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 A252738 #30 Jul 22 2023 14:28:32
%S A252738 1,2,12,2160,2449440000,8488905214204800000000000,
%T A252738 3025568387202006082882734693673523654400000000000000000000000000
%N A252738 Row products of irregular table A005940: a(0) = 1; a(1) = 2; for n > 1: 2^(2^(n-2)) * a(n-1) * A003961(a(n-1)); also row products of A163511, A253563, A253565, and A332977.
%H A252738 Alois P. Heinz, <a href="/A252738/b252738.txt">Table of n, a(n) for n = 0..9</a>
%F A252738 a(0) = 1; a(1) = 2; for n > 1: a(n) = 2^(2^(n-2)) * a(n-1) * A003961(a(n-1)).
%F A252738 a(0) = 1; for n>=1: a(n) = Product_{k=A000079(n-1) .. A000225(n)} A163511(k) = Product_{k=2^(n-1) .. (2^n)-1} A163511(k).
%F A252738 a(0) = 1; a(1) = 2; for n > 1: a(n) = A267096(n-2) * a(n-1)^2. [Compare to the formulas of A191555] - _Antti Karttunen_, Feb 06 2016
%F A252738 From _Michael De Vlieger_, Jul 21 2023: (Start)
%F A252738 a(n) = Product_{k=1..n+1} prime(k)^e(n,k), where e(n,k) = k-th term in row n of A055248.
%F A252738 A067255(a(n)) = row n of A055248. (End)
%e A252738 From _Michael De Vlieger_, Jul 21 2023: (Start)
%e A252738 a(0) = 1 = product of {1},
%e A252738 a(1) = 2^1 = product of {2},
%e A252738 a(2) = 2^2 * 3^1 = product of {3, 2^2},
%e A252738 a(3) = 2^4 * 3^3 * 5^1 = product of {5, 2^1*3^1, 3^2, 2^3},
%e A252738 a(4) = 2^8 * 3^7 * 5^4 * 7^1 = product of
%e A252738 {7, 2^1*5^1, 3^1*5^1, 2^2*3^1, 5^2, 2^1*3^2, 3^3, 2^4},
%e A252738 ...
%e A252738 Table of e(n,k) where a(n) = Product_{k=1..n+1} prime(k)^e(n,k):
%e A252738 prime(k)| 2 3 5 7 11 13 17 19 23 29 31 ...
%e A252738 n\k | 1 2 3 4 5 6 7 8 9 10 11 ...
%e A252738 ----------------------------------------------------
%e A252738 0 | 1
%e A252738 1 | 2 1
%e A252738 2 | 4 3 1
%e A252738 3 | 8 7 4 1
%e A252738 4 | 16 15 11 5 1
%e A252738 5 | 32 31 26 16 6 1
%e A252738 6 | 64 63 57 42 22 7 1
%e A252738 7 | 128 127 120 99 64 29 8 1
%e A252738 8 | 256 255 247 219 163 93 37 9 1
%e A252738 9 | 512 511 502 466 382 256 130 46 10 1
%e A252738 10 | 1024 1023 1013 968 848 638 386 176 56 11 1
%e A252738 ... (End)
%t A252738 Table[Times @@ Array[Prime[# + 1]^Sum[Binomial[n, # + j], {j, 0, n}] &, n + 1, 0], {n, 0, 5}] (* _Michael De Vlieger_, Jul 21 2023 *)
%o A252738 (PARI)
%o A252738 allocatemem(234567890);
%o A252738 A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of _Michel Marcus_
%o A252738 A252738print(up_to_n) = { my(s, i=0, n=0); for(n=0, up_to_n, if(0 == n, s = 1, if(1 == n, s = 2; lev = vector(1); lev[1] = 2, oldlev = lev; lev = vector(2*length(oldlev)); s = 1; for(i = 0, (2^(n-1))-1, lev[i+1] = if((i%2),A003961(oldlev[(i\2)+1]),2*oldlev[(i\2)+1]); s *= lev[i+1]))); write("b252738.txt", n, " ", s)); }; \\ Counts them empirically.
%o A252738 A252738print(7);
%o A252738 (Scheme)
%o A252738 (definec (A252738rec n) (if (<= n 1) (+ 1 n) (* (A000079 (A000079 (- n 2))) (A252738rec (- n 1)) (A003961 (A252738rec (- n 1)))))) ;; Implements the given recurrence; uses the memoizing definec-macro.
%o A252738 (define (A252738 n) (if (zero? n) 1 (mul A163511 (A000079 (- n 1)) (A000225 n))))
%o A252738 (define (mul intfun lowlim uplim) (let multloop ((i lowlim) (res 1)) (cond ((> i uplim) res) (else (multloop (+ 1 i) (* res (intfun i)))))))
%o A252738 ;; Another alternative, implementing the new recurrence:
%o A252738 (definec (A252738 n) (if (<= n 1) (+ 1 n) (* (A267096 (- n 2)) (A000290 (A252738 (- n 1)))))) ;; _Antti Karttunen_, Feb 06 2016
%Y A252738 These are row products of irregular tables A005940, A163511, A253563 and A253565, which all are shaped like a binary tree.
%Y A252738 Partial products of A252740.
%Y A252738 Cf. A252737 (row sums), A252739 (divided by n), A252741 (divided by n!).
%Y A252738 Cf. A000079, A000225, A000290, A003961, A163511, A191555, A266639, A267096, A332977.
%K A252738 nonn
%O A252738 0,2
%A A252738 _Antti Karttunen_, Dec 21 2014
%E A252738 Typos in the second formula corrected by _Antti Karttunen_, Feb 06 2016