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 A374848 #34 May 30 2025 23:16:34
%S A374848 0,1,2,16,162,3600,147456,12320100,2058386904,701841817600,
%T A374848 488286500625000,696425232679321600,2038348954317776486400,
%U A374848 12259459134020160144810000,151596002479762016373851690400,3855806813438155578522841251840000
%N A374848 Obverse convolution A000045**A000045; see Comments.
%C A374848 The obverse convolution of sequences
%C A374848 s = (s(0), s(1), ...) and t = (t(0), t(1), ...)
%C A374848 is introduced here as the sequence s**t given by
%C A374848 s**t(n) = (s(0)+t(n)) * (s(1)+t(n-1)) * ... * (s(n)+t(0)).
%C A374848 Swapping * and + in the representation s(0)*t(n) + s(1)*t(n-1) + ... + s(n)*t(0)
%C A374848 of ordinary convolution yields s**t.
%C A374848 If x is an indeterminate or real (or complex) variable, then for every sequence t of real (or complex) numbers, s**t is a sequence of polynomials p(n) in x, and the zeros of p(n) are the numbers -t(0), -t(1), ..., -t(n).
%C A374848 Following are abbreviations in the guide below for triples (s, t, s**t):
%C A374848 F = (0,1,1,2,3,5,...) = A000045, Fibonacci numbers
%C A374848 L = (2,1,3,4,7,11,...) = A000032, Lucas numbers
%C A374848 P = (2,3,5,7,11,...) = A000040, primes
%C A374848 T = (1,3,6,10,15,...) = A000217, triangular numbers
%C A374848 C = (1,2,6,20,70, ...) = A000984, central binomial coefficients
%C A374848 LW = (1,3,4,6,8,9,...) = A000201, lower Wythoff sequence
%C A374848 UW = (2,5,7,10,13,...) = A001950, upper Wythoff sequence
%C A374848 [ ] = floor
%C A374848 In the guide below, sequences s**t are identified with index numbers Axxxxxx; in some cases, s**t and Axxxxxx differ in one or two initial terms.
%C A374848 Table 1. s = A000012 = (1,1,1,1...) = (1);
%C A374848 t = A000012; 1 s**t = A000079; 2^(n+1)
%C A374848 t = A000027; n s**t = A000142; (n+1)!
%C A374848 t = A000040, P s**t = A054640
%C A374848 t = A000040, P (1/3) s**t = A374852
%C A374848 t = A000079, 2^n s**t = A028361
%C A374848 t = A000079, 2^n (1/3) s**t = A028362
%C A374848 t = A000045, F s**t = A082480
%C A374848 t = A000032, L s**t = A374890
%C A374848 t = A000201, LW s**t = A374860
%C A374848 t = A001950, UW s**t = A374864
%C A374848 t = A005408, 2*n+1 s**t = A000165, 2^n*n!
%C A374848 t = A016777, 3*n+1 s**t = A008544
%C A374848 t = A016789, 3*n+2 s**t = A032031
%C A374848 t = A000142, n! s**t = A217757
%C A374848 t = A000051, 2^n+1 s**t = A139486
%C A374848 t = A000225, 2^n-1 s**t = A006125
%C A374848 t = A032766, [3*n/2] s**t = A111394
%C A374848 t = A034472, 3^n+1 s**t = A153280
%C A374848 t = A024023, 3^n-1 s**t = A047656
%C A374848 t = A000217, T s**t = A128814
%C A374848 t = A000984, C s**t = A374891
%C A374848 t = A279019, n^2-n s**t = A130032
%C A374848 t = A004526, 1+[n/2] s**t = A010551
%C A374848 t = A002264, 1+[n/3] s**t = A264557
%C A374848 t = A002265, 1+[n/4] s**t = A264635
%C A374848 Sequences (c)**L, for c=2..4: A374656 to A374661
%C A374848 Sequences (c)**F, for c=2..6: A374662, A374662, A374982 to A374855
%C A374848 The obverse convolutions listed in Table 1 are, trivially, divisibility sequences. Likewise, if s = (-1,-1,-1,...) instead of s = (1,1,1,...), then s**t is a divisibility sequence for every choice of t; e.g. if s = (-1,-1,-1,...) and t = A279019, then s**t = A130031.
%C A374848 Table 2. s = A000027 = (0,1,2,3,4,5,...) = (n);
%C A374848 t = A000027, n s**t = A007778, n^(n+1)
%C A374848 t = A000290, n^2 s**t = A374881
%C A374848 t = A000040, P s**t = A374853
%C A374848 t = A000045, F s**t = A374857
%C A374848 t = A000032, L s**t = A374858
%C A374848 t = A000079, 2^n s**t = A374859
%C A374848 t = A000201, LW s**t = A374861
%C A374848 t = A005408, 2*n+1 s**t = A000407, (2*n+1)! / n!
