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A370262
Triangle read by rows: T(n, k) = binomial(n+k, n-k)/(2*k + 1) * (2*n + 1)^k.
2
1, 1, 1, 1, 5, 5, 1, 14, 49, 49, 1, 30, 243, 729, 729, 1, 55, 847, 5324, 14641, 14641, 1, 91, 2366, 26364, 142805, 371293, 371293, 1, 140, 5670, 101250, 928125, 4556250, 11390625, 11390625, 1, 204, 12138, 324258, 4593655, 36916282, 168962983, 410338673, 410338673
OFFSET
0,5
COMMENTS
The table entries are integers since a(n, k) := binomial(n+k, n-k)/(2*k + 1) * (2*n + 1) gives the entries of the transpose of triangle A082985.
FORMULA
n-th row polynomial R(n, x) = Sum_{k = 0..n} T(n, k)*x^k = sqrt( 2* Sum_{k = 0..2*n} (2*n + 1)^(k-1) *binomial(2*n+k+2, 2*k+2)/(2*n + k + 2) * x^k ).
R(n, x)^2 = 2/(x*(2*n + 1)^3) * ( ChebyshevT(2*n+1, 1 + (2*n+1)*x/2) - 1 ).
R(n, 2) = A370260(n).
EXAMPLE
Triangle begins
n\k | 0 1 2 3 4 5 6
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
0 | 1
1 | 1 1
2 | 1 5 5
3 | 1 14 49 49
4 | 1 30 243 729 729
5 | 1 55 847 5324 14641 14641
6 | 1 91 2366 26364 142805 371293 371293
...
MAPLE
seq(seq(binomial(n+k, n-k)/(2*k + 1) * (2*n + 1)^k, k = 0..n), n = 0..10);
MATHEMATICA
Table[Binomial[n + k, n - k] / (2*k + 1) * (2*n + 1)^k, {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Apr 17 2024 *)
CROSSREFS
A371697 (row sums), A052750 (main diagonal and subdiagonal), A000330 (column 1).
Sequence in context: A319569 A204005 A075298 * A060058 A092766 A288389
KEYWORD
nonn,tabl,easy
AUTHOR
Peter Bala, Mar 12 2024
STATUS
approved