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A206294 Riordan array (1, x/(1-x)^3). 1
1, 0, 1, 0, 3, 1, 0, 6, 6, 1, 0, 10, 21, 9, 1, 0, 15, 56, 45, 12, 1, 0, 21, 126, 165, 78, 15, 1, 0, 28, 252, 495, 364, 120, 18, 1, 0, 36, 462, 1287, 1365, 680, 171, 21, 1, 0, 45, 792, 3003, 4368, 3060, 1140, 231, 24, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Triangle T(n,k), read by rows, given by (0, 3, -1, 2/3, -1/6, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Row sums are A052529(n).

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

FORMULA

T(n,0) = 0^n, T(n,k) = C(n-1+2k, n-k) for k > 0.

T(n,n) = 1, T(k+1,k) = 3*k = A008585(k), T(k+2,k) = A081266(k).

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A052529(n), A052910(n) for x = 0, 1, 2 respectively.

G.f.: (1-x)^3/((1-x)^3-y*x).

EXAMPLE

Triangle begins:

1

0, 1

0, 3, 1

0, 6, 6, 1

0, 10, 21, 9, 1

0, 15, 56, 45, 12, 1

0, 21, 126, 165, 78, 15, 1

0, 28, 252, 495, 364, 120, 18, 1

0, 36, 462, 1287, 1365, 680, 171, 21, 1

0, 45, 792, 3003, 4368, 3060, 1140, 231, 24, 1

0, 55, 1287, 6435, 12376, 11628, 5985, 1771, 300, 27, 1

0, 66, 2002, 12870, 31824, 38760, 26324, 10626, 2600, 378, 30, 1

MATHEMATICA

Table[If[n == 0 && k == 0 , 1, Binomial[n - 1 + 2 k, n - k]], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 25 2017 *)

PROG

(PARI) {T(n, k)=polcoeff(1/(1-x+x*O(x^(n-k)))^(3*k), n-k)}

(PARI) {T(n, k)=polcoeff(polcoeff((1-x)^3/((1-x)^3-y*x +x*O(x^n)), n, x), k, y)}

for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

CROSSREFS

Cf. Columns: A000007, A000217 (triangular numbers), A000389, A000581, A001288, A010967..(+3)..A011000, A017714..(+3)..A017762.

Cf. A127893.

Sequence in context: A213668 A256595 A004590 * A058150 A058151 A257673

Adjacent sequences:  A206291 A206292 A206293 * A206295 A206296 A206297

KEYWORD

easy,nonn,tabl

AUTHOR

Philippe Deléham, Feb 05 2012

STATUS

approved

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Last modified October 15 11:59 EDT 2018. Contains 316231 sequences. (Running on oeis4.)