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A067001
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Triangle T(n,k) = d(n-k,n), 0 <= k <= n, where d(l,m) = Sum_{k=l..m} 2^k * binomial(2*m-2*k, m-k) * binomial(m+k, m) * binomial(k, l).
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8
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1, 4, 6, 24, 60, 42, 160, 560, 688, 308, 1120, 5040, 8760, 7080, 2310, 8064, 44352, 99456, 114576, 68712, 17556, 59136, 384384, 1055040, 1572480, 1351840, 642824, 134596, 439296, 3294720, 10695168, 19536000, 21778560, 14912064, 5864640, 1038312
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OFFSET
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0,2
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COMMENTS
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For an explanation on how this triangular array is related to the Boros-Moll polynomial P_n(x) and the theory in Comtet (1967), see my comments in A223549. For example, the bivariate o.g.f. below follows from the theory in Comtet (1967). - Petros Hadjicostas, May 24 2020
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LINKS
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FORMULA
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Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = sqrt((1 + y)/(1 - 8*x*(1 + y))/(1 + y*sqrt(1 - 8*x*(1 + y)))). (End)
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EXAMPLE
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Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) starts:
1;
4, 6;
24, 60, 42;
160, 560, 688, 308;
1120, 5040, 8760, 7080, 2310;
...
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MAPLE
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d := proc(l, m) local k; add(2^k*binomial(2*m-2*k, m-k)*binomial(m+k, m)*binomial(k, l), k=l..m); end:
T:= (n, k)-> d(n-k, n):
seq(seq(T(n, k), k=0..n), n=0..10);
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MATHEMATICA
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T[n_, k_] := SeriesCoefficient[Sqrt[(1+y)/(1 - 8x (1+y))/(1 + y Sqrt[1 - 8x (1+y)])], {x, 0, n}, {y, 0, k}];
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PROG
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(PARI) d(l, m) = sum(kk=l, m, 2^kk*binomial(2*m-2*kk, m-kk)*binomial(m+kk, m)*binomial(kk, l));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(d(n-k, n), ", "); ); print(); ); } \\ Michel Marcus, Jul 18 2015
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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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