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
Examples
Triangle starts: 1; 13, 13; 28, 44, 28; 46, 90, 90, 46; 67, 154, 198, 154, 67; 91, 239, 370, 370, 239, 91; ...
Links
- Indranil Ghosh, Rows 0..125, flattened
Crossrefs
Programs
-
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
-
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 *) -
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
-
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
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)