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A283321 Triangle read by rows: T(n,k) (0 <= k <= n) = number of elements of alternating semigroup A_n of height k. 2
1, 1, 1, 1, 2, 1, 1, 9, 9, 3, 1, 16, 72, 48, 12, 1, 25, 200, 600, 300, 60, 1, 36, 450, 2400, 5400, 2160, 360, 1, 49, 882, 7350, 29400, 52920, 17640, 2520, 1, 64, 1568, 18816, 117600, 376320, 564480, 161280, 20160, 1, 81, 2592, 42336, 381024, 1905120, 5080320, 6531840, 1632960, 181440 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

T(n,1) = A000290(n) for n>0, except T(2,1) which equals 2. - Indranil Ghosh, Mar 16 2017

LINKS

Indranil Ghosh, Rows 0..100, flattened

G. N. Bakare, S. O. Makanjuola, Some Results on Properties of Alternating Semigroups, Nigerian Journal of Mathematics and Applications Volume 24,(2015), 184-192.

FORMULA

Bakare et al. give a formula, see Theorem 3.2.

EXAMPLE

Triangle begins:

1,

1,1,

1,2,1,

1,9,9,3,

1,16,72,48,12,

1,25,200,600,300,60,

1,36,450,2400,5400,2160,360,

...

MATHEMATICA

T[n_, k_]:=If[k==n, (n!/2), If[k==n - 1, n^2*(n - 1)!/2, Binomial[n, k]^2 * k!]]; Column[Table[If[n<2, 1, T[n, k]], {n, 0, 10}, {k, 0, n}]] (* Indranil Ghosh, Mar 16 2017 *)

PROG

(PARI) T(n, k) = if(k==n, (n!/2), if(k==n - 1, n^2*(n - 1)!/2,  binomial(n, k)^2 * k!));

tabl(nn) = {for(n=0, nn, for(k=0, n, print1(if(n<2, 1, T(n, k)), ", "); ); print(); ); };

tabl(10); \\ Indranil Ghosh, Mar 16 2017

(Python)

import math

f=math.factorial

def C(n, r): return f(n)/f(r)/f(n - r)

def T(n, k):

....if k==n: return f(n)/2

....elif k==n-1: return n**2 * f(n - 1) / 2

....else: return C(n, k)**2 * f(k)

i=0

for n in range(0, 100):

....for k in range(0, n+1):

........if n<2: print str(i)+" 1"

........else: print str(i)+" "+str(T(n, k))

....i+=1 # Indranil Ghosh, Mar 16 2017

CROSSREFS

For row sums see A283322.

Sequence in context: A128434 A176417 A119731 * A155718 A256168 A054768

Adjacent sequences:  A283318 A283319 A283320 * A283322 A283323 A283324

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Mar 15 2017

STATUS

approved

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Last modified October 20 21:58 EDT 2018. Contains 316404 sequences. (Running on oeis4.)