A201635 Triangle formed by T(n,n) = (-1)^n*Sum_{j=0..n} C(-n,j), T(n,k) = Sum_{j=0..k} T(n-1,j) for k=0..n-1, and n>=0, read by rows.

1, 1, 0, 1, 1, 2, 1, 2, 4, 6, 1, 3, 7, 13, 22, 1, 4, 11, 24, 46, 80, 1, 5, 16, 40, 86, 166, 296, 1, 6, 22, 62, 148, 314, 610, 1106, 1, 7, 29, 91, 239, 553, 1163, 2269, 4166, 1, 8, 37, 128, 367, 920, 2083, 4352, 8518, 15792, 1, 9, 46, 174, 541, 1461, 3544, 7896

Offset: 0

Author

Peter Luschny, Nov 14 2012

Keywords

• nonn
• tabl

Comments

Notation: If a sequence id is starred then the offset and/or some terms are different. Starred terms indicate the variance.

Row sums: [A026641 ] [1, 1, 4, 13, 46, 166, 610]

--

T(j+2, 2) [A000124*] [1*, 2 , 4, 7, 11, 16, 22]

T(j+3, 3) [A003600*] [1*, 2*, 6, 13, 24, 40, 62]

--

T(j , j) [A072547 ] [1, 0, 2, 6, 22, 80, 296]

T(j+1, j) [A026641 ] [1, 1, 4, 13, 46, 166, 610]

T(j+2, j) [A014300 ] [1, 2, 7, 24, 86, 314, 1163]

T(j+3, j) [A014301*] [1, 3, 11, 40, 148, 553, 2083]

T(j+4, j) [A172025 ] [1, 4, 16, 62, 239, 920, 3544]

T(j+5, j) [A172061 ] [1, 5, 22, 91, 367, 1461, 5776]

T(j+6, j) [A172062 ] [1, 6, 29, 128, 541, 2232, 9076]

T(j+7, j) [A172063 ] [1, 7, 37, 174, 771, 3300, 13820]

--

T(2j ,j) [Central ] [1, 1, 7, 40, 239, 1461, 9076]

T(2j+1,j) [A183160 ] [1, 2, 11, 62, 367, 2232, 13820]

T(2j+2,j) [ ] [1, 3, 16, 91, 541, 3300, 20476]

T(2j+3,j) [A199033*] [1, 4, 22, 128, 771, 4744, 29618]

Examples

			Triangle begins as:
[n]|k->
[0] 1
[1] 1, 0
[2] 1, 1,  2
[3] 1, 2,  4,  6
[4] 1, 3,  7, 13,  22
[5] 1, 4, 11, 24,  46,  80
[6] 1, 5, 16, 40,  86, 166, 296
[7] 1, 6, 22, 62, 148, 314, 610, 1106.
		

Links

• G. C. Greubel, Rows n=0..100 of triangle, flattened

Programs

Maple

A201635 := proc(n,k) option remember; local j;
if n=k then (-1)^n*add(binomial(-n,j), j=0..n)
else add(A201635(n-1,j), j=0..k) fi end:
for n from 0 to 7 do seq(A(n,k), k=0..n) od;

Mathematica

T[n_, k_]:= T[n, k]= If[k==n, (-1)^n*Sum[Binomial[-n, j], {j, 0, n}], Sum[T[n-1, j], {j, 0, k}]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 27 2019 *)

PARI

{T(n,k) = if(k==n, (-1)^n*sum(j=0,n, binomial(-n,j)), sum(j=0,k, T(n-1,j)))};
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 27 2019

Sage

@CachedFunction
def A201635(n, k):
    if n==k: return (-1)^n*add(binomial(-n, j) for j in (0..n))
    return add(A201635(n-1, j) for j in (0..k))
for n in (0..7) : [A201635(n, k) for k in (0..n)]