cp's OEIS Frontend

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.

Showing 1-5 of 5 results.

A374848 Obverse convolution A000045**A000045; see Comments.

Original entry on oeis.org

0, 1, 2, 16, 162, 3600, 147456, 12320100, 2058386904, 701841817600, 488286500625000, 696425232679321600, 2038348954317776486400, 12259459134020160144810000, 151596002479762016373851690400, 3855806813438155578522841251840000
Offset: 0

Views

Author

Clark Kimberling, Jul 31 2024

Keywords

Comments

The obverse convolution of sequences
s = (s(0), s(1), ...) and t = (t(0), t(1), ...)
is introduced here as the sequence s**t given by
s**t(n) = (s(0)+t(n)) * (s(1)+t(n-1)) * ... * (s(n)+t(0)).
Swapping * and + in the representation s(0)*t(n) + s(1)*t(n-1) + ... + s(n)*t(0)
of ordinary convolution yields s**t.
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).
Following are abbreviations in the guide below for triples (s, t, s**t):
F = (0,1,1,2,3,5,...) = A000045, Fibonacci numbers
L = (2,1,3,4,7,11,...) = A000032, Lucas numbers
P = (2,3,5,7,11,...) = A000040, primes
T = (1,3,6,10,15,...) = A000217, triangular numbers
C = (1,2,6,20,70, ...) = A000984, central binomial coefficients
LW = (1,3,4,6,8,9,...) = A000201, lower Wythoff sequence
UW = (2,5,7,10,13,...) = A001950, upper Wythoff sequence
[ ] = floor
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.
Table 1. s = A000012 = (1,1,1,1...) = (1);
t = A000012; 1 s**t = A000079; 2^(n+1)
t = A000027; n s**t = A000142; (n+1)!
t = A000040, P s**t = A054640
t = A000040, P (1/3) s**t = A374852
t = A000079, 2^n s**t = A028361
t = A000079, 2^n (1/3) s**t = A028362
t = A000045, F s**t = A082480
t = A000032, L s**t = A374890
t = A000201, LW s**t = A374860
t = A001950, UW s**t = A374864
t = A005408, 2*n+1 s**t = A000165, 2^n*n!
t = A016777, 3*n+1 s**t = A008544
t = A016789, 3*n+2 s**t = A032031
t = A000142, n! s**t = A217757
t = A000051, 2^n+1 s**t = A139486
t = A000225, 2^n-1 s**t = A006125
t = A032766, [3*n/2] s**t = A111394
t = A034472, 3^n+1 s**t = A153280
t = A024023, 3^n-1 s**t = A047656
t = A000217, T s**t = A128814
t = A000984, C s**t = A374891
t = A279019, n^2-n s**t = A130032
t = A004526, 1+[n/2] s**t = A010551
t = A002264, 1+[n/3] s**t = A264557
t = A002265, 1+[n/4] s**t = A264635
Sequences (c)**L, for c=2..4: A374656 to A374661
Sequences (c)**F, for c=2..6: A374662, A374662, A374982 to A374855
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.
Table 2. s = A000027 = (0,1,2,3,4,5,...) = (n);
t = A000027, n s**t = A007778, n^(n+1)
t = A000290, n^2 s**t = A374881
t = A000040, P s**t = A374853
t = A000045, F s**t = A374857
t = A000032, L s**t = A374858
