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Obverse convolution A000045**A000045; see Comments.
58

%I #30 Oct 19 2024 13:11:59

%S 0,1,2,16,162,3600,147456,12320100,2058386904,701841817600,

%T 488286500625000,696425232679321600,2038348954317776486400,

%U 12259459134020160144810000,151596002479762016373851690400,3855806813438155578522841251840000

%N Obverse convolution A000045**A000045; see Comments.

%C The obverse convolution of sequences

%C s = (s(0), s(1), ...) and t = (t(0), t(1), ...)

%C is introduced here as the sequence s**t given by

%C s**t(n) = (s(0)+t(n)) * (s(1)+t(n-1)) * ... * (s(n)+t(0)).

%C Swapping * and + in the representation s(0)*t(n) + s(1)*t(n-1) + ... + s(n)*t(0)

%C of ordinary convolution yields s**t.

%C 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 Following are abbreviations in the guide below for triples (s, t, s**t):

%C F = (0,1,1,2,3,5,...) = A000045, Fibonacci numbers

%C L = (2,1,3,4,7,11,...) = A000032, Lucas numbers

%C P = (2,3,5,7,11,...) = A000040, primes

%C T = (1,3,6,10,15,...) = A000217, triangular numbers

%C C = (1,2,6,20,70, ...) = A000984, central binomial coefficients

%C LW = (1,3,4,6,8,9,...) = A000201, lower Wythoff sequence

%C UW = (2,5,7,10,13,...) = A001950, upper Wythoff sequence

%C [ ] = floor

%C 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 Table 1. s = A000012 = (1,1,1,1...) = (1);

%C t = A000012; 1 s**t = A000079; 2^(n+1)

%C t = A000027; n s**t = A000142; (n+1)!

%C t = A000040, P s**t = A054640

%C t = A000040, P (1/3) s**t = A374852

%C t = A000079, 2^n s**t = A028361

%C t = A000079, 2^n (1/3) s**t = A028362

%C t = A000045, F s**t = A082480

%C t = A000032, L s**t = A374890

%C t = A000201, LW s**t = A374860

%C t = A001950, UW s**t = A374864

%C t = A005408, 2n+1 s**t = A000165, 2^n n!

%C t = A016777, 3n+1 s**t = A008544

%C t = A016789, 3n+2 s**t = A032031

%C t = A000142, n! s**t = A217757

%C t = A000051, 2^n+1 s**t = A139486

%C t = A000225, 2^n-1 s**t = A006125

%C t = A032766, [3n/2] s**t = A111394

%C t = A034472, 3^n+1 s**t = A153280

%C t = A024023, 3^n-1 s**t = A047656

%C t = A000217, T s**t = A128814

%C t = A000984, C s**t = A374891

%C t = A279019, n^2-n s**t = A130032

%C t = A004526, 1+[n/2] s**t = A010551

%C t = A002264, 1+[n/3] s**t = A264557

%C t = A002265, 1+[n/4] s**t = A264635

%C Sequences (c)**L, for c=2..4: A374656 to A374661

%C Sequences (c)**F, for c=2..6: A374662, A374662, A374982 to A374855

%C 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 Table 2. s = A000027 = (0,1,2,3,4,5,...) = (n);

%C t = A000027, n s**t = A007778, n^(n+1)

%C t = A000290, n^2 s**t = A374881

%C t = A000040, P s**t = A374853

%C t = A000045, F s**t = A374857

%C t = A000032, L s**t = A374858

%C t = A000079, 2^n s**t = A374859

%C t = A000201, LW s**t = A374861

%C t = A005408, 2n+1 s**t = A000407, (2n+1)! / n!

%C t = A016777, 3n+1 s**t = A113551

%C t = A016789, 3n+2 s**t = A374866

%C t = A000142, n! s**t = A374871

%C t = A032766, [3n/2] s**t = A374879

%C t = A000217, T s**t = A374892

%C t = A000984, C s**t = A374893

%C t = A038608, (-1)^n n s**t = A374894

%C Table 3. s = A000290 = (0,1,4,9,16,...) = (n^2);

%C t = A000290, n^2 s**t = A323540

%C t = A002522, n^2+1 s**t = A374884

%C t = A000217, T s**t = A374885

%C t = A000578, n^3 s**t = A374886

%C t = A000079, 2^n s**t = A374887

%C t = A000225, 2^n-1 s**t = A374888

%C t = A005408, 2n+1 s**t = A374889

%C t = A000045, F s**t = A374890

%C Table 4. s = t;

