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A091044 One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318. 4
1, 2, 2, 3, 10, 3, 4, 28, 28, 4, 5, 60, 126, 60, 5, 6, 110, 396, 396, 110, 6, 7, 182, 1001, 1716, 1001, 182, 7, 8, 280, 2184, 5720, 5720, 2184, 280, 8, 9, 408, 4284, 15912, 24310, 15912, 4284, 408, 9, 10, 570, 7752, 38760, 83980, 83980, 38760, 7752, 570, 10, 11 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The odd-numbered columns of this triangle can be reduced: see triangle A091043.

The odd-numbered rows coincide with the ones of the reduced triangle A091043.

binomial(2*n,2*m+1) is even for n >= m + 1 >= 1, hence every T(n,m) is a positive integer.

The GCD (greatest common divisor) of the entries of each odd-numbered row n=2*k+1, k>=0, is 1.

The GCD of the entries of the even-numbered row n=2*k, k>=1, is A006519(n) (highest power of 2 in n=2*k).

LINKS

Indranil Ghosh, Rows 1..125, flattened

Kevin Buchin, Man-Kwun Chiu, Stefan Felsner, Günter Rote, André Schulz, The Number of Convex Polyominoes with Given Height and Width, arXiv:1903.01095 [math.CO], 2019.

Hernan de Alba, W. Carballosa, J. Leaños, L. M. Rivera, Independence and matching numbers of some token graphs, arXiv preprint arXiv:1606.06370 [math.CO], 2016.

W. Lang, First 9 rows.

FORMULA

T(n, m)= binomial(2*n, 2*m+1)/2, n >= m + 1 >= 1, else 0.

  Put a(n) = n!*(n+1/2)!/(1/2)!. T(n+1,k) = (n+1)*a(n)/(a(k)*a(n-k)).

  T(n-1,k-1)*T(n,k+1)*T(n+1,k) = T(n-1,k)*T(n,k-1)*T(n+1,k+1). Cf. A111910. - Peter Bala, Oct 13 2011

From Peter Bala, Jul 29 2013: (Start)

O.g.f.: 1/(1 - 2*t*(x + 1) + t^2*(x - 1)^2)= 1 + (2 + 2*x)*t + (3 + 10*x + 3*x^2)*t^2 + ....

The n-th row polynomial R(n,x) = 1/(4*sqrt(x))*( (1 + sqrt(x))^(2*n) - (sqrt(x) - 1)^(2*n) ) and has n-1 real zeros given by the formula -cot^2(k*Pi/(2*n)) for k = 1,2,...,n-1. Cf A091042.

The row polynomial R(n,x) satisfies (x - 1)^n*R(n,x/(x - 1)) = U(n,2*x - 1), the n-th row polynomial of A053124.

Row sums A000302. Sum {k = 0..n-1} 2^k*T(n,k) = A001109(n). (End)

From Werner Schulte, Jan 13 2017: (Start)

(1) T(n,m) = T(n-1,m) + T(n-1,m-1)*(2*n-1-m)/m for 0 < m < n-1 with T(n,0) = n and T(n,n) = 0;

(2) T(n,m) = 2*T(n-1,m) + 2*T(n-1,m-1) - T(n-2,m) + 2*T(n-2,m-1) - T(n-2,m-2) for 0 < m < n-1 with T(n,0) = T(n,n-1) = n and T(n,m) = 0 if m < 0 or m >= n;

(3) The row polynomials p(n,x) = Sum_{m=0..n-1} T(n,m)*x^m satisfy the recurrence equation p(n+2,x) = (2+2*x)*p(n+1,x) - (x-1)^2*p(n,x) for n >= 1 with initial values p(1,x) = 1 and p(2,x) = 2+2*x.

(End)

G.f.: x*y /(1 - 2*(x+y) + (x-y)^2) with the entries regarded as an infinite square array A(i, j) read by antidiagonals. - Michael Somos, Oct 15 2017

EXAMPLE

Triangle begins:

  [1];

  [2,2];

  [3,10,3];

  [4,28,28,4];

  [5,60,126,60,5];

  [6,110,396,396,110,6];

  ...

n = 6 = 2*3: gcd(6,110,396) = 2 = A006519(6);

n = 5: gcd(5,60,126) = 1 = A006519(5).

MATHEMATICA

Flatten[Table[Binomial[2n, 2m+1]/2, {n, 1, 11}, {m, 0, n-1}]] (* Indranil Ghosh, Feb 22 2017 *)

PROG

(PARI) {A(i, j) = binomial(2*i + 2*j - 2, 2*i - 1) / 2}; /* Michael Somos, Oct 15 2017 */

CROSSREFS

Cf. A000302 (row sums), A001109, A053124, A091042, A111910.

Sequence in context: A019234 A032172 A032103 * A079661 A220644 A153920

Adjacent sequences:  A091041 A091042 A091043 * A091045 A091046 A091047

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Jan 23 2004

STATUS

approved

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Last modified November 13 04:20 EST 2019. Contains 329085 sequences. (Running on oeis4.)