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A124960 Triangle read by rows: T(n,k) = p(k)*T(n-1,k) + T(n-1,k-1) (1 <= k <= n), where p(k) denotes the k-th prime.

%I A124960 #12 Sep 08 2022 08:45:28
%S A124960 1,2,1,4,5,1,8,19,10,1,16,65,69,17,1,32,211,410,188,28,1,64,665,2261,
%T A124960 1726,496,41,1,128,2059,11970,14343,7182,1029,58,1,256,6305,61909,
%U A124960 112371,93345,20559,2015,77,1,512,19171,315850,848506,1139166,360612,54814,3478,100,1
%N A124960 Triangle read by rows: T(n,k) = p(k)*T(n-1,k) + T(n-1,k-1) (1 <= k <= n), where p(k) denotes the k-th prime.
%H A124960 G. C. Greubel, <a href="/A124960/b124960.txt">Rows n = 1..100 of triangle, flattened</a>
%e A124960 Triangle starts:
%e A124960    1;
%e A124960    2,   1;
%e A124960    4,   5,   1;
%e A124960    8,  19,  10,   1;
%e A124960   16,  65,  69,  17,  1;
%e A124960   32, 211, 410, 188, 28, 1;
%p A124960 T:=proc(n,k): if n=1 and k=1 then 1 elif k<1 or k>n then 0 else ithprime(k)*T(n-1,k)+T(n-1,k-1) fi end: for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
%t A124960 T[n_, k_]:= T[n, k]= If[n==1 && k==1 , 1, If[k<1 || k>n, 0, Prime[k]*T[n-1, k] + T[n-1, k-1] ]]; Table[T[n, k], {n,12}, {k, n}]//Flatten (* _G. C. Greubel_, Nov 19 2019 *)
%o A124960 (PARI) T(n,k) = if(n==1 && k==1, 1, if(k<1 || k>n, 0, prime(k)*T(n-1, k) + T(n-1, k-1) )); \\ _G. C. Greubel_, Nov 19 2019
%o A124960 (Magma)
%o A124960 function T(n,k)
%o A124960   if k lt 1 or k gt n then return 0;
%o A124960   elif n eq 1 and k eq 1 then return 1;
%o A124960   else return NthPrime(k)*T(n-1,k) + T(n-1,k-1);
%o A124960   end if;
%o A124960   return T;
%o A124960 end function;
%o A124960 [T(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Nov 19 2019
%o A124960 (Sage)
%o A124960 @CachedFunction
%o A124960 def T(n,k):
%o A124960     if (k<1 or k>n): return 0
%o A124960     elif (n==1 and k==1): return 1
%o A124960     else: return nth_prime(k)*T(n-1, k) + T(n-1, k-1)
%o A124960 [[T(n,k) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Nov 19 2019
%Y A124960 T(2n,n) gives A332967 (for n>0).
%K A124960 nonn,tabl
%O A124960 1,2
%A A124960 _Gary W. Adamson_, Nov 13 2006
%E A124960 Edited by _N. J. A. Sloane_, Nov 29 2006