cp's OEIS Frontend

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.

%I A026907 #28 Aug 26 2025 09:51:45
%S A026907 1,13,13,28,44,28,46,90,90,46,67,154,198,154,67,91,239,370,370,239,91,
%T A026907 118,348,627,758,627,348,118,148,484,993,1403,1403,993,484,148,181,
%U A026907 650,1495,2414,2824,2414,1495,650,181,217,849,2163,3927,5256,5256,3927,2163,849,217
%N 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.
%H A026907 Indranil Ghosh, <a href="/A026907/b026907.txt">Rows 0..125, flattened</a>
%F A026907 From _G. C. Greubel_, Aug 21 2025: (Start)
%F A026907 T(n, k) = binomial(n,k) + 3*binomial(n+4, k+2) - 18.
%F A026907 Sum_{k=0..n} (-1)^k*T(n, k) = 6*(1+(-1)^n)* floor((n+1)/2) + [n=0]. (End)
%e A026907 Triangle starts:
%e A026907    1;
%e A026907   13,  13;
%e A026907   28,  44,  28;
%e A026907   46,  90,  90,  46;
%e A026907   67, 154, 198, 154,  67;
%e A026907   91, 239, 370, 370, 239,  91;
%e A026907   ...
%t A026907 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 *)
%o A026907 (PARI) alias(C, binomial);
%o A026907 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);
%o A026907 T(n,k) = t(n,k)-t(0,0)+1;
%o A026907 tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print());
%o A026907 \\ _Michel Marcus_, Mar 13 2017
%o A026907 (Magma)
%o A026907 A026907:= func< n,k | Binomial(n,k) + 3*Binomial(n+4,k+2) - 18 >;
%o A026907 [A026907(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Aug 21 2025
%o A026907 (SageMath)
%o A026907 def A026907(n,k): return binomial(n,k) +3*binomial(n+4,k+2) -18
%o A026907 print(flatten([[A026907(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Aug 21 2025
%Y A026907 Cf. A007318, A027170.
%Y A026907 Sums: A026915 (row), A026916, A026917, A026918 (diagonal).
%Y A026907 Diagonals: A026908, A026909, A026910, A026911, A026912, A026913, A026914.
%K A026907 nonn,tabl,changed
%O A026907 0,2
%A A026907 _Clark Kimberling_