A163840 Triangle interpolating the binomial transform of the swinging factorial (A163865) with the swinging factorial (A056040).

1, 2, 1, 5, 3, 2, 16, 11, 8, 6, 47, 31, 20, 12, 6, 146, 99, 68, 48, 36, 30, 447, 301, 202, 134, 86, 50, 20, 1380, 933, 632, 430, 296, 210, 160, 140, 4251, 2871, 1938, 1306, 876, 580, 370, 210, 70, 13102, 8851, 5980, 4042, 2736, 1860, 1280, 910, 700, 630

Offset

0,2

Comments

Triangle read by rows.

An analog to the binomial triangle of the factorials (A076571).

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

Peter Luschny, Swinging Factorial.

Formula

T(n,k) = Sum_{i=k..n} binomial(n-k,n-i)*i$ where i$ denotes the swinging factorial of i (A056040), for n >= 0, k >= 0.

Example

Triangle begins
    1;
    2,   1;
    5,   3,   2;
   16,  11,   8,   6;
   47,  31,  20,  12,  6;
  146,  99,  68,  48, 36, 30;
  447, 301, 202, 134, 86, 50, 20;

Maple

SumTria := proc(f, n, display) local m, A, j, i, T; T:=f(0);
for m from 0 by 1 to n-1 do A[m] := f(m);
for j from m by -1 to 1 do A[j-1] := A[j-1] + A[j] od;
for i from 0 to m do T := T, A[i] od;
if display then print(seq(T[i], i=nops([T])-m..nops([T]))) fi;
od; subsop(1=NULL, [T]) end:
swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
# Computes n rows of the triangle:
A163840 := n -> SumTria(swing, n, true);

Mathematica

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[Binomial[n - k, n - i]*sf[i], {i, k, n}]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

Crossrefs

Row sums are A163843.

Cf. A056040, A163865, A163841, A163842, A163650.

Sequence in context: A067323 A106534 A123346 * A122833 A193692 A344028

Adjacent sequences: A163837 A163838 A163839 * A163841 A163842 A163843

Keyword

nonn,tabl

Author

Peter Luschny, Aug 06 2009

Status

approved