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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.

%I A217234 #23 Nov 04 2023 05:33:52
%S A217234 1,1,4,1,12,6,1,40,20,8,1,140,70,30,10,1,504,252,112,42,12,1,1848,924,
%T A217234 420,168,56,14,1,6864,3432,1584,660,240,72,16,1,25740,12870,6006,2574,
%U A217234 990,330,90,18,1,97240,48620,22880,10010,4004,1430,440,110,20
%N 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.
%C A217234 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).
%C A217234 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).
%C A217234 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.
%e A217234 1;
%e A217234 1,4;
%e A217234 1,12,6;
%e A217234 1,40,20,8;
%e A217234 1,140,70,30,10;
%e A217234 1,504,252,112,42,12;
%e A217234 1,1848,924,420,168,56,14;
%e A217234 1,6864,3432,1584,660,240,72,16;
%e A217234 1,25740,12870,6006,2574,990,330,90,18;
%e A217234 1,97240,48620,22880,10010,4004,1430,440,110,20;
%p A217234 A217234_row := proc(n)
%p A217234     local A,r,c,s ;
%p A217234     A := array({},1..n,1..n) ;
%p A217234     for r from 2 to n do
%p A217234         A[r,1] := A[1,r] ;
%p A217234     end do:
%p A217234     for r from 2 to n do
%p A217234         for c from 2 to n do
%p A217234             A[r,c] := A[r,c-1]+A[r-1,c] ;
%p A217234         end do:
%p A217234     end do:
%p A217234     s := add(add( A[r,c],c=1..n) ,r=1..n) ;
%p A217234     for c from 1 to n do
%p A217234         printf("%d,", coeff(s,A[1,c]) ) ;
%p A217234     end do:
%p A217234     return ;
%p A217234 end proc:
%p A217234 for n from 1 to 10 do
%p A217234     A217234_row(n) ;
%p A217234     printf(";\n") ;
%p A217234 end do; # _R. J. Mathar_, Oct 13 2012
%t A217234 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]];
%t A217234 Table[A217234row[n], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Nov 04 2023, after _R. J. Mathar_ *)
%Y A217234 Cf. A100320 (2nd column), A000984 (third column), A162551 (third column), A024483 (4th column), A006659 (5th column), A002058 (6th column), A030662 (row sums).
%K A217234 nonn,tabl
%O A217234 1,3
%A A217234 _J. M. Bergot_, Sep 28 2012
%E A217234 Edited by _R. J. Mathar_, Oct 13 2012
%E A217234 Typo in data corrected by _Jean-François Alcover_, Nov 04 2023