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URL: https://oeis.org/A026082

⇱ A026082 - OEIS

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A026082
Irregular triangular array T read by rows: T(n,k) = C(n,k) for k=0..n for n = 0,1,2,3. For n >= 4, T(n,0) = T(n,2n)=1, T(n,1) = T(n,2n-1) = n - 3, T(4,2) = 4, T(4,3) = 3, T(4,4) = 6; T(4,5) = 3, T(4,6)=4; for n >= 5, T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) for k=2..2n-2.
21
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 1, 4, 3, 6, 3, 4, 1, 1, 1, 2, 6, 8, 13, 12, 13, 8, 6, 2, 1, 1, 3, 9, 16, 27, 33, 38, 33, 27, 16, 9, 3, 1, 1, 4, 13, 28, 52, 76, 98, 104, 98, 76, 52, 28, 13, 4, 1, 1, 5, 18, 45, 93, 156, 226, 278, 300, 278, 226, 156, 93, 45, 18, 5, 1, 1, 6, 24, 68, 156, 294, 475, 660, 804
OFFSET
1,5
COMMENTS
For n >= 4, T(n,k) = number of strings s(0)..s(n) such that s(n) = n - k, s(0) = 0, |s(i)-s(i-1)| = 1 for i=1,2,3 and |s(i)-s(i-1)| <= 1 for i >= 4.
FORMULA
G.f.: (1-y*z)^3 / (1-z*(1+y+y^2)).
EXAMPLE
First 6 rows:
1
1 1
1 2 1
1 3 3 1
1 1 4 3 6 3 4 1 1
1 2 6 8 12 12 13 8 6 2 1
MAPLE
A026082 := proc(n, k)
option remember;
if n < 0 or k < 0 or k > 2*n then
0 ;
elif n <= 3 then
binomial(n, k) ;
elif n = 4 then
op(k+1, [1, 1, 4, 3, 6, 3, 4, 1, 1]) ;
elif k =0 or k=2*n then
1 ;
else
procname(n-1, k-2)+procname(n-1, k-1)+procname(n-1, k) ;
end if;
end proc: # R. J. Mathar, Jun 23 2013
MATHEMATICA
z = 15; t[n_, 0] := 1 /; n >= 4; t[n_, 1] := n - 3 /; n >= 4;
t[4, 2] = 4; t[4, 3] = 3; t[4, 4] = 6; t[4, 5] = 3; t[4, 6] = 4;
t[n_, k_] := t[n, k] = Which[0 <= k <= n && 0 <= n <= 3, Binomial[n, k], n
>= 4 && k == 2 n, 1, k == 2 n - 1, n - 3, 2 <= k <= 2 n - 2, t[n - 1, k -
2] + t[n - 1, k - 1] + t[n - 1, k]]; s = Table[Binomial[n, k], {n, 0, 3},
{k, 0, n}]; u = Join[s, Table[t[n, k], {n, 4, z}, {k, 0, 2 n}]];
TableForm[u] (* A026082 array *)
Flatten[u] (* A026082 sequence *)
CROSSREFS
First differences of A024996.
Sequence in context: A307116 A212626 A090402 * A117185 A129181 A157694
KEYWORD
nonn,tabf
EXTENSIONS
Updated by Clark Kimberling, Aug 28 2014
STATUS
approved