A163865 -id:A163865 - cp's OEIS frontend

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

Peter Luschny, Aug 06 2009

Triangle read by rows.

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

			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;

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.

Row sums are A163843.

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

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);

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 *)

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.

A163649 Triangle interpolating between (-1)^n (A033999) and A056040(n), read by rows.

Original entry on oeis.org

1, -1, 1, 1, -2, 2, -1, 3, -6, 6, 1, -4, 12, -24, 6, -1, 5, -20, 60, -30, 30, 1, -6, 30, -120, 90, -180, 20, -1, 7, -42, 210, -210, 630, -140, 140, 1, -8, 56, -336, 420, -1680, 560, -1120, 70

Offset: 0

Peter Luschny, Aug 02 2009

Given T(n,k) = (-1)^(n-k)*floor(k/2)!^(-2)*n!/(n-k)!, let A(n,k) = abs(T(n,k)) be the coefficients of the polynomials Sum_{k=0..n} binomial(n,k)*A056040(k)*q^k. Substituting q^k -> 1/(floor(k/2)+1) in the polynomials gives the extended Motzkin numbers A189912. (See A089627 for the Motzkin numbers and A194586 for the complementary Motzkin numbers.)

			1
-1, 1
1, -2, 2
-1, 3, -6, 6
1, -4, 12, -24, 6
-1, 5, -20, 60, -30, 30
1, -6, 30, -120, 90, -180, 20
-1, 7, -42, 210, -210, 630, -140, 140
1, -8, 56, -336, 420, -1680, 560, -1120, 70

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, The lost Catalan numbers.

Row sums give A163650, row sums of absolute values give A163865.

Aerated versions A194586 (odd case) and A089627 (even case).

a := proc(n,k) (-1)^(n-k)*floor(k/2)!^(-2)*n!/(n-k)! end:
seq(print(seq(a(n,k),k=0..n)),n=0..8);

t[n_, k_] := (-1)^(n - k)*Floor[k/2]!^(-2)*n!/(n - k)!; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

for(n=0,10, for(k=0,n, print1((-1)^(n -k)*( (floor(k/2))! )^(-2)*(n!/(n - k)!), ", "))) \\ G. C. Greubel, Aug 01 2017

T(n,k) = (-1)^(n-k)*floor(k/2)!^(-2)*n!/(n-k)!.

E.g.f.: egf(x,y) = exp(-x)*BesselI(0,2*x*y)*(1+x*y).

cp's OEIS Frontend

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

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