This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A331432 #22 Oct 18 2022 19:14:02
%S A331432 1,0,3,1,5,10,0,10,35,35,1,14,91,189,126,0,21,189,651,924,462,1,27,
%T A331432 351,1749,4026,4290,1716,0,36,594,4026,13299,22737,19305,6435,1,44,
%U A331432 946,8294,36751,89375,120835,85085,24310,0,55,1430,15730,89375,289003,551837,615043,369512,92378,1,65,2080,27950,197275,811733,2047123,3203837,3031678,1587222,352716
%N A331432 Triangle T(n,k) (n >= k >= 0) read by rows: T(n,0) = (1+(-1)^n)/2; for k>=1, set T(0,k) = 0, S(n,k) = binomial(n,k)*binomial(n+k+1,k), and for n>=1, T(n,k) = S(n,k)-T(n-1,k).
%C A331432 The scanned pages of Ser are essentially illegible, and the book is out of print and hard to locate.
%C A331432 For Table IV on page 93, it is simplest to ignore the minus signs. The present triangle then matches all the given terms in that triangle, so it seems best to define the triangle by the recurrences given here, and to conjecture (strongly) that this is the same as Ser's triangle.
%D A331432 J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93.
%H A331432 G. C. Greubel, <a href="/A331432/b331432.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A331432 J. Ser, <a href="/A002720/a002720_4.pdf">Les Calculs Formels des Séries de Factorielles</a>, Gauthier-Villars, Paris, 1933 [Local copy].
%H A331432 J. Ser, <a href="/A002720/a002720.pdf">Les Calculs Formels des Séries de Factorielles</a> (Annotated scans of some selected pages)
%F A331432 T(n, k) = binomial(n,k)*binomial(n+k+1,k) - T(n-1, k), with T(n, 0) = (1 + (-1)^n)/2.
%F A331432 T(n, 0) = A000035(n+1).
%F A331432 T(n, 1) = A176222(n).
%F A331432 T(n, 2) = A331429(n).
%F A331432 T(n, n-2) = A002739(n).
%F A331432 T(n, n-1) = A002737(n).
%F A331432 T(n, n) = A001700(n).
%e A331432 Triangle begins:
%e A331432 1;
%e A331432 0, 3;
%e A331432 1, 5, 10;
%e A331432 0, 10, 35, 35;
%e A331432 1, 14, 91, 189, 126;
%e A331432 0, 21, 189, 651, 924, 462;
%e A331432 1, 27, 351, 1749, 4026, 4290, 1716;
%e A331432 0, 36, 594, 4026, 13299, 22737, 19305, 6435;
%e A331432 1, 44, 946, 8294, 36751, 89375, 120835, 85085, 24310;
%e A331432 0, 55, 1430, 15730, 89375, 289003, 551837, 615043, 369512, 92378;
%e A331432 1, 65, 2080, 27950, 197275, 811733, 2047123, 3203837, 3031678, 1587222, 352716;
%p A331432 SS := (n,k)->binomial(n,k)*binomial(n+k+1,k);
%p A331432 T4:=proc(n,k) local i; global SS; option remember;
%p A331432 if k=0 then return((1+(-1)^n)/2); fi;
%p A331432 if n=0 then 0 else SS(n,k)-T4(n-1,k); fi; end;
%p A331432 rho:=n->[seq(T4(n,k),k=0..n)];
%p A331432 for n from 0 to 14 do lprint(rho(n)); od:
%t A331432 T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0, (1 + (-1)^n)/2, Binomial[n, k]*Binomial[n+k+1, k] - T[n-1, k]]];
%t A331432 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 21 2022 *)
%o A331432 (Sage)
%o A331432 def T(n,k): # A331432
%o A331432 if (n<0): return 0
%o A331432 elif (k==0): return ((n+1)%2)
%o A331432 else: return binomial(n,k)*binomial(n+k+1,k) - T(n-1,k)
%o A331432 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 21 2022
%Y A331432 Columns 1 and 2 are A176222 and A331429; the last three diagonals are A002739, A002737, A001700.
%Y A331432 Taking the component-wise sums of the rows by pairs give the triangle in A178303.
%Y A331432 Ser's tables I and III are A331430 and A331431 (both are still mysterious).
%K A331432 nonn,tabl
%O A331432 0,3
%A A331432 _N. J. A. Sloane_, Jan 17 2020