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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 January 18 11:25 EST 2019. Contains 319271 sequences. (Running on oeis4.)