t = A000079, 2^n s**t = A374859
t = A000201, LW s**t = A374861
t = A005408, 2*n+1 s**t = A000407, (2*n+1)! / n!
t = A016777, 3*n+1 s**t = A113551
t = A016789, 3*n+2 s**t = A374866
t = A000142, n! s**t = A374871
t = A032766, [3*n/2] s**t = A374879
t = A000217, T s**t = A374892
t = A000984, C s**t = A374893
t = A038608, n*(-1)^n s**t = A374894
Table 3. s = A000290 = (0,1,4,9,16,...) = (n^2);
t = A000290, n^2 s**t = A323540
t = A002522, n^2+1 s**t = A374884
t = A000217, T s**t = A374885
t = A000578, n^3 s**t = A374886
t = A000079, 2^n s**t = A374887
t = A000225, 2^n-1 s**t = A374888
t = A005408, 2*n+1 s**t = A374889
t = A000045, F s**t = A374890
Table 4. s = t;
s = t = A000012, 1 s**s = A000079; 2^(n+1)
s = t = A000027, n s**s = A007778, n^(n+1)
s = t = A000290, n^2 s**s = A323540
s = t = A000045, F s**s = this sequence
s = t = A000032, L s**s = A374850
s = t = A000079, 2^n s**s = A369673
s = t = A000244, 3^n s**s = A369674
s = t = A000040, P s**s = A374851
s = t = A000201, LW s**s = A374862
s = t = A005408, 2*n+1 s**s = A062971
s = t = A016777, 3*n+1 s**s = A374877
s = t = A016789, 3*n+2 s**s = A374878
s = t = A032766, [3*n/2] s**s = A374880
s = t = A000217, T s**s = A375050
s = t = A005563, n^2-1 s**s = A375051
s = t = A279019, n^2-n s**s = A375056
s = t = A002398, n^2+n s**s = A375058
s = t = A002061, n^2+n+1 s**s = A375059
If n = 2*k+1, then s**s(n) is a square; specifically,
s**s(n) = ((s(0)+s(n))*(s(1)+s(n-1))*...*(s(k)+s(k+1)))^2.
If n = 2*k, then s**s(n) has the form 2*s(k)*m^2, where m is an integer.
Table 5. Others
s = A000201, LW t = A001950, UW s**t = A374863
s = A000045, F t = A000032, L s**t = A374865
s = A005843, 2*n t = A005408, 2*n+1 s**t = A085528, (2*n+1)^(n+1)
s = A016777, 3*n+1 t = A016789, 3*n+2 s**t = A091482
s = A005408, 2*n+1 t = A000045, F s**t = A374867
s = A005408, 2*n+1 t = A000032, L s**t = A374868
s = A005408, 2*n+1 t = A000079, 2^n s**t = A374869
s = A000027, n t = A000142, n! s**t = A374871
s = A005408, 2*n+1 t = A000142, n! s**t = A374872
s = A000079, 2^n t = A000142, n! s**t = A374874
s = A000142, n! t = A000045, F s**t = A374875
s = A000142, n! t = A000032, L s**t = A374876
s = A005408, 2*n+1 t = A016777, 3*n+1 s**t = A352601
s = A005408, 2*n+1 t = A016789, 3*n+2 s**t = A064352
Table 6. Arrays of coefficients of s(x)**t(x), where s(x) and t(x) are polynomials
s(x) t(x) s(x)**t(x)
n x A132393
n^2 x A269944
x+1 x+1 A038220
x+2 x+2 A038244
x x+3 A038220
nx x+1 A094638
1 x^2+x+1 A336996
n^2 x x+1 A375041
n^2 x 2x+1 A375042
n^2 x x+2 A375043
2^n x x+1 A375044
2^n 2x+1 A375045
2^n x+2 A375046
x+1 F(n) A375047
x+1 x+F(n) A375048
x+F(n) x+F(n) A375049

