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A194586 Triangle read by rows, T(n,k) the coefficients of the polynomials Sum_{k=0..n} binomial(n,k)*A056040(k)*(k mod 2)*q^k. 2
0, 0, 1, 0, 2, 0, 0, 3, 0, 6, 0, 4, 0, 24, 0, 0, 5, 0, 60, 0, 30, 0, 6, 0, 120, 0, 180, 0, 0, 7, 0, 210, 0, 630, 0, 140, 0, 8, 0, 336, 0, 1680, 0, 1120, 0, 0, 9, 0, 504, 0, 3780, 0, 5040, 0, 630, 0, 10, 0, 720, 0, 7560, 0, 16800, 0, 6300, 0, 0, 11, 0, 990, 0, 13860, 0, 46200, 0, 34650, 0, 2772, 0, 12 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Substituting q^k -> 1/(floor(k/2)+1) in the polynomials gives the complementary Motzkin numbers A005717. (See A089627 for the Motzkin numbers and A163649 for the extended Motzkin numbers.)

LINKS

Table of n, a(n) for n=0..79.

Peter Luschny, The lost Catalan numbers.

FORMULA

egf(x,y) = x*y*exp(x)*BesselI(0,2*x*y).

EXAMPLE

               0

              0, 1

            0, 2, 0

           0, 3, 0, 6

         0, 4, 0, 24, 0

       0, 5, 0, 60, 0, 30

    0, 6, 0, 120, 0, 180, 0

  0, 7, 0, 210, 0, 630, 0, 140

                0

                q

               2 q

            3 q + 6 q^3

           4 q + 24 q^3

       5 q + 60 q^3  + 30 q^5

      6 q + 120 q^3  + 180 q^5

  7 q + 210 q^3  + 630 q^5  + 140 q^7

MAPLE

A194586 := proc(n, k) local j, swing; swing := n -> n!/iquo(n, 2)!^2:

add(binomial(n, j)*swing(j)*q^j*(j mod 2), j=0..n); coeff(%, q, k) end:

seq(print(seq(A194586(n, k), k=0..n)), n=0..8);

CROSSREFS

Row sums are A109188. Cf. A163649, A089627.

Sequence in context: A298203 A298209 A211871 * A288437 A287736 A180969

Adjacent sequences:  A194583 A194584 A194585 * A194587 A194588 A194589

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Aug 29 2011

STATUS

approved

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Last modified March 18 12:10 EDT 2019. Contains 321283 sequences. (Running on oeis4.)