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Terms have the same parity as those of Pascal's triangle.

Coefficients of polynomials (1/2)*((1 + x^(1/2))^(2n) + (1 - x^(1/2))^(2n)).

Number of compositions of 2n having k parts greater than 1; example: T(3, 2) = 15 because we have 4+2, 2+4, 3+2+1, 3+1+2, 2+3+1, 2+1+3, 1+3+2, 1+2+3, 2+2+1+1, 2+1+2+1, 2+1+1+2, 1+2+2+1, 1+2+1+2, 1+1+2+2, 3+3. - Philippe Deléham, May 18 2005

Number of binary words of length 2n - 1 having k runs of consecutive 1's; example: T(3,2) = 15 because we have 00101, 01001, 01010, 01011, 01101, 10001, 10010, 10011, 10100, 10110, 10111, 11001, 11010, 11011, 11101. - Philippe Deléham, May 18 2005

Let M_n be the n X n matrix M_n(i, j) = T(i, j-1); then for n > 0, det(M_n) = A000364(n), Euler numbers; example: det([1, 1, 0, 0; 1, 6, 1, 0; 1, 15, 15, 1; 1, 28, 70, 28 ]) = 1385 = A000364(4). - Philippe Deléham, Sep 04 2005

Equals ConvOffsStoT transform of the hexagonal numbers, A000384: (1, 6, 15, 28, 45, ...); e.g., ConvOffs transform of (1, 6, 15, 28) = (1, 28, 70, 28, 1). - Gary W. Adamson, Apr 22 2008

From Peter Bala, Oct 23 2008: (Start)

Let C_n be the root lattice generated as a monoid by {+-2*e_i: 1 <= i <= n; +-e_i +- e_j: 1 <= i not equal to j <= n}. Let P(C_n) be the polytope formed by the convex hull of this generating set. Then the rows of this array are the h-vectors of a unimodular triangulation of P(C_n) [Ardila et al.]. See A127674 for (a signed version of) the corresponding array of f-vectors for these type C_n polytopes. See A008459 for the array of h-vectors for type A_n polytopes and A108558 for the array of h-vectors associated with type D_n polytopes.

The Hilbert transform of this triangle is A142992 (see A145905 for the definition of this term).

(End)

Diagonal sums: A108479. - Philippe Deléham, Sep 08 2009

Coefficients of Product_{k=1..n} (cot(k*Pi/(2n+1))^2 - x) = Sum_{k=0..n} (-1)^k*binomial(2n,2k)*x^k/(2n+1-2k). - David Ingerman (daviddavifree(AT)gmail.com), Mar 30 2010

Generalized Narayana triangle for 4^n (or cosh(2x)). - Paul Barry, Sep 28 2010

Coefficients of the matrix inverse appear to be T^(-1)(n,k) = (-1)^(n+k)*A086646(n,k). - R. J. Mathar, Mar 12 2013

Let E(y) = Sum_{n>=0} y^n/(2*n)! = cosh(sqrt(y)). Then this triangle is the generalized Riordan array (E(y), y) with respect to the sequence (2*n)! as defined in Wang and Wang. Cf. A103327. - Peter Bala, Aug 06 2013

Row 6, (1,66,495,924,495,66,1), plays a role in expansions of powers of the Dedekind eta function. See the Chan link, p. 534, and A034839. - Tom Copeland, Dec 12 2016

The row polynomials Pe(n, x) := Sum_{m=0..n} a(n, m)*x^m appear as numerators of the generating functions for the even numbered column sequences of array A034870.

Elements have the same parity as those of Pascal's triangle.

All zeros of polynomial Pe(n, x) are negative. They are -tan^2(Pi/2*n+1), -tan^2(2*Pi/2*n+1), ..., -tan^2(n*Pi/2*n+1). Moreover, for m >= 1, Pe(m, -x^2) is the characteristic polynomial of the linear difference equation with constant coefficients for differences between multiples of 2*m+1 with even and odd digit sum in base 2*m in the interval [0,(2*m)^n). - Vladimir Shevelev and Peter J. C. Moses, May 22 2012

Row reverse of A103327. - Peter Bala, Jul 29 2013

The row polynomial Pe(d, x), multiplied by (2*d)!/d! = A001813(d), gives the numerator polynomial of the o.g.f. of the sequence of the diagonal d, for d >= 0, of the Sheffer triangle Lah[4,1] given in A048854. - Wolfdieter Lang, Oct 12 2017

A subset of Pascal's triangle A007318.

