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A348869 Triangle T(n,c) counting Motzkin Paths of length n with c sections starting with an up-step at level 0.

%I A348869 #8 Aug 12 2023 04:38:04
%S A348869 1,2,4,1,8,4,17,12,1,38,32,6,89,82,24,1,216,208,80,8,539,530,243,40,1,
%T A348869 1374,1364,702,160,10,3562,3551,1975,564,60,1,9360,9348,5484,1840,280,
%U A348869 12,24871,24858,15144,5716,1125,84,1,66706,66692,41768,17208,4102,448,14
%N A348869 Triangle T(n,c) counting Motzkin Paths of length n with c sections starting with an up-step at level 0.
%C A348869 This is a Sequence Transform of A086615. A086615(n-2) counts the Motzkin Paths of length n which start with an u-step, return to the horizontal level once with a d-step and remain there (with any number of trailing h-steps). These might be called single-return M-Paths. The path of length n=2 is ud. The paths of length 3 are udh, uhd. The Paths of length 4 are uudd, udhh, uhdh and uhhd. A Motzkin Path can be chopped into subpaths of that type by splitting it at each u-step that starts from the horizontal line. [The exception is the path that consists entirely of h-steps.] The triangle of the Sequence Transform T(n,c) counts how many Motzkin Paths of length n which start with an u-step are concatenations of c of these single-return M-paths. T(n,1) are the single-return M-Paths. Row sums and column 1 are an INVERT transform pair.
%F A348869 G.f.: 1/(1-y*g086615(x)) where g086615(x) = x^2 +2*x^3 +4*x^4 +8*x^5 +17*x^6 +....
%e A348869 The triangle starts
%e A348869       1
%e A348869       2
%e A348869       4     1
%e A348869       8     4
%e A348869      17    12     1
%e A348869      38    32     6
%e A348869      89    82    24     1
%e A348869     216   208    80     8
%e A348869     539   530   243    40    1
%e A348869    1374  1364   702   160   10
%e A348869    3562  3551  1975   564   60   1
%e A348869    9360  9348  5484  1840  280  12
%e A348869   24871 24858 15144  5716 1125  84  1
%e A348869   66706 66692 41768 17208 4102 448 14
%e A348869 T(4,2)=1 counts udud.
%e A348869 T(5,1)=8 counts uuddh uudhd uuhdd udhhh uhudd uhdhh uhhdh uhhhd.
%e A348869 T(5,2)=4 counts ududh uduhd udhud uhdud.
%e A348869 T(2n,n) = 1 counts udududu... (ud repeated n times).
%p A348869 A348869 := proc(n,c)
%p A348869     local g,x,y ;
%p A348869     g := add( A086615(i)*x^(i+2),i=0..n) ;
%p A348869     1/(1-y*g) ;
%p A348869     coeftayl(%,x=0,n) ;
%p A348869     coeftayl(%,y=0,c) ;
%p A348869 end proc:
%p A348869 seq(seq( A348869(n,c),c=1..n/2),n=2..10) ;
%t A348869 b[n_] := b[n] = If[n <= 3, 2^n, (3*(n+1)*b[n-1] + (n-4)*b[n-2] - 3*(n-1)*b[n-3])/(n+2)];
%t A348869 T[n_, c_] := Module[{g, x, y}, g = Sum[b[i]*x^(i+2), {i, 0, n}]; 1/(1-y*g) // SeriesCoefficient[#, {x, 0, n}]& // SeriesCoefficient[#, {y, 0, c}]&];
%t A348869 Table[T[n, c], {n, 2, 15}, {c, 1, n/2}] // Flatten (* _Jean-François Alcover_, Aug 12 2023, after Maple code *)
%Y A348869 Cf. A086615 (column c=1), A002026 (row sums)
%K A348869 nonn,tabf
%O A348869 2,2
%A A348869 _R. J. Mathar_, Nov 02 2021