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A071944 Triangle read by rows giving numbers of paths in a lattice satisfying certain conditions. 4
1, 1, 1, 1, 2, 2, 1, 3, 5, 6, 1, 4, 9, 16, 19, 1, 5, 14, 31, 54, 63, 1, 6, 20, 52, 111, 188, 219, 1, 7, 27, 80, 197, 405, 676, 787, 1, 8, 35, 116, 320, 752, 1508, 2492, 2897, 1, 9, 44, 161, 489, 1276, 2900, 5712, 9361, 10869, 1, 10, 54, 216, 714, 2034, 5095, 11296, 21933, 35702, 41414 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

G. C. Greubel, Rows n = 0..100 of triangle, flattened

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)/(n+1))*Sum_{i=0..k/3} binomial(n+1, i)*binomial(n+k -3*i, n), for k <= n.

EXAMPLE

Triangle begins with:

  1;

  1,   1;

  1,   2,   2;

  1,   3,   5,   6;

  1,   4,   9,  16,  19;

  1,   5,  14,  31,  54,  63;

  1,   6,  20,  52, 111, 188, 219;

  1,   7,  27,  80, 197, 405, 676, 787;

  ...

MAPLE

a := proc(n, k) if k<=n then (n-k+1)*sum(binomial(n+1, i)*binomial(n+k-3*i, n), i=0..k/3)/(n+1) else 0 fi end;

MATHEMATICA

Table[((n-k+1)/(n+1))*Sum[Binomial[n+1, j]*Binomial[n+k-3*j, n], {j, 0, k/3}], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 17 2019 *)

PROG

(PARI) {T(n, k) = ((n-k+1)/(n+1))*sum(j=0, floor(k/3), binomial(n+1, j)* binomial(n+k -3*j, n))}; \\ G. C. Greubel, Mar 17 2019

(MAGMA) [[((n-k+1)/(n+1))*(&+[Binomial(n+1, j)*Binomial(n+k -3*j, n): j in [0..Floor(k/3)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Mar 17 2019

(Sage) [[((n-k+1)/(n+1))*sum(binomial(n+1, j)*binomial(n+k-3*j, n) for j in (0..floor(k/3))) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 17 2019

CROSSREFS

Diagonal entries form A071969.

Sequence in context: A059718 A076038 A095788 * A309495 A080955 A125231

Adjacent sequences:  A071941 A071942 A071943 * A071945 A071946 A071947

KEYWORD

nonn,easy,tabl

AUTHOR

N. J. A. Sloane, Jun 15 2002

EXTENSIONS

More terms from Emeric Deutsch, Dec 19 2003

STATUS

approved

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Last modified November 12 09:01 EST 2019. Contains 329052 sequences. (Running on oeis4.)