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Irregular triangle read by rows: T(n,k) = number of "pat" permutations of [1..n] with k descents.
1

%I #17 Feb 26 2024 11:23:34

%S 1,0,1,0,2,0,1,4,0,0,12,2,0,0,12,30,0,0,4,100,28,0,0,0,140,280,9,0,0,

%T 0,90,980,360,0,0,0,22,1680,2940,220,0,0,0,0,1540,10584,4620,52,0,0,0,

%U 0,728,20790,33264,4004,0,0,0,0,140,24024,121968,60060,1820,0,0,0,0,0,16380,264264,396396,65520,340,0,0,0,0,0,6120

%N Irregular triangle read by rows: T(n,k) = number of "pat" permutations of [1..n] with k descents.

%C Row sums are Catalan numbers A000108.

%H D. Callan, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.119.05.415">Flexagons lead to a Catalan number identity</a>, Amer. Math. Monthly, 119 (May 2012), 415-419.

%H Tad White, <a href="https://arxiv.org/abs/2401.01462">Quota Trees</a>, arXiv:2401.01462 [math.CO], 2024. See p. 20.

%F T(n,k) = binomial(2n-2k-1,k)*binomial(2k,n-k-1)/(2n-2k-1).

%e Triangle begins:

%e 1

%e 0 1

%e 0 2

%e 0 1 4

%e 0 0 12 2

%e 0 0 12 30

%e 0 0 4 100 28

%e 0 0 0 140 280 9

%e 0 0 0 90 980 360

%e 0 0 0 22 1680 2940 220

%e ...

%p A212206 := proc(n,k)

%p binomial(2*n-2*k-1,k)*binomial(2*k,n-k-1)/(2*n-2*k-1) ;

%p end proc:

%p for n from 0 to 15 do

%p for k from 0 to floor((2*n-1)/3) do

%p printf("%d,",A212206(n,k)) ;

%p end do:

%p end do; # _R. J. Mathar_, Jun 26 2012

%Y Cf. A000108.

%K nonn,tabf

%O 1,5

%A _N. J. A. Sloane_, May 15 2012