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A259525
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First differences of A007318, when Pascal's triangle is seen as flattened list.
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3
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0, 0, 0, 1, -1, 0, 2, 0, -2, 0, 3, 2, -2, -3, 0, 4, 5, 0, -5, -4, 0, 5, 9, 5, -5, -9, -5, 0, 6, 14, 14, 0, -14, -14, -6, 0, 7, 20, 28, 14, -14, -28, -20, -7, 0, 8, 27, 48, 42, 0, -42, -48, -27, -8, 0, 9, 35, 75, 90, 42, -42, -90, -75, -35, -9, 0, 10, 44, 110
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OFFSET
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0,7
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COMMENTS
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A214292 gives first differences per row in Pascal's triangle.
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LINKS
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FORMULA
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If viewed as a triangle then:
T(n, k) = binomial(n, k+1) - binomial(n, k), with T(n, n) = 0.
T(n, n-k) = - T(n, k), for 0 <= k < n.
Sum_{k=0..n} T(n, k) = 0 (row sums).
Sum_{k=0..floor(n/2)} T(n, k) = A047171(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A021499(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A074331(n-1). (End)
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MATHEMATICA
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Table[If[k==n, 0, ((n-2*k-1)/(n-k))*Binomial[n, k+1]], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 25 2024 *)
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PROG
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(Haskell)
a259525 n = a259525_list !! n
a259525_list = zipWith (-) (tail pascal) pascal
where pascal = concat a007318_tabl
(Magma)
[k eq n select 0 else (n-2*k-1)*Binomial(n, k+1)/(n-k): k in [0..n], n in [0..14]]; // G. C. Greubel, Apr 25 2024
(SageMath)
flatten([[binomial(n, k+1) -binomial(n, k) +int(k==n) for k in range(n+1)] for n in range(15)]) # G. C. Greubel, Apr 25 2024
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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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