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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, 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]
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
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Nov 14 2012
STATUS
approved