Search: a163649 -id:a163649
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A163650
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Subswing - the inverse binomial transform of the swinging factorial (A056040).
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+10
9
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1, 0, 1, 2, -9, 44, -165, 594, -2037, 6824, -22437, 72830, -234047, 746316, -2364947, 7455798, -23405085, 73207728, -228275949, 709906518, -2202557691, 6819616020, -21076580511, 65032888998, -200369138571, 616531573224, -1894784517675, 5816886949874
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OFFSET
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0,4
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COMMENTS
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Analog to the subfactorial A000166.
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LINKS
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FORMULA
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E.g.f.: exp(-x)*BesselI(0,2*x)*(1+x). - Peter Luschny, Aug 26 2012
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k)*(k!/(floor(k/2)!)^2). - G. C. Greubel, Aug 01 2017
a(n) ~ -(-1)^n * sqrt(n) * 3^(n - 1/2) / (2*sqrt(Pi)). - Vaclav Kotesovec, Oct 31 2017
D-finite with recurrence n*a(n) +5*(n-1)*a(n-1) +(n-4)*a(n-2) +(-13*n+23)*a(n-3) +6*(n-3)*a(n-4)=0. - R. J. Mathar, Jul 04 2023
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MAPLE
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a := proc(n) local k: add((-1)^(n-k)*binomial(n, k)*(k!/iquo(k, 2)!^2), k=0..n) end:
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MATHEMATICA
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sf[n_] := n!/Quotient[n, 2]!^2; a[n_] := Sum[(-1)^(n-k)*Binomial[n, k]*sf[k], {k, 0, n}]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jun 28 2013 *)
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PROG
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(PARI) for(n=0, 50, print1(sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(k!/((k\2)!)^2)), ", ")) \\ G. C. Greubel, Aug 01 2017
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A194586
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Triangle read by rows, T(n,k) the coefficients of the polynomials Sum_{k=0..n} binomial(n,k)*A056040(k)*(k mod 2)*q^k.
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+10
2
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0, 0, 1, 0, 2, 0, 0, 3, 0, 6, 0, 4, 0, 24, 0, 0, 5, 0, 60, 0, 30, 0, 6, 0, 120, 0, 180, 0, 0, 7, 0, 210, 0, 630, 0, 140, 0, 8, 0, 336, 0, 1680, 0, 1120, 0, 0, 9, 0, 504, 0, 3780, 0, 5040, 0, 630, 0, 10, 0, 720, 0, 7560, 0, 16800, 0, 6300, 0, 0, 11, 0, 990, 0, 13860, 0, 46200, 0, 34650, 0, 2772, 0, 12
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OFFSET
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0,5
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COMMENTS
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Substituting q^k -> 1/(floor(k/2)+1) in the polynomials gives the complementary Motzkin numbers A005717. (See A089627 for the Motzkin numbers and A163649 for the extended Motzkin numbers.)
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LINKS
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FORMULA
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egf(x,y) = x*y*exp(x)*BesselI(0,2*x*y).
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EXAMPLE
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0
0, 1
0, 2, 0
0, 3, 0, 6
0, 4, 0, 24, 0
0, 5, 0, 60, 0, 30
0, 6, 0, 120, 0, 180, 0
0, 7, 0, 210, 0, 630, 0, 140
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q
2 q
3 q + 6 q^3
4 q + 24 q^3
5 q + 60 q^3 + 30 q^5
6 q + 120 q^3 + 180 q^5
7 q + 210 q^3 + 630 q^5 + 140 q^7
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MAPLE
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A194586 := proc(n, k) local j, swing; swing := n -> n!/iquo(n, 2)!^2:
add(binomial(n, j)*swing(j)*q^j*(j mod 2), j=0..n); coeff(%, q, k) end:
seq(print(seq(A194586(n, k), k=0..n)), n=0..8);
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MATHEMATICA
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sf[n_] := n!/Quotient[n, 2]!^2;
row[n_] := Sum[Binomial[n, j] sf[j] q^j Mod[j, 2], {j, 0, n}] // CoefficientList[#, q]& // PadRight[#, n+1]&;
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A163945
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Triangle interpolating between (-1)^n (A033999) and the swinging factorial function (A056040) restricted to odd indices (2n+1)$ (A002457).
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+10
0
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1, -1, 6, 1, -12, 30, -1, 18, -90, 140, 1, -24, 180, -560, 630, -1, 30, -300, 1400, -3150, 2772, 1, -36, 450, -2800, 9450, -16632, 12012, -1, 42, -630, 4900, -22050, 58212, -84084, 51480, 1, -48, 840, -7840, 44100, -155232, 336336, -411840, 218790
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OFFSET
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0,3
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COMMENTS
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Triangle read by rows.
For n >= 0, k >= 0 let T(n, k) = (-1)^(n-k) binomial(n,k) (2*k+1)$ where i$ denotes the swinging factorial of i (A056040).
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LINKS
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FORMULA
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Conjectural g.f.: sqrt(1 + t)/(1 + (1 - 4*x)*t)^(3/2) = 1 + (-1 + 6*x)*t + (1 - 12*x + 30*x^2)*t^2 + .... - Peter Bala, Nov 10 2013
T(n, k) = ((-1)^(k mod 2) + n)*((2*k + 1)!/(k!)^2)*binomial(n, n - k). - Detlef Meya, Oct 07 2023
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EXAMPLE
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1;
-1, 6;
1, -12, 30;
-1, 18, -90, 140;
1, -24, 180, -560, 630;
-1, 30, -300, 1400, -3150, 2772;
1, -36, 450, -2800, 9450, -16632, 12012;
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MAPLE
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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:
a := proc(n, k) (-1)^(n-k)*binomial(n, k)*swing(2*k+1) end:
seq(print(seq(a(n, k), k=0..n)), n=0..8);
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MATHEMATICA
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T[n_, k_] := ((-1)^(Mod[k, 2]+n)*((2*k+1)!/(k!)^2)*Binomial[n, n-k]);
Flatten[Table[T[n, k], {n, 0, 8}, {k, 0, n}]] (*End*)
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CROSSREFS
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Row sums are the inverse binomial transform of the beta numbers (A163872).
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KEYWORD
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AUTHOR
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STATUS
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approved
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