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A228336
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Triangle read by rows: the Z-transformation of the Catalan triangle A033184.
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3
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1, 1, 1, 2, 2, 1, 4, 6, 3, 1, 10, 15, 12, 4, 1, 25, 45, 36, 20, 5, 1, 70, 126, 126, 70, 30, 6, 1, 196, 392, 392, 280, 120, 42, 7, 1, 588, 1176, 1344, 960, 540, 189, 56, 8, 1, 1764, 3780, 4320, 3600, 2025, 945, 280, 72, 9, 1, 5544, 11880, 14850, 12375, 8250, 3850, 1540, 396, 90, 10, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,4
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LINKS
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Table of n, a(n) for n=0..65.
Yidong Sun and Fei Ma, Four transformations on the Catalan triangle, arXiv preprint arXiv:1305.2017 [math.CO], 2013.
Yidong Sun and Fei Ma, Some new binomial sums related to the Catalan triangle, Electronic Journal of Combinatorics 21(1) (2014), #P1.33.
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EXAMPLE
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Triangle begins:
1;
1, 1;
2, 2, 1;
4, 6, 3, 1;
10, 15, 12, 4, 1;
25, 45, 36, 20, 5, 1;
70, 126, 126, 70, 30, 6, 1;
...
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MATHEMATICA
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c[n_, k_] := Boole[k <= n] Binomial[2n - k, n] (k + 1)/(n + 1);
T[n_, k_] := Module[{nn, kk}, If[OddQ[n], nn = (n + 1)/2, nn = n/2]; If[OddQ[k], kk = (k - 1)/2, kk = k/2]; If [OddQ[n], If[OddQ[k], c[nn + kk, 2kk + 1] c[nn + kk + 1, 2kk + 2], c[nn + kk, 2kk] c[nn + kk, 2kk + 1]], If[OddQ[k], c[nn + kk + 1, 2kk + 1] c[nn + kk + 1, 2kk + 2], c[nn + kk, 2kk] c[nn + kk + 1, 2kk + 1]]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 04 2018, from PARI *)
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PROG
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(PARI) C(n, k) = (k<=n)*binomial(2*n-k, n)*(k+1)/(n+1);
T(n, k) = {my(nn, kk); if (n % 2, nn = (n+1)/2, nn = n/2); if (k % 2, kk = (k-1)/2, kk = k/2); if ((n % 2), if (k % 2, C(nn+kk, 2*kk+1)*C(nn+kk+1, 2*kk+2), C(nn+kk, 2*kk)*C(nn+kk, 2*kk+1)), if (k % 2, C(nn+kk+1, 2*kk+1)*C(nn+kk+1, 2*kk+2), C(nn+kk, 2*kk)*C(nn+kk+1, 2*kk+1))); } \\ Michel Marcus, Feb 13 2014
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CROSSREFS
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Cf. A033184, A228334, A228335, A228337.
Sequence in context: A175136 A091869 A112307 * A111062 A193597 A191490
Adjacent sequences: A228333 A228334 A228335 * A228337 A228338 A228339
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KEYWORD
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nonn,tabl
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AUTHOR
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N. J. A. Sloane, Aug 26 2013
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EXTENSIONS
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More terms from Michel Marcus, Feb 13 2014
A-number for Catalan triangle changed by Michel Marcus, Feb 13 2014
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STATUS
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approved
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