A125165 - OEIS

%I #27 Sep 08 2022 08:45:28

%S 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,

%T 1,64,140,196,182,112,44,10,1,81,204,336,378,294,156,54,11,1,100,285,

%U 540,714,672,450,210,65,12,1,121,385,825,1254,1386,1122,660,275,77,13,1,144

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

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

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

%C Riordan array ((1+x)/(1-x)^3, x/(1-x)). - _Philippe Deléham_, Dec 07 2013

%H G. C. Greubel, <a href="/A125165/b125165.txt">Rows n=0..100 of triangle, flattened</a>

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

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

%F 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

%F 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

%e Triangle starts:

%e 1;

%e 4, 1;

%e 9, 5, 1;

%e 16, 14, 6, 1;

%e 25, 30, 20, 7, 1;

%e 36, 55, 50, 27, 8, 1;

%e 49, 91, 105, 77, 35, 9, 1;

%p 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

%t 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 *)

%o (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

%o (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

%Y Cf. A050488, A000290.

%K nonn,tabl

%O 0,2

%A _Gary W. Adamson_, Nov 21 2006

%E Edited by _N. J. A. Sloane_, Dec 02 2006