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A258820 Reversed rows of A178252 presented as diagonals of an irregular triangle. 3
1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 5, 2, 1, 1, 3, 10, 1, 1, 7, 5, 5, 1, 1, 4, 7, 5, 1, 1, 9, 28, 35, 3, 1, 1, 5, 12, 14, 7, 1, 1, 11, 15, 21, 14, 7, 1, 1, 6, 55, 30, 126, 28, 1, 1, 13, 22, 165, 42, 21, 4, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

The diagonals of T are the reversed rows of A178252. E.g., the fifth diagonal of T is (1,2,2,1,1) from the example below, which is the fifth reversed row of A178252.

Factoring out the greatest common divisor (gcd) of the coefficients of the sub-polynomials in the indeterminate q of the polynomials in s presented on p. 12 of the Alexeev et al. link and then evaluating the sub-polynomials at q=1 gives the first nine rows of T given in the example below. E.g., for k=6 (the seventh row), q*s^6 + (6*q + 9*q^2) s^4 + (15*q + 15*q^2) s^2 + 5  = q*s^6 + 3*(2*q + 3*q^2)*s^4 + 15*(q + q^2)*s^2 + 5 generates (1,2+3,1+1,1)=(1,5,2,1).

The row length sequence of this irregular triangle is A008619(n) = 1 + floor(n/2). - Wolfdieter Lang, Aug 25 2015

LINKS

Table of n, a(n) for n=0..63.

N. Alexeev, J. Andersen, R. Penner, P. Zograf, Enumeration of chord diagrams on many intervals and their non-orientable analogs, arXiv:1307.0967 [math.CO], 2013-2014.

FORMULA

T(n,k) =  A178252(n-k,n-2k) = A055151(n,k) / A161642(n,k) = A007318(n,2k) * A000108(k) / A161642(n,k) = n! / [(n-2k)! k! (k+1)! A161642(n,k)] =  A003989(n-k+1,k+1) * (n-k)! / [ (n-2k)! (k+1)! ], where A003989(j,k) = gcd(j,k).

EXAMPLE

The irregular triangle T(n,k) starts

n\k  0 1  2  3 4 5 ...

0:   1

1:   1

2:   1 1

3:   1 1

4:   1 3  1

5:   1 2  1

6:   1 5  2  1

7:   1 3 10  1

8:   1 7  5  5 1

9:   1 4  7  5 1

10:  1 9 28 35 3 1

... reformatted. - Wolfdieter Lang, Aug 25 2015

MATHEMATICA

max = 15; coes = Table[ PadRight[ CoefficientList[ BernoulliB[n, x], x], max], {n, 0, max-1}]; inv = Inverse[coes] // Numerator; t[n_, k_] := inv[[n, k]]; t[n_, k_] /; k == n+1 = 1; Table[t[n-k+1, k], {n, 2, max+1}, {k, 2, Floor[n/2]+1}] // Flatten (* Jean-Fran├žois Alcover, Jul 22 2015 *)

CROSSREFS

Cf. A050169, A161642, A055151, A088617, A178252, A107711, A165661, A003989, A008619.

Sequence in context: A318440 A307790 A189965 * A030347 A010275 A176494

Adjacent sequences:  A258817 A258818 A258819 * A258821 A258822 A258823

KEYWORD

nonn,tabf,easy

AUTHOR

Tom Copeland, Jun 18 2015

STATUS

approved

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Last modified January 21 13:55 EST 2020. Contains 331113 sequences. (Running on oeis4.)