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A108428 Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have k doubledescents (i.e., dd's). 0
1, 1, 1, 1, 4, 4, 1, 1, 9, 23, 23, 9, 1, 1, 16, 76, 156, 156, 76, 16, 1, 1, 25, 190, 650, 1167, 1167, 650, 190, 25, 1, 1, 36, 400, 2045, 5685, 9318, 9318, 5685, 2045, 400, 36, 1, 1, 49, 749, 5341, 21133, 50813, 77947, 77947, 50813, 21133, 5341, 749, 49, 1, 1, 64, 1288 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row n contains 2n terms (n > 0).

Row sums yield A027307.

T(n,1) = T(n,2n-2) = n^2*T(n,k) = T(n,2n-k-1) (mirror symmetry).

LINKS

Table of n, a(n) for n=0..59.

Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.

FORMULA

T(n, k) = (1/n)*Sum_{j=0..k} binomial(n, j)*binomial(n, k-j)*binomial(n+j, k+1).

G.f.: G = G(t, z) satisfies t^2*zG^3 - t^2*zG^2 - (1 + z - tz)G + 1 = 0.

EXAMPLE

T(2,1)=4 because we have udUdd, uudd, Uddud and Ududd.

Triangle begins:

  1;

  1,  1;

  1,  4,  4,   1;

  1,  9, 23,  23,   9,  1;

  1, 16, 76, 156, 156, 76, 16, 1;

  ...

MAPLE

a:=proc(n, k) if n=0 and k=0 then 1 elif n=0 then 0 else (1/n)*sum(binomial(n, j)*binomial(n, k-j)*binomial(n+j, k+1), j=0..k) fi end: print(1); for n from 1 to 8 do seq(a(n, k), k=0..2*n-1) od; # yields sequence in triangular form

CROSSREFS

Cf. A027307.

Sequence in context: A046539 A198929 A172347 * A174126 A075613 A221837

Adjacent sequences:  A108425 A108426 A108427 * A108429 A108430 A108431

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Jun 03 2005

STATUS

approved

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Last modified October 20 22:44 EDT 2019. Contains 328291 sequences. (Running on oeis4.)