A119304 -id:A119304 - cp's OEIS frontend

A119301 Triangle read by rows: T(n,k) = binomial(3*n-k,n-k).

1, 3, 1, 15, 5, 1, 84, 28, 7, 1, 495, 165, 45, 9, 1, 3003, 1001, 286, 66, 11, 1, 18564, 6188, 1820, 455, 91, 13, 1, 116280, 38760, 11628, 3060, 680, 120, 15, 1, 735471, 245157, 74613, 20349, 4845, 969, 153, 17, 1, 4686825, 1562275, 480700, 134596, 33649, 7315

Offset: 0

Paul Barry, May 13 2006

First column is A005809. Second column is A025174.

Row sums are A045721. Inverse is Riordan array (1-3x,x(1-x)^2), A119302.

			Triangle begins
  1,
  3, 1,
  15, 5, 1,
  84, 28, 7, 1,
  495, 165, 45, 9, 1,
  3003, 1001, 286, 66, 11, 1,
  18564, 6188, 1820, 455, 91, 13, 1,
  116280, 38760, 11628, 3060, 680, 120, 15, 1
  ...
Horizontal recurrence: T(4,1) = 1*84 + 2*28 + 3*7 + 4*1 = 165. - _Peter Bala_, Dec 29 2014

Cf. A092392, A119304.

Cf. A005809, A025174, A045721, A119302.

T := proc(n,k) option remember;
`if`(n = 0, 1, add(i*T(n-1,k-2+i),i=1..n+1-k)) end:
for n from 0 to 9 do print(seq(T(n,k),k=0..n)) od; # Peter Luschny, Dec 30 2014

Flatten[Table[Binomial[3n-k,n-k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jul 28 2012 *)

G.f. g(x) = 2*sin(arcsin(3*sqrt(3*x)/2)/3)/sqrt(3*x) satisfies g(x) = 1/(1-x*g(x)^2).
Riordan array (1/(1-3*x*g(x)^2),x*g(x)^2) where g(x)=1+x*g(x)^3.
'Horizontal' recurrence equation: T(n,0) = binomial(3*n,n) and for k >= 1, T(n,k) = Sum_{i = 1..n+1-k} i*T(n-1,k-2+i). - Peter Bala, Dec 28 2014
T(n, k) = Sum_{j = 0..n} binomial(n+j-1, j)*binomial(2*n-k-j, n). - Peter Bala, Jun 04 2024

A119305 Riordan array (1-4x, x(1-x)^3).

Original entry on oeis.org

1, -4, 1, 0, -7, 1, 0, 15, -10, 1, 0, -13, 39, -13, 1, 0, 4, -80, 72, -16, 1, 0, 0, 95, -228, 114, -19, 1, 0, 0, -66, 462, -484, 165, -22, 1, 0, 0, 25, -630, 1375, -875, 225, -25, 1, 0, 0, -4, 588, -2772, 3185, -1428, 294, -28, 1, 0, 0, 0, -372, 4092, -8463, 6324, -2170, 372

Offset: 0

Paul Barry, May 13 2006

Inverse of number triangle binomial(4n-k, n-k), A119304. Row sums are A119306.

			Triangle begins
   1;
  -4,    1;
   0,   -7,    1;
   0,   15,  -10,    1;
   0,  -13,   39,  -13,    1;
   0,    4,  -80,   72,  -16,    1;
   0,    0,   95, -228,  114,  -19,    1;

Indranil Ghosh, Rows 0..101, flattened

Flatten[Table[(Binomial[3k,n-k]+4Binomial[3k,n-k-1])*(-1)^(n-k),{n,0,11},{k,0,n}]] (* Indranil Ghosh, Feb 26 2017 *)

tabl(nn) = {for (n=0,nn,for (k=0,n,print1((binomial(3*k,n-k)+4*binomial(3*k,n-k-1))*(-1)^(n-k),", "););print(););} \\ Indranil Ghosh, Feb 26 2017

Number triangle T(n,k) = (C(3k, n-k) + 4*C(3k, n-k-1))(-1)^(n-k).

cp's OEIS Frontend

A119301 Triangle read by rows: T(n,k) = binomial(3*n-k,n-k).

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