamma function and e is the exponential constant 2.71828...

1, 1, 1, 2, 2, 1, 6, 5, 3, 1, 24, 16, 10, 4, 1, 120, 65, 38, 17, 5, 1, 720, 326, 168, 78, 26, 6, 1, 5040, 1957, 872, 393, 142, 37, 7, 1, 40320, 13700, 5296, 2208, 824, 236, 50, 8, 1, 362880, 109601, 37200, 13977, 5144, 1569, 366, 65, 9, 1, 3628800, 986410, 297856

Offset

0,4

Links

Table of n, a(n) for n=0..57.

Eric Weisstein's World of Mathematics, Incomplete Gamma Function.

Wikipedia, Incomplete gamma function.

Formula

a(n,k) = gamma(n+1,k)*e^k = Sum_{m=0..n} m!*binomial(n,m)*k^(n-m).

a(n,k) = n*a(n-1,k) + k^n for n,k > 0.

E.g.f. (by columns) is e^(kx)/(1-x).

a(n,k) = the binomial transform by columns of a(n,k-1).

Conjecture: a(n,k) is the permanent of the n X n matrix with k+1 on the main diagonal and 1 elsewhere.

Example

Square array begins:
    1,    1,    1,     1,     1,     1,      1, ...
    1,    2,    3,     4,     5,     6,      7, ...
    2,    5,   10,    17,    26,    37,     50, ...
    6,   16,   38,    78,   142,   236,    366, ...
   24,   65,  168,   393,   824,  1569,   2760, ...
  120,  326,  872,  2208,  5144, 10970,  21576, ...
  720, 1957, 5296, 13977, 34960, 81445, 176112, ...

Mathematica

T[n_, k_] := Gamma[n+1, k]*E^k; Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] //Flatten (* Amiram Eldar, Jun 27 2020 *)

Crossrefs

Cf. a(n, 0) = A000142(n); a(n, 1) = A000522(n); a(n, 2) = A010842(n); a(n, 3) = A053486(n); a(n, 4) = A053487(n); a(n, 5) = A080954(n); a(n, 6) = A108869(n); a(1, k) = A000027(k+1); a(2, k) = A002522(k+1); a(n, n) = A063170(n); a(n, n+1) = A001865(n+1); a(n, n+2) = A001863(n+2).

Another version: A089258.

A transposed version: A080955.

Cf. A001113.

Sequence in context: A171486 A127743 A125278 * A344639 A230420 A137381

Adjacent sequences: A134555 A134556 A134557 * A134559 A134560 A134561

Keyword

nonn,tabl

Author

Ross La Haye, Jan 22 2008

Extensions

More terms from Amiram Eldar, Jun 27 2020

Status

approved