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A287919 Square array T(0,n) = prime(n) and T(m+1,n) = T(m,n) + T(m,n+1), m >= 0, n >= 1, read by falling antidiagonals.

%I A287919 #22 Oct 31 2023 17:38:31
%S A287919 2,3,5,5,8,13,7,12,20,33,11,18,30,50,83,13,24,42,72,122,205,17,30,54,
%T A287919 96,168,290,495,19,36,66,120,216,384,674,1169,23,42,78,144,264,480,
%U A287919 864,1538,2707,29,52,94,172,316,580,1060,1924,3462,6169
%N A287919 Square array T(0,n) = prime(n) and T(m+1,n) = T(m,n) + T(m,n+1), m >= 0, n >= 1, read by falling antidiagonals.
%H A287919 Seiichi Manyama, <a href="/A287919/b287919.txt">Antidiagonals n = 0..139, flattened</a>
%e A287919 The array starts:
%e A287919 [0]    2    3    5     7    11    13    17    19    23    29 ... (A000040)
%e A287919 [1]    5    8   12    18    24    30    36    42    52    60 ... (A001043)
%e A287919 [2]   13   20   30    42    54    66    78    94   112   128 ... (A096277)
%e A287919 [3]   33   50   72    96   120   144   172   206   240   274 ... (A096278)
%e A287919 [4]   83  122  168   216   264   316   378   446   514   582 ... (A096279)
%e A287919 [5]  205  290  384   480   580   694   824   960  1096  1226 ...
%e A287919 [6]  495  674  864  1060  1274  1518  1784  2056  2322  2570 ...
%e A287919 [7] 1169 1538 1924  2334  2792  3302  3840  4378  4892  5380 ...
%e A287919 ...
%e A287919 First column is A007443: binomial transform of primes.
%e A287919 Second column is A178167: binomial transform of odd primes.
%t A287919 A287919list[dmax_]:=With[{a=Reverse[NestList[ListConvolve[{1,1},#]&,Prime[Range[dmax]],dmax-1]]},Array[Reverse[Diagonal[a,#]]&,dmax,1-dmax]];
%t A287919 A287919list[10] (* Generates 10 antidiagonals *) (* _Paolo Xausa_, Oct 31 2023 *)
%o A287919 (PARI) A287919(m,n)=sum(k=0,m,prime(n+k)*binomial(m,k))
%o A287919 /* read by antidiagonals */ for(m=0,13,for(n=0,m,print1(A287919(n,m-n+1)",")))
%Y A287919 Rows and columns include A001043, A096277-A096279, A007443, A178167.
%K A287919 nonn,easy,tabl
%O A287919 0,1
%A A287919 _M. F. Hasler_, Jun 02 2017