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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 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); MATHEMATICA sf[n_] := n!/Quotient[n, 2]!^2; row[n_] := Sum[Binomial[n, j] sf[j] q^j Mod[j, 2], {j, 0, n}] // CoefficientList[#, q]& // PadRight[#, n+1]&; Table[row[n], {n, 0, 12}] (* Jean-François Alcover, Jun 26 2019 *) CROSSREFS Row sums are A109188. Cf. A056040, A005717, 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 September 30 22:26 EDT 2020. Contains 337440 sequences. (Running on oeis4.)