A026907 Triangular array T read by rows (9-diamondization of Pascal's triangle). Step 1: t(n,k) = sum of 9 entries in diamond-shaped subarray of Pascal's triangle having vertices C(n,k), C(n+4,k+2), C(n+2,k), C(n+2,k+2). Step 2: T(n,k) = t(n,k) - t(0,0) + 1.

1, 13, 13, 28, 44, 28, 46, 90, 90, 46, 67, 154, 198, 154, 67, 91, 239, 370, 370, 239, 91, 118, 348, 627, 758, 627, 348, 118, 148, 484, 993, 1403, 1403, 993, 484, 148, 181, 650, 1495, 2414, 2824, 2414, 1495, 650, 181, 217, 849, 2163, 3927, 5256, 5256, 3927, 2163, 849, 217

Offset

0,2

Links

Indranil Ghosh, Rows 0..125, flattened

Formula

From G. C. Greubel, Aug 21 2025: (Start)

T(n, k) = binomial(n,k) + 3*binomial(n+4, k+2) - 18.

Sum_{k=0..n} (-1)^k*T(n, k) = 6*(1+(-1)^n)* floor((n+1)/2) + [n=0]. (End)

Example

Triangle starts:
   1;
  13,  13;
  28,  44,  28;
  46,  90,  90,  46;
  67, 154, 198, 154,  67;
  91, 239, 370, 370, 239,  91;
  ...

Mathematica

t[n_, k_]:=Binomial[n + 4, k + 2 ] + Binomial[n + 3, k + 1] + Binomial[n + 3, k + 2] + Binomial[n + 2, k] + Binomial[n + 2, k + 1] + Binomial[n + 2, k + 2] + Binomial[n + 1, k] + Binomial[n + 1, k + 1] + Binomial[n, k] ; T[n_, k_]:=t[n, k] - t[0, 0] + 1; Flatten[Table[T[n, k], {n, 0, 9}, {k, 0, n}]] (* Indranil Ghosh, Mar 13 2017 *)

Prog

(PARI) alias(C, binomial);
t(n, k) = C(n+4, k+2) + C(n+3, k+1) + C(n+3, k+2) + C(n+2, k) + C(n+2, k+1) + C(n+2, k+2) + C(n+1, k) + C(n+1, k+1) + C(n, k);
T(n, k) = t(n, k)-t(0, 0)+1;
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print());
\\ Michel Marcus, Mar 13 2017
(Magma)
A026907:= func< n, k | Binomial(n, k) + 3*Binomial(n+4, k+2) - 18 >;
[A026907(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 21 2025
(SageMath)
def A026907(n, k): return binomial(n, k) +3*binomial(n+4, k+2) -18
print(flatten([[A026907(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Aug 21 2025

Crossrefs

Cf. A007318, A027170.

Sums: A026915 (row), A026916, A026917, A026918 (diagonal).

Diagonals: A026908, A026909, A026910, A026911, A026912, A026913, A026914.

Sequence in context: A003889 A166545 A022347 * A210475 A379220 A394298

Adjacent sequences: A026904 A026905 A026906 * A026908 A026909 A026910

Keyword

nonn,tabl

Author

Clark Kimberling

Status

approved