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A071949 Triangle read by rows of numbers of paths in a lattice satisfying certain conditions. 0
1, 1, 2, 1, 4, 10, 1, 6, 24, 66, 1, 8, 42, 172, 498, 1, 10, 64, 326, 1360, 4066, 1, 12, 90, 536, 2706, 11444, 34970, 1, 14, 120, 810, 4672, 23526, 100520, 312066, 1, 16, 154, 1156, 7410, 42024, 211546, 911068, 2862562, 1, 18, 192, 1582, 11088, 69002, 387456, 1951494, 8457504, 26824386 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..54.

D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad J. Math., 49 (1997), 301-320.

FORMULA

T(n, k) = (n-k+1)*(Sum_{j=0..k-1} (2^(j+1)*binomial(k, j+1)*binomial(n+k, j)))/k for 0<k<=n; T(n, 0)=1; T(n, k)=0 for k>n.

T(n,0) = 1, T(n,n) = T(n,n-1) + T(n+1,n-1), otherwise T(n,k) = T(n,k-1) + T(n+1,k-1) + T(n-1,k). [Gerald McGarvey, Oct 09 2008]

EXAMPLE

Triangle begins:

  1;

  1, 2;

  1, 4, 10;

  1, 6, 24,  66;

  1, 8, 42, 172, 498;

  ...

MAPLE

T := proc(n, k) if k>0 and k<=n then (n-k+1)*sum(2^(j+1)*binomial(k, j+1)*binomial(n+k, j), j=0..k-1)/k elif k=0 then 1 else 0 fi end: seq(seq(T(n, k), k=0..n), n=0..10);

MATHEMATICA

T[_, 0] = 1;

T[n_, n_] := T[n, n] = T[n, n-1] + T[n+1, n-1];

T[n_, k_] /; 0 <= k < n := T[n, k] = T[n, k-1] + T[n+1, k-1] + T[n-1, k];

T[_, _] = 0;

Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Nov 15 2019 *)

CROSSREFS

T(n, n)=A027307(n).

Sequence in context: A279927 A137634 A100229 * A297506 A297720 A297654

Adjacent sequences:  A071946 A071947 A071948 * A071950 A071951 A071952

KEYWORD

nonn,easy,tabl

AUTHOR

N. J. A. Sloane, Jun 15 2002

EXTENSIONS

Edited by Emeric Deutsch, Mar 04 2004

STATUS

approved

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Last modified November 28 13:42 EST 2021. Contains 349412 sequences. (Running on oeis4.)