Elements have the same parity as those of Pascal's triangle.

Matrix inverse is A104033. - Paul D. Hanna, Feb 28 2005

Row reverse of A091042. - Peter Bala, Jul 29 2013

Let E(y) = cosh(sqrt(y)) = 1 + 3*y/3! + 5*y^2/5! + 7*y^3/7! + .... Then this triangle is the generalized Riordan array (E(y), y) with respect to the sequence (2*n+1)! as defined in Wang and Wang. Cf. A086645. - Peter Bala, Aug 06 2013

The row polynomial P(d, x) = Sum_{k=0..d} T(d, k)*x^k, multiplied by (2*d)!/d! = A001813(d), gives the numerator polynomial of the o.g.f. of the sequence of the diagonal d, for d >= 0, of the Sheffer triangle Lah[4,3] given in A292219. - Wolfdieter Lang, Oct 12 2017

The ratio k(n) /(2*j(n)) tends to sqrt(2) as n increases.

The squares of the numbers in this sequence are one less than a triangular number: a(n)^2 = A164080(n). For example, 18^2 is 324, and 325 is a triangular number. a(n)^2 + 1 = A164055(n). a(n)^2 = A072221(n)(A072221(n)+1)/2 - 1. - Tanya Khovanova & Alexey Radul, Aug 09 2009

For n > 0, a(n+1) is the n-th almost balancing number of first type (see Tekcan and Erdem). - Stefano Spezia, Nov 25 2022

The ratio a(n)/(2*j(n)) tends to sqrt(2) as n increases.

After 3, first differences of A301383. - Bruno Berselli, Mar 22 2018

For n > 0, a(n+1) is the n-th almost Lucas-balancing number of first type (see Tekcan and Erdem). - Stefano Spezia, Nov 25 2022

Row n has length 1 for n = 1 and 2^(n-1) = A000079(n-1) for n >= 1.

The triangle T gives the non-vanishing coefficients of the polynomial q(0,x) = 1 and q(n,x) = 2^(-n)*Sum_{k=0..2^(n-1)-1} A281122(n,k) * x^(2^n-1-2*k), n >= 1.

The polynomial q(n,x) = product_{k=0..n-1} p(k,x) with polynomial p(n,x) = ((x+1)^(2^n) + (x-1)^(2^n))/2, whose coefficients are tabulated in A201461.

The algorithm r(n) = 1/2*(r(n-1) + A/r(n-1)), starting with r(0) = A, used for approximating sqrt(A), which is known as the Babylonian method or Hero's method after the first-century Greek mathematician Hero of Alexandria and which can be derived from Newton's method, generates fractions beginning with (A+1)/2, (A^2 + 6*A + 1)/(4*(A + 1)), (A^4 + 28*A^3 + 70*A^2 + 28*A + 1)/(8*(A^3 + 7*A^2 + 7*A + 1)), ... This is sqrt(A)*p(n,sqrt(A))/(2^n*q(n,sqrt(A))) with the given polynomials p(n,x) and q(n,x).

The length of row n is 1 for n = 0 and 2^(n-1) = A000079(n-1) for n >= 1.

The polynomial q(n,x) = 2^n*Product_{k=0..n-1} p(k,x) with polynomial p(n,x) = ((x+1)^(2^n) + (x-1)^(2^n))/2, whose coefficients are tabulated in A201461.

The row polynomials are q(n, x) = Sum_{k=0..2^(n-1)-1} T(n, k)*x^(2^n-1-2*k) for n >= 1, and q(0,x) = 1. - Wolfdieter Lang, Jan 19 2017

A201461 and T(n,k) are a bisection of row 2^n of Pascal's triangle A007318.

The algorithm r(n) = 1/2*(r(n-1) + A/r(n-1)), starting with r(0) = A, used for approximating sqrt(A), which is known as the Babylonian method or Hero's method after the first-century Greek mathematician Hero of Alexandria and which can be derived from Newton's method, generates fractions beginning with (A+1)/2, (A^2 + 6*A + 1)/(4*A + 4), (A^4 + 28*A^3 + 70*A^2 + 28*A + 1)/(8*A^3 + 56*A^2 + 56*A + 8), ... This is sqrt(A)*p(n,sqrt(A))/q(n,sqrt(A)) with the given polynomials p(n,x) and q(n,x).