%C A374848 t = A016777, 3*n+1 s**t = A113551
%C A374848 t = A016789, 3*n+2 s**t = A374866
%C A374848 t = A000142, n! s**t = A374871
%C A374848 t = A032766, [3*n/2] s**t = A374879
%C A374848 t = A000217, T s**t = A374892
%C A374848 t = A000984, C s**t = A374893
%C A374848 t = A038608, n*(-1)^n s**t = A374894
%C A374848 Table 3. s = A000290 = (0,1,4,9,16,...) = (n^2);
%C A374848 t = A000290, n^2 s**t = A323540
%C A374848 t = A002522, n^2+1 s**t = A374884
%C A374848 t = A000217, T s**t = A374885
%C A374848 t = A000578, n^3 s**t = A374886
%C A374848 t = A000079, 2^n s**t = A374887
%C A374848 t = A000225, 2^n-1 s**t = A374888
%C A374848 t = A005408, 2*n+1 s**t = A374889
%C A374848 t = A000045, F s**t = A374890
%C A374848 Table 4. s = t;
%C A374848 s = t = A000012, 1 s**s = A000079; 2^(n+1)
%C A374848 s = t = A000027, n s**s = A007778, n^(n+1)
%C A374848 s = t = A000290, n^2 s**s = A323540
%C A374848 s = t = A000045, F s**s = this sequence
%C A374848 s = t = A000032, L s**s = A374850
%C A374848 s = t = A000079, 2^n s**s = A369673
%C A374848 s = t = A000244, 3^n s**s = A369674
%C A374848 s = t = A000040, P s**s = A374851
%C A374848 s = t = A000201, LW s**s = A374862
%C A374848 s = t = A005408, 2*n+1 s**s = A062971
%C A374848 s = t = A016777, 3*n+1 s**s = A374877
%C A374848 s = t = A016789, 3*n+2 s**s = A374878
%C A374848 s = t = A032766, [3*n/2] s**s = A374880
%C A374848 s = t = A000217, T s**s = A375050
%C A374848 s = t = A005563, n^2-1 s**s = A375051
%C A374848 s = t = A279019, n^2-n s**s = A375056
%C A374848 s = t = A002398, n^2+n s**s = A375058
%C A374848 s = t = A002061, n^2+n+1 s**s = A375059
%C A374848 If n = 2*k+1, then s**s(n) is a square; specifically,
%C A374848 s**s(n) = ((s(0)+s(n))*(s(1)+s(n-1))*...*(s(k)+s(k+1)))^2.
%C A374848 If n = 2*k, then s**s(n) has the form 2*s(k)*m^2, where m is an integer.
%C A374848 Table 5. Others
%C A374848 s = A000201, LW t = A001950, UW s**t = A374863
%C A374848 s = A000045, F t = A000032, L s**t = A374865
%C A374848 s = A005843, 2*n t = A005408, 2*n+1 s**t = A085528, (2*n+1)^(n+1)
%C A374848 s = A016777, 3*n+1 t = A016789, 3*n+2 s**t = A091482
%C A374848 s = A005408, 2*n+1 t = A000045, F s**t = A374867
%C A374848 s = A005408, 2*n+1 t = A000032, L s**t = A374868
%C A374848 s = A005408, 2*n+1 t = A000079, 2^n s**t = A374869
%C A374848 s = A000027, n t = A000142, n! s**t = A374871
%C A374848 s = A005408, 2*n+1 t = A000142, n! s**t = A374872
%C A374848 s = A000079, 2^n t = A000142, n! s**t = A374874
%C A374848 s = A000142, n! t = A000045, F s**t = A374875
%C A374848 s = A000142, n! t = A000032, L s**t = A374876
%C A374848 s = A005408, 2*n+1 t = A016777, 3*n+1 s**t = A352601
%C A374848 s = A005408, 2*n+1 t = A016789, 3*n+2 s**t = A064352
%C A374848 Table 6. Arrays of coefficients of s(x)**t(x), where s(x) and t(x) are polynomials
%C A374848 s(x) t(x) s(x)**t(x)
%C A374848 n x A132393
%C A374848 n^2 x A269944
%C A374848 x+1 x+1 A038220
%C A374848 x+2 x+2 A038244
%C A374848 x x+3 A038220
%C A374848 nx x+1 A094638
%C A374848 1 x^2+x+1 A336996
%C A374848 n^2 x x+1 A375041
%C A374848 n^2 x 2x+1 A375042
%C A374848 n^2 x x+2 A375043
%C A374848 2^n x x+1 A375044
%C A374848 2^n 2x+1 A375045
%C A374848 2^n x+2 A375046
%C A374848 x+1 F(n) A375047
%C A374848 x+1 x+F(n) A375048
%C A374848 x+F(n) x+F(n) A375049
%F A374848 a(n) ~ c * phi^(3*n^2/4 + n) / 5^((n+1)/2), where c = QPochhammer(-1, 1/phi^2)^2/2 if n is even and c = phi^(1/4) * QPochhammer(-phi, 1/phi^2)^2 / (phi + 1)^2 if n is odd, and phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Aug 01 2024
%e A374848 a(0) = 0 + 0 = 0
%e A374848 a(1) = (0+1) * (1+0) = 1
%e A374848 a(2) = (0+1) * (1+1) * (1+0) = 2
%e A374848 a(3) = (0+2) * (1+1) * (1+1) * (2+0) = 16
%e A374848 As noted above, a(2*k+1) is a square for k>=0. The first 5 squares are 1, 16, 3600, 12320100, 701841817600, with corresponding square roots 1, 4, 60, 3510, 837760.
%e A374848 If n = 2*k, then s**s(n) has the form 2*F(k)*m^2, where m is an integer and F(k) is the k-th Fibonacci number; e.g., a(6) = 2*F(3)*(192)^2.
%p A374848 a:= n-> (F-> mul(F(n-j)+F(j), j=0..n))(combinat[fibonacci]):
%p A374848 seq(a(n), n=0..15); # _Alois P. Heinz_, Aug 02 2024
%t A374848 s[n_] := Fibonacci[n]; t[n_] := Fibonacci[n];
%t A374848 u[n_] := Product[s[k] + t[n - k], {k, 0, n}];
%t A374848 Table[u[n], {n, 0, 20}]
%o A374848 (PARI) a(n)=prod(k=0, n, fibonacci(k) + fibonacci(n-k)) \\ _Andrew Howroyd_, Jul 31 2024
%Y A374848 Cf. A000045, A374850-to-A374881, A374884-to-A374894, A375041-to-A375049.
%K A374848 nonn
%O A374848 0,3
%A A374848 _Clark Kimberling_, Jul 31 2024