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A163770
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Triangle read by rows interpolating the swinging subfactorial (A163650) with the swinging factorial (A056040).
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4
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1, 0, 1, 1, 1, 2, 2, 3, 4, 6, -9, -7, -4, 0, 6, 44, 35, 28, 24, 24, 30, -165, -121, -86, -58, -34, -10, 20, 594, 429, 308, 222, 164, 130, 120, 140, -2037, -1443, -1014, -706, -484, -320, -190, -70, 70, 6824, 4787, 3344, 2330, 1624, 1140, 820, 630, 560, 630
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OFFSET
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0,6
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COMMENTS
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An analog to the derangement triangle (A068106).
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LINKS
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FORMULA
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T(n,k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*i$ where i$ denotes the swinging factorial of i (A056040).
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EXAMPLE
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1
0, 1
1, 1, 2
2, 3, 4, 6
-9, -7, -4, 0, 6
44, 35, 28, 24, 24, 30
-165, -121, -86, -58, -34, -10, 20
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MAPLE
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DiffTria := 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, (-1)^(m-i)*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.
A163770 := n -> DiffTria(k->swing(k), n, true);
A068106 := n -> DiffTria(k->factorial(k), n, true);
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MATHEMATICA
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sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*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 *)
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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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