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A079508 Triangle T(n,k) (n >= 2, k >= 1) of Raney numbers read by rows. 2
1, 0, 1, 0, 2, 1, 0, 0, 5, 1, 0, 0, 5, 9, 1, 0, 0, 0, 21, 14, 1, 0, 0, 0, 14, 56, 20, 1, 0, 0, 0, 0, 84, 120, 27, 1, 0, 0, 0, 0, 42, 300, 225, 35, 1, 0, 0, 0, 0, 0, 330, 825, 385, 44, 1, 0, 0, 0, 0, 0, 132, 1485, 1925, 616, 54, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,5

COMMENTS

There are only m nonzero entries in the m-th column.

Related to A033282: shift row n of A033282 triangle n places to the right and transpose the resulting table. - Michel Marcus, Feb 04 2014

LINKS

G. C. Greubel, Rows n=2..100 of triangle, flattened

G. Kreweras, Les preordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30 (see definition p. 26 and table p. 27).

G. N. Raney, Functional composition patterns and power series reversion, Trans. Amer. Math. Soc., 94 (1960), pp. 441-451.

FORMULA

T(n,k) = C(k, n-k) * C(n, k+1)/k. - Michel Marcus, Feb 04 2014

EXAMPLE

From Michel Marcus, Feb 04 2014: (Start)

Triangle starts:

  1;

  0, 1;

  0, 2, 1;

  0, 0, 5,  1;

  0, 0, 5,  9,  1;

  0, 0, 0, 21, 14,   1;

  0, 0, 0, 14, 56,  20,    1;

  0, 0, 0,  0, 84, 120,   27,    1;

  0, 0, 0,  0, 42, 300,  225,   35,   1;

  0, 0, 0,  0,  0, 330,  825,  385,  44,  1;

  0, 0, 0,  0,  0, 132, 1485, 1925, 616, 54, 1;

  ... (End)

MATHEMATICA

Table[Binomial[k, n-k]*Binomial[n, k+1]/k, {n, 2, 10}, {k, 1, n-1}]//Flatten (* G. C. Greubel, Jan 17 2019 *)

PROG

(PARI) tabl(nn) = {for (n = 2, nn, for (k = 1, n-1, print1(binomial(k, n-k)*binomial(n, k+1)/k, ", "); ); print(); ); } \\ Michel Marcus, Feb 04 2014

(MAGMA) [[Binomial(k, n-k)*Binomial(n, k+1)/k: k in [1..n-1]]: n in [2..10]]; // G. C. Greubel, Jan 17 2019

(Sage) [[binomial(k, n-k)*binomial(n, k+1)/k for k in (1..n-1)] for n in (2..10)] # G. C. Greubel, Jan 17 2019

(GAP) Flat(List([1..10], n->List([1..n-1], k-> Binomial(k, n-k)*Binomial(n , k+1)/k ))); # G. C. Greubel, Jan 17 2019

CROSSREFS

Sum of nonzero entries in each column gives A001003. Alternating sum of each column is 1. Second diagonal on right gives A000096.

Leftmost diagonal is A000108.

Sequence in context: A278213 A034093 A246187 * A057150 A185663 A262125

Adjacent sequences:  A079505 A079506 A079507 * A079509 A079510 A079511

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Jan 21 2003

EXTENSIONS

Corrected and extended by Michel Marcus, Feb 04 2014

STATUS

approved

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Last modified November 19 06:26 EST 2019. Contains 329310 sequences. (Running on oeis4.)