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A178655
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Triangle which contains the first differences of the Catalan triangle A001263 constructed along rows.
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1
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1, 1, -1, 1, 0, -1, 1, 2, -2, -1, 1, 5, 0, -5, -1, 1, 9, 10, -10, -9, -1, 1, 14, 35, 0, -35, -14, -1, 1, 20, 84, 70, -70, -84, -20, -1, 1, 27, 168, 294, 0, -294, -168, -27, -1, 1, 35, 300, 840, 588, -588, -840, -300, -35, -1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,8
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LINKS
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G. C. Greubel, Rows n = 0..100 of triangle, flattened
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FORMULA
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T(n,k) = -T(n,n-k), n > 0.
T(n,k) = A001263(n,k+1) - A001263(n,k), n > 0. - R. J. Mathar, Jun 16 2015
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EXAMPLE
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Triangle begins
1;
1, -1;
1, 0, -1;
1, 2, -2, -1;
1, 5, 0, -5, -1;
1, 9, 10, -10, -9, -1;
1, 14, 35, 0, -35, -14, -1;
1, 20, 84, 70, -70, -84, -20, -1;
1, 27, 168, 294, 0, -294, -168, -27, -1;
1, 35, 300, 840, 588, -588, -840, -300, -35, -1;
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MATHEMATICA
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Join[{1}, Table[((n+1)*(n-2*k)/(n*(k+1)*(n-k+1)))*Binomial[n, k]^2, {n, 1, 10}, {k, 0, n}]//Flatten] (* G. C. Greubel, Jan 28 2019 *)
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PROG
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(PARI) {T(n, k) = if(n==0, 1, ((n+1)*(n-2*k)/(n*(k+1)*(n-k+1)))* binomial(n, k)^2)};
for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jan 28 2019
(Magma) [[n le 0 select 1 else ((n+1)*(n-2*k)/(n*(k+1)*(n-k+1)))*Binomial(n, k)^2: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jan 28 2019
(Sage) [1] + [[((n+1)*(n-2*k)/(n*(k+1)*(n-k+1)))* binomial(n, k)^2 for k in (0..n)] for n in (1..10)] # G. C. Greubel, Jan 28 2019
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CROSSREFS
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Cf. A001263, A000007 (row sums).
Sequence in context: A361014 A064552 A209543 * A337278 A178304 A123585
Adjacent sequences: A178652 A178653 A178654 * A178656 A178657 A178658
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KEYWORD
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sign,tabl,easy
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AUTHOR
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Roger L. Bagula, Jun 01 2010
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STATUS
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approved
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