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A127893 Riordan array (1/(1-x)^3, x/(1-x)^3). 7
1, 3, 1, 6, 6, 1, 10, 21, 9, 1, 15, 56, 45, 12, 1, 21, 126, 165, 78, 15, 1, 28, 252, 495, 364, 120, 18, 1, 36, 462, 1287, 1365, 680, 171, 21, 1, 45, 792, 3003, 4368, 3060, 1140, 231, 24, 1, 55, 1287, 6435, 12376, 11628, 5985, 1771, 300, 27, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Row sums are A052529(n+1). Diagonal sums are A095263. Inverse is A127894.

From Peter Bala, Jul 22 2014: (Start)

Let M denote the unsigned version of the lower unit triangular array A122432 and for k = 0,1,2,... define M(k) to be the lower unit triangular block array

/I_k 0\

\ 0  M/

having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle equals the infinite matrix product M(0)*M(1)*M(2)*... (which is clearly well-defined). See the Example section. (End)

LINKS

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

FORMULA

T(n,k) = C(n+2*k+2, n-k).

Recurrence: T(n+1,k+1) = sum {i = 0..n-k} 1/2*(i+1)*(i+2)*T(n-i,k). - Peter Bala, Jul 22 2014

G.f.: 1/((1-x)^3-x*y). - Vladimir Kruchinin, Apr 27 2015

EXAMPLE

Triangle begins

  1,

  3, 1,

  6, 6, 1,

  10, 21, 9, 1,

  15, 56, 45, 12, 1,

  21, 126, 165, 78, 15, 1,

  28, 252, 495, 364, 120, 18, 1,

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

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

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

  66, 2002, 12870, 31824, 38760, 26334, 10626, 2600, 378, 30, 1

  ...

From Peter Bala, Jul 22 2014: (Start)

With the arrays M(k) as defined in the Comments section, the infinite product M(0*)M(1)*M(2)*... begins

/ 1         \/1         \/1       \       / 1       \

| 3  1      ||0  1      ||0 1      |      | 3  1    |

| 6  3 1    ||0  3 1    ||0 0 1    |... = | 6  6 1  |

|10  6 3 1  ||0  6 3 1  ||0 0 3 1  |      |10 21 9 1|

|15 10 6 3 1||0 10 6 3 1||0 0 6 3 1|      |...      |

|...        ||...       ||...      |      |...      |

(End)

MAPLE

seq(seq(binomial(n+2*k+2, n-k), k=0..n), n=0..10); # Robert Israel, Apr 28 2015

MATHEMATICA

Flatten@ Table[Binomial[n + 2 k - 1, n - k], {n, 10}, {k, n}] (* Michael De Vlieger, Apr 27 2015 *)

PROG

(PARI) for(n=0, 10, for(k=0, n, print1(binomial(n +2*k + 2, n-k), ", "))) \\ G. C. Greubel, Apr 29 2018

(MAGMA) /* As triangle */ [[Binomial(n +2*k + 2, n-k): k in [0..n]]: n in [0..10]];  // G. C. Greubel, Apr 29 2018

(GAP) Flat(List([0..10], n->List([0..n], k->Binomial(n+2*k+2, n-k)))); # Muniru A Asiru, Apr 30 2018

CROSSREFS

Cf. A052529, A095263, A122432, A127894.

Sequence in context: A235706 A124847 A249251 * A127895 A325005 A325013

Adjacent sequences:  A127890 A127891 A127892 * A127894 A127895 A127896

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Feb 04 2007

STATUS

approved

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Last modified March 7 10:30 EST 2021. Contains 341869 sequences. (Running on oeis4.)