The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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, Rows n = 0..99, flattened Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2. FORMULA T(n,k) = binomial(n+2*k+2, n-k). Sum_{k=0..n} T(n, k) = A052529(n+1) (row sums). Sum_{k=0..floor(n/2)} T(n-k, k) = A095263(n+1) (diagonal sums). Recurrence: T(n+1, k+1) = Sum_{i = 0..n-k} binomial(i+2, 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+2k-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) [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 (Sage) flatten([[binomial(n+2*k+2, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 16 2021 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 5 15:27 EST 2022. Contains 358588 sequences. (Running on oeis4.)