A082985 -id:A082985 - cp's OEIS frontend

A370262 Triangle read by rows: T(n, k) = binomial(n+k, n-k)/(2k + 1) (2*n + 1)^k.

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

Peter Bala, Mar 12 2024

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.

			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
  ...

A371697 (row sums), A052750 (main diagonal and subdiagonal), A000330 (column 1).

Cf. A008310, A082985, A258708, A370259, A370260.

seq(seq(binomial(n+k, n-k)/(2*k + 1) * (2*n + 1)^k, k = 0..n), n = 0..10);

Table[Binomial[n + k, n - k] / (2*k + 1) * (2*n + 1)^k, {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Apr 17 2024 *)

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).

A111126 Triangle read by rows: T(k,s) = binomial(k+s,2s+1)(2k-1)(2k+1)/(2s+3), k >= 1, 0 <= s <= k-1.

Original entry on oeis.org

1, 10, 3, 35, 28, 5, 84, 126, 54, 7, 165, 396, 297, 88, 9, 286, 1001, 1144, 572, 130, 11, 455, 2184, 3510, 2600, 975, 180, 13, 680, 4284, 9180, 9350, 5100, 1530, 238, 15, 969, 7752, 21318, 28424, 20995, 9044, 2261, 304, 17, 1330, 13167, 45144, 76076, 72618

Offset: 1

N. J. A. Sloane, Oct 16 2005

			Triangle starts:
1;
10,3;
35,28,5;
84,126,54,7;
165,396,297,88,9;

K. Dilcher and K. B. Stolarsky, A Pascal-type triangle characterizing twin primes, Amer. Math. Monthly, 112 (2005), 673-681.

Mirror image of A111127. Cf. A111125, A082985, A100218, A098599.

T:=(k,s)->binomial(k+s,2*s+1)*(2*k-1)*(2*k+1)/(2*s+3): for k from 1 to 10 do seq(T(k,s),s=0..k-1) od; # yields sequence in triangular form; Emeric Deutsch, Feb 01 2006

More terms from Emeric Deutsch, Feb 01 2006

cp's OEIS Frontend

A370262 Triangle read by rows: T(n, k) = binomial(n+k, n-k)/(2k + 1) (2*n + 1)^k.

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A111126 Triangle read by rows: T(k,s) = binomial(k+s,2s+1)(2k-1)(2k+1)/(2s+3), k >= 1, 0 <= s <= k-1.

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cp's OEIS Frontend

A370262 Triangle read by rows: T(n, k) = binomial(n+k, n-k)/(2*k + 1) * (2*n + 1)^k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A111126 Triangle read by rows: T(k,s) = binomial(k+s,2s+1)*(2k-1)*(2k+1)/(2s+3), k >= 1, 0 <= s <= k-1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Extensions

A370262 Triangle read by rows: T(n, k) = binomial(n+k, n-k)/(2k + 1) (2*n + 1)^k.

A111126 Triangle read by rows: T(k,s) = binomial(k+s,2s+1)(2k-1)(2k+1)/(2s+3), k >= 1, 0 <= s <= k-1.