Examples

			a(0) = 0 + 0 = 0
a(1) = (0+1) * (1+0) = 1
a(2) = (0+1) * (1+1) * (1+0) = 2
a(3) = (0+2) * (1+1) * (1+1) * (2+0) = 16
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.
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.

Crossrefs

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

Programs

  • Maple
    a:= n-> (F-> mul(F(n-j)+F(j), j=0..n))(combinat[fibonacci]):
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 02 2024
  • Mathematica
    s[n_] := Fibonacci[n]; t[n_] := Fibonacci[n];
u[n_] := Product[s[k] + t[n - k], {k, 0, n}];
Table[u[n], {n, 0, 20}]
  • PARI
    a(n)=prod(k=0, n, fibonacci(k) + fibonacci(n-k)) \\ Andrew Howroyd, Jul 31 2024

Formula

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

A129467 Orthogonal polynomials with all zeros integers from 2*A000217.

Original entry on oeis.org

1, 0, 1, 0, -2, 1, 0, 12, -8, 1, 0, -144, 108, -20, 1, 0, 2880, -2304, 508, -40, 1, 0, -86400, 72000, -17544, 1708, -70, 1, 0, 3628800, -3110400, 808848, -89280, 4648, -112, 1, 0, -203212800, 177811200, -48405888, 5808528, -349568, 10920, -168, 1, 0, 14631321600, -13005619200, 3663035136, -466619904, 30977424, -1135808, 23016, -240, 1
Offset: 0

Views

Author

Wolfdieter Lang, May 04 2007

Keywords

Comments

The row polynomials p(n,x) = Sum_{k=0..n} T(n,k)*x^k have the n integer zeros 2*A000217(j), j=0..n-1.
The row polynomials satisfy a three-term recurrence relation which qualify them as orthogonal polynomials w.r.t. some (as yet unknown) positive measure.
Column sequences (without leading zeros) give A000007, A010790(n-1)*(-1)^(n-1), A084915(n-1)*(-1)^(n-2), A130033 for m=0..3.
Apparently this is the triangle read by rows of Legendre-Stirling numbers of the first kind. See the Andrews-Gawronski-Littlejohn paper, table 2. The mirror version is the triangle A191936. - Omar E. Pol, Jan 10 2012

Examples

			Triangle starts:
  1;
  0,    1;
  0,   -2,     1;
  0,   12,    -8,   1;
  0, -144,   108, -20,   1;
  0, 2880, -2304, 508, -40,  1;
  ...
n=3: [0,12,-8,1]. p(3,x) = x*(12-8*x+x^2) = x*(x-2)*(x-6).
n=5: [0,2880,-2304,508,-40,1]. p(5,x) = x*(2880-2304*x+508*x^2-40*x^3 +x^4) = x*(x-2)*(x-6)*(x-12)*(x-20).

Links

Crossrefs

Cf. A129462 (v=2 member), A129065 (v=1 member), A191936 (row reversed?).
Cf. A000217, A130031 (row sums), A130032 (unsigned row sums), A191936.
Column sequences (without leading zeros): A000007 (k=0), (-1)^(n-1)*A010790(n-1) (k=1), (-1)^n*A084915(n-1) (k=2), A130033 (k=3).
Cf. A008275.

Programs

  • Magma
    f:= func< n,k | (&+[Binomial(2*k+j,j)*StirlingFirst(2*n,2*k+j)*n^j: j in [0..2*(n-k)]]) >;
function T(n,k) // T = A129467
  if k eq n then return 1;
  else return f(n,k) -  (&+[Binomial(j,2*(j-k))*T(n,j): j in [k+1..n]]);
end if;
end function;
[[T(n,k): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Feb 09 2024
  • Mathematica
    T[n_, k_, m_]:= T[n,k,m]= If[k<0 || k>n, 0, If[n==0, 1, (2*(n-1)*(n-m) -(m-1))*T[n-1,k,m] -((n-1)*(n-m-1))^2*T[n-2,k,m] +T[n-1,k-1,m]]]; (* T=A129467 *)
Table[T[n,k,n], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 09 2024 *)
  • SageMath
    @CachedFunction
def f(n,k): return sum(binomial(2*k+j,j)*(-1)^j*stirling_number1(2*n,2*k+j)*n^j for j in range(2*n-2*k+1))
def T(n,k): # T = A129467
    if n==0: return 1
    else: return - sum(binomial(j,2*j-2*k)*T(n,j) for j in range(k+1,n+1)) + f(n,k)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 09 2024

Formula

Row polynomials p(n,x) = Product_{m=1..n} (x - m*(m-1)), n>=1, with p(0,x) = 1.
Row polynomials p(n,x) = p(n, v=n, x) with the recurrence: p(n,v,x) = (x + 2*(n-1)^2 - 2*(v-1)*(n-1) - v + 1)*p(n-1,v,x) - (n-1)^2*(n-1-v)^2*p(n-2,v,x) with p(-1,v,x) = 0 and p(0,v,x) = 1.
T(n, k) = [x^k] p(n, n, x), n >= k >= 0, otherwise 0.
T(n, k) = Sum_{j=0..2*(n-k)} ( binomial(2*k+j, j)*s(n,k)*n^j ) - Sum_{j=k+1..n} binomial(j, 2*(j-k))*T(n, j) (See Coffey and Lettington formula (4.7)). - G. C. Greubel, Feb 09 2024

A130032 Row sums of unsigned triangle A129467.

