

A107230


A number triangle of inverse Chebyshev transforms.


5



1, 1, 1, 2, 2, 1, 3, 6, 3, 1, 6, 12, 12, 4, 1, 10, 30, 30, 20, 5, 1, 20, 60, 90, 60, 30, 6, 1, 35, 140, 210, 210, 105, 42, 7, 1, 70, 280, 560, 560, 420, 168, 56, 8, 1, 126, 630, 1260, 1680, 1260, 756, 252, 72, 9, 1, 252, 1260, 3150, 4200, 4200, 2520, 1260, 360, 90, 10, 1
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OFFSET

0,4


COMMENTS

First column is A001405, second column is A100071, third column is A107231. Row sums are A005773(n+1), diagonal sums are A026003. The inverse Chebyshev transform concerned takes a g.f. g(x)>(1/sqrt(14x^2))g(xc(x^2)) where c(x) is the g.f. of A000108. It transforms a(n) to b(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*a(n2k). Then a(n) = Sum_{k=0..floor(n/2)} (n/(nk))*(1)^k*binomial(nk,k) *b(n2k).
Triangle read by rows: T(n,k) is the number of paths of length n with steps U=(1,1), D=(1,1) and H=(1,0), starting at (0,0), staying weakly above the xaxis (i.e., left factors of Motzkin paths) and having k H steps. Example: T(3,1)=6 because we have HUD. HUU, UDH, UHD, UHU and UUH. Sum_{k=0..n} k*T(n,k) = A132894(n).  Emeric Deutsch, Oct 07 2007


LINKS



FORMULA

T(n,k) = binomial(n,k)*binomial(nk, floor((nk)/2)).
G.f.: G=G(t,z) satisfies z*(12*zt*z)*G^2+(12*zt*z)*G1=0.  Emeric Deutsch, Oct 07 2007
E.g.f.: exp(x*y)*(BesselI(0,2*x)+BesselI(1,2*x)).  Vladeta Jovovic, Dec 02 2008
T(n, k) = (n/floor((n+k+1)/2))*(T(n1, k) + T(n1, k1)), n >= k > 0.  Mikhail Kurkov, Feb 10 2019


EXAMPLE

Triangle begins
1;
1, 1;
2, 2, 1;
3, 6, 3, 1;
6, 12, 12, 4, 1;
10, 30, 30, 20, 5, 1;


MAPLE

T:=proc(n, k) options operator, arrow: binomial(n, k)*binomial(nk, floor((1/2)*n(1/2)*k)) end proc: for n from 0 to 11 do seq(T(n, k), k=0..n) end do; # yields sequence in triangular form  Emeric Deutsch, Oct 07 2007


MATHEMATICA

Table[Binomial[n, k]*Binomial[nk, Floor[(nk)/2]], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 11 2019 *)


PROG

(PARI) T(n, k) = binomial(n, k)*binomial(nk, (nk)\2); \\ Michel Marcus, Feb 10 2019
(Magma) [[Binomial(n, k)*Binomial(nk, Floor((nk)/2)): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 11 2019
(Sage) [[binomial(n, k)*binomial(nk, floor((nk)/2)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 11 2019


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KEYWORD



AUTHOR



STATUS

approved



