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A117440
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A cyclically signed version of Pascal's triangle.
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4
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1, 1, 1, -1, 2, 1, -1, -3, 3, 1, 1, -4, -6, 4, 1, 1, 5, -10, -10, 5, 1, -1, 6, 15, -20, -15, 6, 1, -1, -7, 21, 35, -35, -21, 7, 1, 1, -8, -28, 56, 70, -56, -28, 8, 1, 1, 9, -36, -84, 126, 126, -84, -36, 9, 1, -1, 10, 45, -120, -210, 252, 210, -120, -45, 10, 1
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OFFSET
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0,5
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LINKS
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FORMULA
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Column k has e.g.f.: (x^k/k!)*(cos(x) + sin(x)).
T(n, k) = binomial(n,k)*(cos(Pi*(n-k)/2) + sin(Pi*(n-k)/2).
Sum_{k=0..n} T(n, k) = A009545(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A117441(n) (upward diagonal sums).
G.f.: (1 + x - x*y)/(1 - 2*x*y + x^2*(1 + y^2)). - Stefano Spezia, Mar 10 2024
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EXAMPLE
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Triangle begins:
1;
1, 1;
-1, 2, 1;
-1, -3, 3, 1;
1, -4, -6, 4, 1;
1, 5, -10, -10, 5, 1;
-1, 6, 15, -20, -15, 6, 1;
-1, -7, 21, 35, -35, -21, 7, 1;
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MATHEMATICA
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Table[Binomial[n, k]*(Cos[Pi*(n-k)/2] +Sin[Pi*(n-k)/2]), {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 01 2021 *)
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PROG
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(Sage) flatten([[binomial(n, k)*( cos(pi*(n-k)/2) + sin(pi*(n-k)/2) ) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 01 2021
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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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