A125027 - OEIS

%I #19 Jan 27 2024 10:34:25

%S 1,3,3,9,11,6,26,32,27,10,72,86,85,54,15,192,222,233,189,95,21,496,

%T 558,597,549,371,153,28,1248,1374,1473,1446,1160,664,231,36,3072,3326,

%U 3549,3600,3203,2246,1107,332,45,7424,7934,8409,8659,8201,6567,4051,1745,459,55,17664,18686,19669,20367,20015,17503,12597,6893,2629,615,66

%N Binomial transform of the "1,2,3,..." triangle.

%F Given the triangle (natural numbers in succession: 1; 2,3; 4,5,6; ...) as an infinite matrix M and P = Pascal's triangle as a lower triangular matrix, perform P*M, deleting the zeros.

%F The row sums s(n) = 1, 6, 26, 95, 312, 952, ... obey (-3*n+2)*s(n) +(9*n+7)*s(n-1) + 2*(-3*n-2)*s(n-2) = 0. - _R. J. Mathar_, May 21 2018

%e First few rows of the triangle:

%e    1;

%e    3,  3;

%e    9, 11,  6;

%e   26, 32, 27, 10;

%e   72, 86, 85, 54, 15;

%e   ...

%p A27 := proc(n,k)

%p     option remember;

%p     if k>= 0 and k <=n then

%p         if k = 0 then

%p             1+procname(n-1,n-1) ;

%p         else

%p             procname(n,0)+k ;

%p         end if;

%p     else

%p         0;

%p     end if;

%p end proc:

%p A125027 := proc(n,k)

%p     add( binomial(n,j)*A27(j,k),j=k..n) ;

%p end proc: # _R. J. Mathar_, May 21 2018

%t A27[n_, k_] := A27[n, k] = If[k >= 0 && k <= n, If[k == 0, 1+A27[n-1, n-1], A27[n, 0]+k], 0];

%t A125027[n_, k_] := Sum[Binomial[n, j]*A27[j, k], {j, k, n}];

%t Table[A125027[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 27 2024, after _R. J. Mathar_ *)

%Y Cf. A072863 (first column), A000217 (diagonal), A164845 (subdiagonal).

%K nonn,tabl,easy

%O 1,2

%A _Gary W. Adamson_, Nov 15 2006