A172025 -id:A172025 - cp's OEIS frontend

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

Peter Luschny, Nov 14 2012

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]

			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.

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

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;

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 *)

{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

@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)]

cp's OEIS Frontend

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.

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