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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 (list; table; graph; refs; listen; history; text; internal format)
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(1-4x^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(n-2k). Then a(n) = Sum_{k=0..floor(n/2)} (n/(n-k))*(-1)^k*binomial(n-k,k) *b(n-2k).

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 x-axis (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

Jinyuan Wang, Rows n=0..200 of triangle, flattened

Paul Barry, The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.

FORMULA

T(n,k) = binomial(n,k)*binomial(n-k, floor((n-k)/2)).

G.f.: G=G(t,z) satisfies z(1-2z-tz)G^2+(1-2z-tz)G-1=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))*(T(n-1, k) + T(n-1, k-1)), 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(n-k, 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[n-k, Floor[(n-k)/2]], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 11 2019 *)

PROG

(PARI) T(n, k) = binomial(n, k)*binomial(n-k, (n-k)\2); \\ Michel Marcus, Feb 10 2019

(MAGMA) [[Binomial(n, k)*Binomial(n-k, Floor((n-k)/2)): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 11 2019

(Sage) [[binomial(n, k)*binomial(n-k, floor((n-k)/2)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 11 2019

CROSSREFS

Cf. A132894.

Sequence in context: A094436 A286012 A094441 * A159830 A293472 A046726

Adjacent sequences:  A107227 A107228 A107229 * A107231 A107232 A107233

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, May 13 2005

STATUS

approved

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Last modified December 14 12:04 EST 2019. Contains 329979 sequences. (Running on oeis4.)