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%I #30 Sep 08 2022 08:45:58
%S 8,15,15,26,30,26,42,56,56,42,64,98,112,98,64,93,162,210,210,162,93,
%T 130,255,372,420,372,255,130,176,385,627,792,792,627,385,176,232,561,
%U 1012,1419,1584,1419,1012,561,232,299,793,1573,2431,3003,3003,2431
%N Triangle read by rows: T(n,k) = Sum_{j=0..3} binomial(n+3, k+j), 0 <= k <= n.
%H G. C. Greubel, <a href="/A192915/b192915.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n,0) = T(n,n) = A000125(n+3); T(n,k) = T(n-1,k-1) + T(n-1,k) for 0 < k < n. - _Georg Fischer_, Nov 26 2021
%F T(n, k) = binomial(n + 4, k + 1) + binomial(n + 4, k + 3). - _Peter Luschny_, Nov 26 2021
%e Northwest corner:
%e 8;
%e 15, 15;
%e 26, 30, 26;
%e 42, 56, 56, 42;
%e 64, 98, 112, 98, 64;
%p A192915 := proc(n,k) binomial(n+3,k) +binomial(n+3,k+1) +binomial(n+3,k+2) +binomial(n+3,k+3) ; end proc: # _R. J. Mathar_, Aug 25 2011
%t T[n_, k_]:= Sum[Binomial[n+3, k+j], {j, 0, 3}]
%t Flatten[Table[T[n, k], {n,0,10}, {k,0,n}]] (* A192915 as a sequence *)
%t TableForm[Table[T[n, k], {n,0,10}, {k,0,n}]] (* A192915 as an array *)
%o (PARI) for(n=0,10, for(k=0,n, print1(sum(j=0,3, binomial(n+3,k+j)), ", "))) \\ _G. C. Greubel_, Jan 12 2019
%o (Magma) [[(&+[Binomial(n+3, k+j): j in [0..3]]): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Jan 12 2019
%o (Sage) [[sum(binomial(n+3,k+j) for j in (0..3)) for k in range(n+1)] for n in range(10)] # _G. C. Greubel_, Jan 12 2019
%o (GAP) T:=Flat(List([0..10], n->List([0..n], k->Sum([0..3], j-> Binomial(n+3, k+j) )))); # _G. C. Greubel_, Jan 12 2019
%Y Cf. A194121, A194122.
%K nonn,tabl
%O 0,1
%A _Clark Kimberling_, Aug 16 2011