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

Author

Peter Luschny, Aug 06 2009

Keywords

• nonn
• tabl
• changed

Comments

Triangle read by rows.

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

Examples

			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;
		

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.

Crossrefs

Row sums are A163843.

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

Programs

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 *)

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.