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A225483
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Triangle T(n, k) = Sum_{j=0..k} (-1)^(k-j)*A159041(2*n+1, j), read by rows.
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2
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1, 1, -26, 1, 1, -120, 1192, -120, 1, 1, -502, 14609, -88736, 14609, -502, 1, 1, -2036, 152638, -2205524, 9890752, -2205524, 152638, -2036, 1, 1, -8178, 1479727, -45541628, 424761262, -1551163136, 424761262, -45541628, 1479727, -8178, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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LINKS
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FORMULA
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T(n, k) = [x^k]( A159041(x,n)/(x+1) ).
T(n, k) = Sum_{j=0..k} (-1)^(k-j)*A159041(2*n+1, j).
T(n, 2*n-k) = T(n, k). (End)
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EXAMPLE
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The triangle begins:
1;
1, -26, 1;
1, -120, 1192, -120, 1;
1, -502, 14609, -88736, 14609, -502, 1;
1, -2036, 152638, -2205524, 9890752, -2205524, 152638, -2036, 1;
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MATHEMATICA
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(* First program *)
Needs["Combinatorica`"];
p[n_, x_]:= p[n, x]= Sum[If[i==Floor[n/2] && Mod[n, 2]==0, 0, If[i<=Floor[n/2], (-1)^i*Eulerian[n+1, i]*x^i, (-1)^(n-i+1)*Eulerian[n+1, i]*x^i]], {i, 0, n}]/(1- x^2);
Table[CoefficientList[p[x, 2*n], x], {n, 0, 10}]//Flatten
(* Second program *)
A008292[n_, k_]:= A008292[n, k]= Sum[(-1)^j*(k-j)^n*Binomial[n+1, j], {j, 0, k}];
f[n_, k_]:= f[n, k]= If[k==0 || k==n, 1, If[k<=Floor[n/2], f[n, k-1] + (-1)^k*A008292[n+2, k+1], f[n, n-k]]]; (* f = A159041 *)
T[n_, k_]:= T[n, k]= Sum[(-1)^(k-j)*f[2*n+1, j], {j, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, 2*n}]//Flatten (* G. C. Greubel, Mar 29 2022 *)
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PROG
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(Sage)
def A008292(n, k): return sum( (-1)^j*(k-j)^n*binomial(n+1, j) for j in (0..k) )
@CachedFunction
if (k==0 or k==n): return 1
elif (k <= (n//2)): return f(n, k-1) + (-1)^k*A008292(n+2, k+1)
else: return f(n, n-k)
def A225483(n, k): return sum( (-1)^(k-j)*f(2*n+1, j) for j in (0..k) )
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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