A108448 -id:A108448 - cp's OEIS frontend

A216182 Riordan array ((1+x)/(1-x)^2, x(1+x)^2/(1-x)^2).

1, 3, 1, 5, 7, 1, 7, 25, 11, 1, 9, 63, 61, 15, 1, 11, 129, 231, 113, 19, 1, 13, 231, 681, 575, 181, 23, 1, 15, 377, 1683, 2241, 1159, 265, 27, 1, 17, 575, 3653, 7183, 5641, 2047, 365, 31, 1, 19, 833, 7183, 19825, 22363, 11969, 3303, 481, 35, 1

Offset: 0

Philippe Deléham, Mar 11 2013

Triangle formed of odd-numbered columns of the Delannoy triangle A008288.

			Triangle begins
   1;
   3,   1;
   5,   7,    1;
   7,  25,   11,    1;
   9,  63,   61,   15,    1;
  11, 129,  231,  113,   19,    1;
  13, 231,  681,  575,  181,   23,   1;
  15, 377, 1683, 2241, 1159,  265,  27,  1;
  17, 575, 3653, 7183, 5641, 2047, 365, 31, 1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. (columns:) A005408, A001845, A001847, A001849, A008419.
Cf. Diagonals: A000012, A004767, A060820.
Cf. A008288 (Delannoy triangle), A114123 (even-numbered columns of A008288).

A216182[n_, k_]:= Hypergeometric2F1[-n +k, -2*k-1, 1, 2];
Table[A216182[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 19 2021 *)

def A216182(n,k): return simplify( hypergeometric([-n+k, -2*k-1], [1], 2) )
flatten([[A216182(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 19 2021

T(2n, n) = A108448(n+1).
Sum_{k=0..n} T(n,k) = A073717(n+1).
From G. C. Greubel, Nov 19 2021: (Start)
T(n, k) = A008288(n+k+1, 2*k+1).
T(n, k) = hypergeometric([-n+k, -2*k-1], [1], 2). (End)

A108446 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 peaks of the form ud.

Original entry on oeis.org

1, 1, 1, 4, 5, 1, 20, 32, 13, 1, 113, 223, 135, 26, 1, 688, 1620, 1300, 412, 45, 1, 4404, 12064, 12050, 5350, 1030, 71, 1, 29219, 91335, 109134, 62450, 17575, 2247, 105, 1, 199140, 699689, 973077, 682234, 254625, 49210, 4438, 148, 1, 1385904, 5407744

Offset: 0

Emeric Deutsch, Jun 10 2005

Row sums yield A027307. Column 0 yields A108447. T(n,n-1) = A008778(n-1) = n(n^2+6n-1)/6. Number of ud peaks in all paths from (0,0) to (3n,0) is given by A108448.

			T(2,1) = 5 because we have udUdd, uudd, Uddud, Ududd and Uuddd.
Triangle begins:
1;
1,1;
4,5,1;
20,32,13,1;
113,223,135,26,1;

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

Cf. A027307, A008778, A108447, A108448, A108425, A108426.

T:=proc(n,k) if n=0 and k=0 then 1 elif n=0 then 0 elif k=n then 1 elif k=n then 1 else (1/n)*binomial(n,k)*sum(binomial(n-k,j)*binomial(n+2*j,k+j-1),j=0..n-k) fi end: for n from 0 to 9 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

T[0, 0] = 1; T[n_, k_] := (1/n) Binomial[n, k]*Sum[Binomial[n-k, j]* Binomial[n+2j, k+j-1], {j, 0, n-k}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 19 2018 *)

T(n,k) = (1/n) binomial(n, k)*sum(binomial(n-k,j)*binomial(n+2j,k+j-1), j=0..n-k).

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

cp's OEIS Frontend

A216182 Riordan array ((1+x)/(1-x)^2, x(1+x)^2/(1-x)^2).

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