Original entry on oeis.org

1, 1, 3, 21, 273, 5733, 177723, 7642089, 435599073, 31798732329, 2893684641939, 321198995255229, 42719466368945457, 6706956219924436749, 1227372988246171925067, 258975700519942276189137, 62413143825306088561582017, 17038788264308562177311890641
Offset: 0

Views

Author

Wolfdieter Lang, May 04 2007

Keywords

Links

Crossrefs

Cf. A130031 (signed row sums), A130559 (unsigned row sums).

Programs

  • Magma
    [1] cat [n le 1 select 1 else (n^2-n+1)*Self(n-1): n in [1..30]]; // G. C. Greubel, Feb 10 2024
  • Mathematica
    Round@Table[Cosh[Sqrt[3] Pi/2] Gamma[n + 1/2 + I Sqrt[3]/2] Gamma[n + 1/2 - I Sqrt[3]/2]/Pi, {n, 0, 20}] (* Vladimir Reshetnikov, Aug 23 2016 *)
Product[k^2-k+1, {k,0,Range[0,30]}] (* G. C. Greubel, Feb 10 2024 *)
  • PARI
    a(n)=prod(k=1,n,k^2-k+1) \\ Charles R Greathouse IV, Mar 04 2012
  • SageMath
    def A130032(n): return 1 if n<2 else (n^2-n+1)*A130032(n-1)
[A130032(n) for n in range(31)] # G. C. Greubel, Feb 10 2024

Formula

a(n) = Sum_{m=0..n} |A129467(n,m)| for n >= 0.
a(n) = Sum_{j=0..n-1} |A130559(n-1, j)|, n >= 1.
For n > 0, a(n) = n! * Product_{k=1..n} [Gamma(k + 1/k)/Gamma(k - 1 + 1/k)]. - Gerald McGarvey, Nov 05 2007
a(n) = Product_{k=0..n} (k^2 - k + 1). - Gary Detlefs, Mar 04 2012
a(n) ~ c*n!*(n-1)! for c = Product_{k>=1} (1+1/(k^2+k)) = 2.428189792... [Charles R Greathouse IV, Mar 04 2012], c = cosh(sqrt(3)*Pi/2)/Pi. - Vaclav Kotesovec, Aug 24 2016
G.f.: 1 + x + 3*x^2/(Q(0)-3*x), where Q(k) = 1 + x*(k^2+3*k+3) - x*(k^2+5*k+7)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 15 2013

Extensions

Definition corrected by Wolfdieter Lang, Jun 04 2010

A130559 Coefficients of the v=n member of a family of certain orthogonal polynomials with Diophantine properties.

Original entry on oeis.org

1, -2, 1, 12, -8, 1, -144, 108, -20, 1, 2880, -2304, 508, -40, 1, -86400, 72000, -17544, 1708, -70, 1, 3628800, -3110400, 808848, -89280, 4648, -112, 1, -203212800, 177811200, -48405888, 5808528, -349568, 10920, -168, 1, 14631321600, -13005619200, 3663035136, -466619904
Offset: 0

Views

Author

Wolfdieter Lang, Jul 13 2007

Keywords

Comments

For v>=1 the orthogonal polynomials pt(n,v,x) have only integer zeros k*(k+1), k=1..n These integer zeros are from 2*A000217.
Coefficients of pt(n,v=n,x) (in the quoted Bruschi et al. paper {\tilde p}^{(\nu)}_n(x) of eqs. (20) and (24a),(24b)) in increasing powers of x.
The v-family pt(n,v,x) consists of characteristic polynomials of the tridiagonal M x M matrix Vt=Vt(M,v) with entries Vt_{m,n} given by 2*m*(v+1-m) if n=m, m=1,...,M; -m*(v+1-m) if n=m-1, m=2,...,M; -m*(v+1-m) if n=m+1, m=1..M-1 and 0 else. pt(n,v,x):=det(x*I_n-Vt(n,v) with the n dimensional unit matrix I_n.
pt(n,v=n,x) has, for every n>=1, the n integer zeros 2,6,12,...,n*(n+1). pt(2,2,x) has therefore only the integer zeros 2 and 6. 12= 2*6 = det(Vt(2,2))=16-4.
This triangle coincides with triangle A129467 without row n=0 and column m=0, taking as offset again [0,0].
Column sequences give for m=0..2: A010790(n-1)*(-1)^(n-1), A084915(n+1)*(-1)^n, A130033.

