A125165 Triangle read by rows: T(n,k) = C(n,k) + 3*C(n,k+1) + 2*C(n,k+2) (0<=k<=n).

1, 4, 1, 9, 5, 1, 16, 14, 6, 1, 25, 30, 20, 7, 1, 36, 55, 50, 27, 8, 1, 49, 91, 105, 77, 35, 9, 1, 64, 140, 196, 182, 112, 44, 10, 1, 81, 204, 336, 378, 294, 156, 54, 11, 1, 100, 285, 540, 714, 672, 450, 210, 65, 12, 1, 121, 385, 825, 1254, 1386, 1122, 660, 275, 77, 13, 1, 144

Offset: 0

Author

Gary W. Adamson, Nov 21 2006

Keywords

• nonn
• tabl

Comments

Binomial transform of the infinite tridiagonal matrix M with main diagonal (1,1,1...), subdiagonal (3,3,3...) and subsubdiagonal (2,2,2...).

Sum of entries in row n = 6*2^n-2n-5 = A050488(n+1).

Riordan array ((1+x)/(1-x)^3, x/(1-x)). - Philippe Deléham, Dec 07 2013

Examples

			Triangle starts:
1;
4, 1;
9, 5, 1;
16, 14, 6, 1;
25, 30, 20, 7, 1;
36, 55, 50, 27, 8, 1;
49, 91, 105, 77, 35, 9, 1;
		

Links

• G. C. Greubel, Rows n=0..100 of triangle, flattened

Crossrefs

Cf. A050488, A000290.

Programs

Magma

[[Binomial(n,k) + 3*Binomial(n,k+1) + 2*Binomial(n,k+2): k in [0..n]]: n in [0..15]]; // G. C. Greubel, Oct 23 2018

Maple

T:=(n,k)->binomial(n,k)+3*binomial(n,k+1)+2*binomial(n,k+2): for n from 0 to 11 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Mathematica

Table[Binomial[n,k]+3Binomial[n,k+1]+2Binomial[n,k+2],{n,0,15},{k,0,n}]//Flatten (* Harvey P. Dale, Nov 20 2016 *)

PARI

for(n=0,15, for(k=0,n, print1(binomial(n,k) + 3*binomial(n,k+1) + 2*binomial(n,k+2), ", "))) \\ G. C. Greubel, Oct 23 2018

Formula

T(n,k) = T(n-1,k) + T(n-1,k-1) for n>=k>=1.

T(n,0) = (n+1)^2 = A000290(n+1).

T(n,k) = 3*T(n-1,k) + T(n-1,k-1)-3*T(n-2,k)-2*T(n-2,k-1)+T(n-3,k)+T(n-2,k-1), T(0,0)=1, T(1,0)=4, T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 10 2014

exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(16 + 14*x + 6*x^2/2! + x^3/3!) = 16 + 30*x + 50*x^2/2! + 77*x^3/3! + 112*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). Cf. A233295. - Peter Bala, Dec 21 2014

Extensions

Edited by N. J. A. Sloane, Dec 02 2006