A025177 Triangular array, read by rows: first differences in n,n direction of trinomial array A027907.

1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 2, 4, 4, 4, 2, 1, 1, 3, 7, 10, 12, 10, 7, 3, 1, 1, 4, 11, 20, 29, 32, 29, 20, 11, 4, 1, 1, 5, 16, 35, 60, 81, 90, 81, 60, 35, 16, 5, 1, 1, 6, 22, 56, 111, 176, 231, 252, 231, 176, 111, 56, 22, 6, 1, 1, 7, 29, 84, 189, 343, 518, 659, 714, 659, 518, 343

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabf
• easy

Comments

The Motzkin transforms of the rows starting (1, 2), (1, 3) and (1, 4), extended by zeros after their last element, are apparently in A026134, A026109 and A026110. - R. J. Mathar, Dec 11 2008

Examples

			               1
            1  0  1
         1  1  2  1  1
      1  2  4  4  4  2  1
   1  3  7 10 12 10  7  3  1
1  4 11 20 29 32 29 20 11  4  1
		

Links

• Georg Fischer, Table of n, a(n) for n = 0..2499 [rows 0..49; first 676 terms from _G. C. Greubel_]

Crossrefs

Columns include A025178, A025179, A025180, A025181, A025182.

Cf. A024996, A025192 (row sums).

Programs

Maple

A025177 := proc(n,k)
    option remember;
    if k < 0 or k > 2*n then
        0;
    elif n = 0 then
        1 ;
    elif n = 1 then
        op(k+1,[1,0,1]) ;
    else
        procname(n-1,k-2)+procname(n-1,k-1)+procname(n-1,k) ;
    end if;
end proc:
seq(seq(A025177(n,k),k=0..2*n),n=0..20)  ; # R. J. Mathar, Feb 25 2015

Mathematica

nmax = 10; CoefficientList[CoefficientList[Series[(1 - y*x)/(1 - x*(1 + y + y^2)), {x, 0, nmax}, {y, 0, 2*nmax}], x], y] // Flatten (* G. C. Greubel, May 22 2017; amended by Georg Fischer, Jun 24 2020 *)

PARI

{T(n, k) = if( k<0 || k>2*n, 0, if( n==0, 1, if( n==1, [1,0,1][k+1], if( n==2, [1,1,2,1,1][k+1], T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)))))};

PARI

T(n,k)=polcoeff(Ser(polcoeff(Ser((1-y*z)/(1-z*(1+y+y^2)),y),k,y),z),n,z)

PARI

{T(n, k) = if( k<0 || k>2*n, 0, if( n==0, 1, polcoeff( (1 + x + x^2)^n, k) - polcoeff( (1 + x + x^2)^(n-1), k-1)))};

PARI

g=matrix(33,65);
for(n=0,32,for(k=0,2*n,g[n+1,k+1]=0));
g[1,1]=1;
g[2,1]=1;g[2,2]=0;g[2,3]=1;
g[3,1]=1;g[3,2]=1;g[3,3]=2;g[3,4]=1;g[3,5]=1;
for(n=0,2,for(k=0,2*n,print(n," ",k," ",g[n+1,k+1])))
for(n=3,32,g[n+1,1]=1;print(n," 1 1");g[n+1,2]=n-1;print(n," 2 ",n-1);for(k=2,2*n,g[n+1,k+1]=g[n,k-1]+g[n,k]+g[n,k+1];print(n," ",k," ",g[n+1,k+1])))
\\ Michael B. Porter, Feb 02 2010

Formula

T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), starting with [1], [1, 0, 1].

G.f.: (1-y*z)/[1-z*(1+y+y^2)].

Extensions

Edited by Ralf Stephan, Jan 09 2005

Offset corrected by R. J. Mathar, Feb 25 2015