A217234 Triangle of expansion coefficients of the sum of an n X n array with equal top row and left column (extended by the rule of Pascal's triangle) in terms of the top row elements.

1, 1, 4, 1, 12, 6, 1, 40, 20, 8, 1, 140, 70, 30, 10, 1, 504, 252, 112, 42, 12, 1, 1848, 924, 420, 168, 56, 14, 1, 6864, 3432, 1584, 660, 240, 72, 16, 1, 25740, 12870, 6006, 2574, 990, 330, 90, 18, 1, 97240, 48620, 22880, 10010, 4004, 1430, 440, 110, 20

Offset: 1

Author

J. M. Bergot, Sep 28 2012

Keywords

• nonn
• tabl

Comments

Define a finite n X n square array with indeterminate elements A(1, c), c=1..n in the top row, the same elements A(r,1 ) = A(1,r) in the first column, r=1..n, and the remaining elements defined by the Pascal triangle rule: A(r,c) = A(r,c-1)+A(r-1,c).

The triangle T(n,m) gives the coefficients in the formula Sum_{r=1..n} Sum_{c=1..n} A(r,c) = Sum_{m=1..n} T(n,m) * A(1,m).

It says how many times the first, second, third, etc. element of the first row (or the first column) contributes to the sum of the n X n array.

Examples

			1;
1,4;
1,12,6;
1,40,20,8;
1,140,70,30,10;
1,504,252,112,42,12;
1,1848,924,420,168,56,14;
1,6864,3432,1584,660,240,72,16;
1,25740,12870,6006,2574,990,330,90,18;
1,97240,48620,22880,10010,4004,1430,440,110,20;
		

Crossrefs

Cf. A100320 (2nd column), A000984 (third column), A162551 (third column), A024483 (4th column), A006659 (5th column), A002058 (6th column), A030662 (row sums).

Programs

Maple

A217234_row := proc(n)
    local A,r,c,s ;
    A := array({},1..n,1..n) ;
    for r from 2 to n do
        A[r,1] := A[1,r] ;
    end do:
    for r from 2 to n do
        for c from 2 to n do
            A[r,c] := A[r,c-1]+A[r-1,c] ;
        end do:
    end do:
    s := add(add( A[r,c],c=1..n) ,r=1..n) ;
    for c from 1 to n do
        printf("%d,", coeff(s,A[1,c]) ) ;
    end do:
    return ;
end proc:
for n from 1 to 10 do
    A217234_row(n) ;
    printf(";\n") ;
end do; # R. J. Mathar, Oct 13 2012

Mathematica

A217234row [n_] := Module[{A, x, r, c, s }, A = Array[x, {n, n}]; Do[A[[r, 1]] = A[[1, r]], {r, 2, n}]; Do[A[[r, c]] = A[[r, c - 1]] + A[[r - 1, c]], {r, 2, n}, {c, 2, n}]; s = Sum[A[[r, c]], {r, 1, n}, {c, 1, n}]; If[n == 1, {1}, List @@ s /. x[, ] -> 1]];
Table[A217234row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Nov 04 2023, after R. J. Mathar *)

Extensions

Edited by R. J. Mathar, Oct 13 2012

Typo in data corrected by Jean-François Alcover, Nov 04 2023