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A125027
Binomial transform of the "1,2,3,..." triangle.
1
1, 3, 3, 9, 11, 6, 26, 32, 27, 10, 72, 86, 85, 54, 15, 192, 222, 233, 189, 95, 21, 496, 558, 597, 549, 371, 153, 28, 1248, 1374, 1473, 1446, 1160, 664, 231, 36, 3072, 3326, 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
OFFSET
1,2
FORMULA
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.
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
EXAMPLE
First few rows of the triangle:
1;
3, 3;
9, 11, 6;
26, 32, 27, 10;
72, 86, 85, 54, 15;
...
MAPLE
A27 := proc(n, k)
option remember;
if k>= 0 and k <=n then
if k = 0 then
1+procname(n-1, n-1) ;
else
procname(n, 0)+k ;
end if;
else
0;
end if;
end proc:
A125027 := proc(n, k)
add( binomial(n, j)*A27(j, k), j=k..n) ;
end proc: # R. J. Mathar, May 21 2018
MATHEMATICA
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];
A125027[n_, k_] := Sum[Binomial[n, j]*A27[j, k], {j, k, n}];
Table[A125027[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 27 2024, after R. J. Mathar *)
CROSSREFS
Cf. A072863 (first column), A000217 (diagonal), A164845 (subdiagonal).
Sequence in context: A121072 A133164 A022156 * A360244 A360242 A005296
KEYWORD
nonn,tabl,easy
AUTHOR
Gary W. Adamson, Nov 15 2006
STATUS
approved