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A158815
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Triangle T(n,k) read by rows, matrix product of A046899(row-reversed) * A130595.
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5
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1, 1, 1, 4, 1, 1, 13, 5, 1, 1, 46, 16, 6, 1, 1, 166, 58, 19, 7, 1, 1, 610, 211, 71, 22, 8, 1, 1, 2269, 781, 261, 85, 25, 9, 1, 1, 8518, 2920, 976, 316, 100, 28, 10, 1, 1, 32206, 11006, 3676, 1196, 376, 116, 31, 11, 1, 1
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OFFSET
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0,4
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COMMENTS
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The left factor of the matrix product is the triangle which starts
1;
2, 1;
6, 3, 1;
20, 10, 4, 1;
a row-reversed version of A046899, equivalent to the triangular view of the array A092392. The right factor is the inverse of the matrix A007318, which is A130595.
T(n,k) is the number of nonnegative paths consisting of upsteps U=(1,1) and downsteps D=(1,-1) of length 2n with k low peaks. (A low peak has its peak vertex at height 1.) Example: T(3,1)=5 counts UDUUUU, UDUUUD, UDUUDU, UDUUDD, UUDDUD. - David Callan, Nov 21 2011
Matrix product P^2 * Q * P^(-2), where P denotes Pascal's triangle A007318 and Q denotes A061554 (formed from P by sorting the rows into descending order). Cf. A158793 and A171243. - Peter Bala, Jul 13 2021
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LINKS
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FORMULA
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T(n, k) = Sum_{j=0..n} binomial(j, k)*binomial(2*n-j, n). - Peter Bala, Jul 13 2021
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EXAMPLE
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The triangle starts
1;
1, 1;
4, 1, 1;
13, 5, 1, 1;
46, 16, 6, 1, 1;
166, 58, 19, 7, 1, 1;
610, 211, 71, 22, 8, 1, 1;
2269, 781, 261, 85, 25, 9, 1, 1;
8518, 2620, 976, 316, 100, 28, 10, 1, 1;
32206, 11006, 3676, 1196, 376, 116, 31, 11, 1, 1;
122464, 41746, 13938, 4544, 1442, 441, 133, 34, 12, 1, 1;
...
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MAPLE
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add((-1)^(j+k)*binomial(2*n-j, n)*binomial(j, k), j = 0..n);
end proc:
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MATHEMATICA
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T[n_, k_]:= T[n, k]= Sum[(-1)^(j+k)*Binomial[j, k]*Binomial[2*n-j, n], {j, 0, n}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 22 2021 *)
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PROG
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(Sage)
def A158815(n, k): return sum( (-1)^(j+k)*binomial(2*n-j, n)*binomial(j, k) for j in (0..n) )
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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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