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A125053
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Variant of triangle A008301, read by rows of 2*n+1 terms, such that the first column is the secant numbers (A000364).
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8
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1, 1, 3, 1, 5, 15, 21, 15, 5, 61, 183, 285, 327, 285, 183, 61, 1385, 4155, 6681, 8475, 9129, 8475, 6681, 4155, 1385, 50521, 151563, 247065, 325947, 378105, 396363, 378105, 325947, 247065, 151563, 50521, 2702765, 8108295, 13311741, 17908935
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OFFSET
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0,3
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COMMENTS
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Foata and Han refer to this as the triangle of Poupard numbers h_n(k). - N. J. A. Sloane, Feb 17 2014
Central terms (A125054) equal the binomial transform of the tangent numbers (A000182).
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LINKS
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FORMULA
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Sum_{k=0..2n} C(2n,k)*T(n,k) = 4^n * A000182(n), where A000182 are the tangent numbers.
Sum_{k=0..2n} (-1)^n*C(2n,k)*T(n,k) = (-4)^n.
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EXAMPLE
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If we write the triangle like this:
......................... ...1;
................... ...1, ...3, ...1;
............. ...5, ..15, ..21, ..15, ...5;
....... ..61, .183, .285, .327, .285, .183, ..61;
. 1385, 4155, 6681, 8475, 9129, 8475, 6681, 4155, 1385;
then the first nonzero term is the sum of the previous row:
1385 = 61 + 183 + 285 + 327 + 285 + 183 + 61,
the next term is 3 times the first:
4155 = 3*1385,
and the remaining terms in each row are obtained by the rule illustrated by:
6681 = 2*4155 - 1385 - 4*61;
8475 = 2*6681 - 4155 - 4*183;
9129 = 2*8475 - 6681 - 4*285;
8475 = 2*9129 - 8475 - 4*327;
6681 = 2*8475 - 9129 - 4*285;
4155 = 2*6681 - 8475 - 4*183;
1385 = 2*4155 - 6681 - 4*61.
An alternate recurrence is illustrated by:
4155 = 1385 + 2*(61 + 183 + 285 + 327 + 285 + 183 + 61);
6681 = 4155 + 2*(183 + 285 + 327 + 285 + 183);
8475 = 6681 + 2*(285 + 327 + 285);
9129 = 8475 + 2*(327);
and then for k>n, T(n,k) = T(n,2*n-k).
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MAPLE
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T := proc(n, k) option remember; local j;
if n = 1 then 1
elif k = 1 then add(T(n-1, j), j=1..2*n-3)
elif k = 2 then 3*T(n, 1)
elif k > n then T(n, 2*n-k)
else 2*T(n, k-1) - T(n, k-2) - 4*T(n-1, k-2)
fi end:
seq(print(seq(T(n, k), k=1..2*n-1)), n=1..5); # Peter Luschny, May 11 2014
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MATHEMATICA
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t[n_, k_] := t[n, k] = If[2*n < k || k < 0, 0, If[n == 0 && k == 0, 1, If[k == 0, Sum[t[n-1, j], {j, 0, 2*n-2}], If[k <= n, t[n, k-1] + 2*Sum[t[n-1, j], {j, k-1, 2*n-1-k}], t[n, 2*n-k]]]]]; Table[t[n, k], {n, 0, 6}, {k, 0, 2*n}] // Flatten (* Jean-François Alcover, Dec 06 2012, translated from Pari *)
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PROG
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(PARI) T(n, k)=if(2*n<k || k<0, 0, if(n==0 && k==0, 1, if(k==0, sum(j=0, 2*n-2, T(n-1, j)), if(k<=n, T(n, k-1)+2*sum(j=k-1, 2*n-1-k, T(n-1, j)), T(n, 2*n-k)))))
(Haskell)
a125053 n k = a125053_tabf !! n !! k
a125053_row n = a125053_tabf !! n
a125053_tabf = iterate f [1] where
f zs = zs' ++ reverse (init zs') where
zs' = (sum zs) : g (map (* 2) zs) (sum zs)
g [x] y = [x + y]
g xs y = y' : g (tail $ init xs) y' where y' = sum xs + y
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CROSSREFS
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KEYWORD
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nonn,tabf,nice
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AUTHOR
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STATUS
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approved
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