OFFSET
0,2
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
Binomial transform of A051340.
From G. C. Greubel, Mar 18 2023: (Start)
T(n, k) = (k+1)*binomial(n,k) + Sum_{j=1..n-k} binomial(n, j+k).
T(n, k) = (k+1)*binomial(n,k) + binomial(n,k+1)*Hypergeometric2F1([1, k-n+1], [k+2], -1).
T(2*n, n) = (1/2)*T(2*n+1, n) = A258431(n+1).
Sum_{k=0..n} T(n, k) = A001787(n+1).
Sum_{k=0..n-1} T(n, k) = A058877(n+1), for n >= 1.
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A084633(n). (End)
EXAMPLE
First few rows of the triangle are:
1;
2, 2;
4, 5, 3;
8, 10, 10, 4;
16, 19, 23, 17, 5;
32, 36, 46, 46, 26, 6;
64, 69, 87, 102, 82, 37, 7;
MAPLE
A051340 := proc(n, k)
if k = n then
n+1 ;
elif k <= n then
1;
else
0;
end if;
end proc:
A130265 := proc(n, k)
add( binomial(n, j)*A051340(j, k), j=k..n) ;
end proc:
seq(seq(A130265(n, k), k=0..n), n=0..15) ; # R. J. Mathar, Aug 06 2016
MATHEMATICA
T[n_, k_]:= (k+1)*Binomial[n, k] + Binomial[n, k+1]*Hypergeometric2F1[1, k-n+1, k+2, -1];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 18 2023 *)
PROG
(Magma)
A130265:= func< n, k | k eq n select n+1 else (k+1)*Binomial(n, k) + (&+[Binomial(n, j+k): j in [1..n-k]]) >;
[A130265(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 18 2023
(SageMath)
def A130265(n, k): return (k+1)*binomial(n, k) + sum(binomial(n, j+k) for j in range(1, n-k+1))
flatten([[A130265(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Mar 18 2023
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, May 18 2007
EXTENSIONS
Missing term inserted by R. J. Mathar, Aug 06 2016
STATUS
approved