A119304 Triangle read by rows: T(n,k) = binomial(4n-k,n-k), 0 <= k <= n.
1, 4, 1, 28, 7, 1, 220, 55, 10, 1, 1820, 455, 91, 13, 1, 15504, 3876, 816, 136, 16, 1, 134596, 33649, 7315, 1330, 190, 19, 1, 1184040, 296010, 65780, 12650, 2024, 253, 22, 1, 10518300, 2629575, 593775, 118755, 20475, 2925, 325, 25, 1, 94143280, 23535820
Offset: 0
Examples
Triangle begins
1;
4, 1;
28, 7, 1;
220, 55, 10, 1;
1820, 455, 91, 13, 1;
15504, 3876, 816, 136, 16, 1;
134596, 33649, 7315, 1330, 190, 19, 1;
Links
- Indranil Ghosh, Rows 0..125, flattened
Programs
-
Mathematica
Flatten[Table[Binomial[4n-k,n-k],{n,0,9},{k,0,n}]] (* Indranil Ghosh, Feb 26 2017 *) -
PARI
tabl(nn) = {for (n=0,nn,for (k=0,n,print1(binomial(4*n-k,n-k),", ");); print(););} \\ Indranil Ghosh, Feb 26 2017 -
Python
from sympy import binomial i=0 for n in range(12): for k in range(n+1): print(str(i)+" "+str(binomial(4*n-k,n-k))) i+=1 # Indranil Ghosh, Feb 26 2017
Formula
Riordan array (1/(1-4f(x)),f(x)) where f(x)(1-f(x))^3 = x.
From Peter Bala, Jun 04 2024: (Start)
'Horizontal' recurrence equation: T(n, 0) = binomial(4*n,n) and for k >= 1, T(n, k) = Sum_{i = 1..n+1-k} i*(i+1)/2 * T(n-1, k-2+i).
T(n, k) = Sum_{j = 0..n} binomial(n+j-1, j)*binomial(3*n-k-j, 2*n). (End)
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