Examples

			n=2: [12,-8,1] stands for pt(2,2,x) = 12-8*x+x^2 = (x-2)*(x-6) with the integer zeros 2*1 and 2*3.
Triangle begins:
  [1];
  [-2,1];
  [12,-8,1];
  [-144,108,-20,1];
  [2880,-2304,508,-40,1];
  ...

Links

Crossrefs

Row sums give A130031(n+1), n>=0. Unsigned row sums give A130032(n+1), n>=1.
Cf. A130182 (v=1 member).

Formula

a(n,m) = [x^m]pt(n,n,x), n>=0, with the three term recurrence for orthogonal polynomial systems of the form pt(n,v,x) = (x + 2*n*(n-1-v))*pt(n-1,v,x) - (n-1)*n*(n-1-v)*(n-2-v)*pt(n-2,v,x), n>=1; pt(-1,v,x) = 0 and pt(0,v,x) = 1. Start with v = n.

A154232 a(2n) = (n^2-n-1) + a(2n-2), a(2n+1) = (n^2-n-1)*a(2n-1), with a(0)=0 and a(1)=1.

Original entry on oeis.org

0, 1, -1, -1, 0, -1, 5, -5, 16, -55, 35, -1045, 64, -30305, 105, -1242505, 160, -68337775, 231, -4851982025, 320, -431826400225, 429, -47069077624525, 560, -6166049168812775, 715, -955737621165980125, 896, -172988509431042402625
Offset: 0

Views

Author

Roger L. Bagula, Jan 05 2009

Keywords

Comments

Essentially A077415 and A130031 interleaved, see formulas.

Links

Crossrefs

Cf. A077415, A130031.

Programs

  • Magma
    function a(n)
  if n lt 2 then return n;
  elif (n mod 2 eq 0) then return ((n^2-2*n-4)/4) + a(n-2);
  else return ((n^2-4*n-1)/4)*a(n-2);
  end if; return a;
end function;
[a(n): n in [0..40]]; // G. C. Greubel, Mar 02 2021
  • Maple
    a[0]:= 0: a[1]:= 1:
for n from 1 to 49 do
  a[2*n]:= (n^2-n-1) +a[2*n-2];
  a[2*n+1]:= (n^2-n-1)*a[2*n-1];
od:
seq(a[i],i=0..99); # Robert Israel, Sep 06 2016
  • Mathematica
    (* First program *)
b[n_]:= b[n]= If[n==0, 0, n^2 -n -1 + b[n-1]];
c[n_]:= c[n]= If[n==0, 1, (n^2 -n -1)*c[n-1]];
Flatten[Table[{b[n], c[n]}, {n, 0, 15}]] (* modified by G. C. Greubel, Mar 02 2021 *)
(* Second program *)
a[n_]:= a[n]= If[n<2, n, If[EvenQ[n], ((n^2-2*n-4)/4) + a[n-2], ((n^2-4*n-1)/4)*a[n-2]]];
Table[a[n], {n,0,40}] (* G. C. Greubel, Mar 02 2021 *)
  • Sage
    def a(n):
    if (n<2): return n
    elif (n%2==0): return ((n^2-2*n-4)/4) + a(n-2)
    else: return ((n^2-4*n-1)/4)*a(n-2)
[a(n) for n in (0..40)] # G. C. Greubel, Mar 02 2021

Formula

From Robert Israel, Sep 06 2016: (Start)
a(2*n) = A077415(n) for n >= 2.
a(2*n+1) = cos(Pi*sqrt(5)/2)*Gamma(n+1/2-sqrt(5)/2)*Gamma(n+1/2+sqrt(5)/2)/Pi.
a(2*n+1) = (-1)^n*A130031(n). (End)

Extensions

Edited by Robert Israel, Sep 06 2016
Showing 1-5 of 5 results.

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