A091043 - OEIS

A091043

Normalized triangle of odd numbered entries of even numbered rows of Pascal's triangle A007318.

1, 1, 1, 3, 10, 3, 1, 7, 7, 1, 5, 60, 126, 60, 5, 3, 55, 198, 198, 55, 3, 7, 182, 1001, 1716, 1001, 182, 7, 1, 35, 273, 715, 715, 273, 35, 1, 9, 408, 4284, 15912, 24310, 15912, 4284, 408, 9, 5, 285, 3876, 19380, 41990, 41990, 19380, 3876, 285, 5, 11, 770, 13167, 85272, 248710, 352716, 248710, 85272, 13167, 770, 11

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OFFSET

1,4

COMMENTS

b(n)= A006519(n), with b(n) defined in the formula. For every odd n b(n)=1.

The row polynomials Po(n,x) := 2*b(n)*Sum_{m=0..n-1} T(n,m)*x^m, n>=1, appear as numerators of the generating functions for the odd numbered column sequences of array A034870. b(n) is defined in the formula below.

LINKS

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of triangle, flattened).

Wolfdieter Lang, First 9 rows.

FORMULA

T(n, m) = binomial(2*n, 2*m+1)/(2*b(n)), n>=m+1>=1, else 0, with b(n) := GCD(seq(binomial(2*n, 2*m+1)/2, m=0..n-1)), where GCD denotes the greatest common divisor of a set of numbers (here one half of the odd numbered entries in the even numbered rows of Pascal's triangle). It suffices to consider m=0..floor((n-1)/2) due to symmetry.

EXAMPLE

Triangle begins

[1];

[1,1];

[3,10,3];

[1,7,7,1];

[5,60,126,60,5];

...

For n=3: GCD(3,10,3) = GCD(3,10) = 1 = b(3) = A006519(3).

For n=4: GCD(4,28,28,4) = GCD(4,28) = 4 = b(4) = A006519(4).

MATHEMATICA

A091043row[n_] := # / GCD @@ # & [Binomial[2*n, 2*Range[n] - 1]];

Array[A091043row, 12] (* Paolo Xausa, May 07 2026 *)

PROG

(PARI) row(n) = my(v=vector(n, k, binomial(2*n, 2*k-1))); v/gcd(v); \\ Michel Marcus, May 06 2026

CROSSREFS

Cf. A006519, A007318, A034870.

Sequence in context: A179397 A111272 A124692 * A321118 A167790 A010708

Adjacent sequences: A091040 A091041 A091042 * A091044 A091045 A091046

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Jan 23 2004

EXTENSIONS

More terms from Michel Marcus, May 06 2026

STATUS

approved