%C s = t = A000012, 1 s**s = A000079; 2^(n+1)

%C s = t = A000027, n s**s = A007778, n^(n+1)

%C s = t = A000290, n^2 s**s = A323540

%C s = t = A000045, F s**s = this sequence

%C s = t = A000032, L s**s = A374850

%C s = t = A000079, 2^n s**s = A369673

%C s = t = A000244, 3^n s**s = A369674

%C s = t = A000040, P s**s = A374851

%C s = t = A000201, LW s**s = A374862

%C s = t = A005408, 2n+1 s**s = A062971

%C s = t = A016777, 3n+1 s**s = A374877

%C s = t = A016789, 3n+2 s**s = A374878

%C s = t = A032766, [3n/2] s**s = A374880

%C s = t = A000217, T s**s = A375050

%C s = t = A005563, n^2-1 s**s = A375051

%C s = t = A279019, n^2-n s**s = A375056

%C s = t = A002398, n^2+n s**s = A375058

%C s = t = A002061, n^2+n+1 s**s = A375059

%C If n = 2k+1, then s**s(n) is a square; specifically,

%C s**s(n) = ((s(0)+s(n))*(s(1)+s(n-1))*...*((s(k)+s(k+1))^2.

%C If n = 2k, then s**s(n) has the form 2*s(k)*m^2, where m is an integer.

%C Table 5. Others

%C s = A000201, LW t = A001950, UW s**t = A374863

%C s = A000045, F t = A000032, L s**t = A374865

%C s = A005843, 2n t = A005408, 2n+1 s**t = A085528, (2n+1)^(n+1)

%C s = A016777, 3n+1 t = A016789, 3n+2 s**t = A091482

%C s = A005408, 2n+1 t = A000045, F s**t = A374867

%C s = A005408, 2n+1 t = A000032, L s**t = A374868

%C s = A005408, 2n+1 t = A000079, 2^n s**t = A374869

%C s = A000027, n t = A000142, n! s**t = A374871

%C s = A005408, 2n+1 t = A000142, n! s**t = A374872

%C s = A000079, 2^n t = A000142, n! s**t = A374874

%C s = A000142, n! t = A000045, F s**t = A374875

%C s = A000142, n! t = A000032, L s**t = A374876

%C s = A005408, 2n+1 t = A016777, 3n+1 s**t = A352601

%C s = A005408, 2n+1 t = A016789, 3n+2 s**t = A064352

%C Table 6. Arrays of coefficients of s(x)**t(x), where s(x) and t(x) are polynomials

%C s(x) t(x) s(x)**t(x)

%C n x A132393

%C n^2 x A269944

%C x+1 x+1 A038220

%C x+2 x+2 A038244

%C x x+3 A038220

%C nx x+1 A094638

%C 1 x^2+x+1 A336996

%C n^2 x x+1 A375041

%C n^2 x 2x+1 A375042

%C n^2 x x+2 A375043

%C 2^n x x+1 A375044

%C 2^n 2x+1 A375045

%C 2^n x+2 A375046

%C x+1 F(n) A375047

%C x+1 x+F(n) A375048

%C x+F(n) x+F(n) A375049

%F 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 a(0) = 0 + 0 = 0

%e a(1) = (0+1) * (1+0) = 1

%e a(2) = (0+1) * (1+1) * (1+0) = 2

%e a(3) = (0+2) * (1+1) * (1+1) * (2+0) = 16

%e As noted above, a(2k+1) is a square for k>=0. The first 5 squares are

%e 1, 16, 3600, 12320100, 701841817600, with corresponding square roots

%e 1, 4, 60, 3510, 837760.

%e If n = 2k, 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 a:= n-> (F-> mul(F(n-j)+F(j), j=0..n))(combinat[fibonacci]):

%p seq(a(n), n=0..15); # _Alois P. Heinz_, Aug 02 2024

%t s[n_] := Fibonacci[n]; t[n_] := Fibonacci[n];

%t u[n_] := Product[s[k] + t[n - k], {k, 0, n}];

%t Table[u[n], {n, 0, 20}]

%o (PARI) a(n)=prod(k=0, n, fibonacci(k) + fibonacci(n-k)) \\ _Andrew Howroyd_, Jul 31 2024

%Y Cf. A000045, A374850-to-A374881, A374884-to-A374894, A375041-to-A375049.

%K nonn

%O 0,3

%A _Clark Kimberling_, Jul 31 2024