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

0,6

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]

Links

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

Example

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.

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 *)

Prog

(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)]
(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

Crossrefs

Sequence in context: A360279 A284001 A139145 * A179787 A358918 A368058

Adjacent sequences: A201632 A201633 A201634 * A201636 A201637 A201638

Keyword

nonn,tabl

Author

Peter Luschny, Nov 14 2012

Status

approved