A129462 -id:A129462 - cp's OEIS frontend

A129464 Second column (m=1) of triangle A129462 (v=2 member of a certain family).

1, -2, -6, -48, -720, -17280, -604800, -29030400, -1828915200, -146313216000, -14485008384000, -1738201006080000, -248562743869440000, -41758540970065920000, -8142915489162854400000, -1824013069572479385600000, -465123332740982243328000000

Offset: 0

Wolfdieter Lang, May 04 2007

See A129462 for the M. Bruschi et al. reference.

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A129462, A129465 (m=2), A129466 (m=3).

Cf. A229020.

[1] cat [-Factorial(n-1)*Factorial(n+1): n in [1..30]]; // G. C. Greubel, Feb 08 2024

A129464 := n -> `if`(n=0,1,-(n-1)!^2*n*(n+1)); # Peter Luschny, Oct 15 2010

Table[If[n==0, 1, -(n-1)!*(n+1)!], {n,0,30}] (* G. C. Greubel, Feb 08 2024 *)

[1]+[-factorial(n-1)*factorial(n+1) for n in range(1,31)] # G. C. Greubel, Feb 08 2024

a(n) = A129462(n+1,1), n >= 0.
a(n) = -(n-1)!^2*n*(n+1), n > 0. - Peter Luschny, Oct 15 2010
From Amiram Eldar, May 17 2022: (Start)
Sum_{n>=1} 1/a(n) = -BesselI(2, 2) = -A229020.
Sum_{n>=1} (-1)^n/a(n) = BesselJ(2, 2). (End)

A129465 Third column (m=2) sequence of triangle A129462 (v=2 member of a certain family).

Original entry on oeis.org

1, 1, -4, -204, -7776, -358560, -20839680, -1516112640, -135920332800, -14772931891200, -1917601910784000, -293337284308992000, -52263416690343936000, -10734227287227924480000, -2518467729187335045120000, -669569466986357627289600000

Offset: 0

Wolfdieter Lang, May 04 2007

See A129462 for the M. Bruschi et al. reference.

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A129462, A129464 (m=1), A129466(m=3).

A129465:= func< n | n eq 0 select 1 else -Factorial(n)*Factorial(n+2)*(HarmonicNumber(n+2) -2) >;
[A129465(n): n in [0..30]]; // G. C. Greubel, Feb 08 2024

A129465[n_]:= If[n==0, 1, -n!*(n+2)!*(HarmonicNumber[n+2] -2)];
Table[A129465[n], {n,0,30}] (* G. C. Greubel, Feb 08 2024 *)

def A129465(n): return 1 if (n==0) else -factorial(n)*factorial(n+2)*( harmonic_number(n+2) -2)
[A129465(n) for n in range(31)] # G. C. Greubel, Feb 08 2024

a(n) = A129462(n+2, 2), n >= 0.

a(n) = (-1)*n!*(n+2)!*(HarmonicNumber(n+2) - 2), for n >= 1, otherwise a(0) = 1. - G. C. Greubel, Feb 08 2024

A129466 Fourth column (m=3) sequence of triangle A129462 (v=2 member of a certain family).

Original entry on oeis.org

1, 12, 208, 5208, 179688, 8175744, 472666752, 33625704960, 2858013642240, 281566521446400, 30978996781363200, 3583376917637529600, 374151199254884352000, 9777217907401555968000, -16608590925355066982400000, -10323797933882945175552000000

Offset: 0

Wolfdieter Lang, May 04 2007

See A129462 for the M. Bruschi et al. reference.

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A129462, A129465 (m=2).

function T(n, k) // T = A129462
  if k lt 0 or k gt n then return 0;
  elif n eq 0 then return 1;
  else return (2*(n-1)*(n-2)-1)*T(n-1, k) - ((n-1)*(n-3))^2*T(n-2, k) + T(n-1, k-1);
  end if;
end function;
A129466:= func< n | T(n+3,3) >;
[A129466(n): n in [0..20]]; // G. C. Greubel, Feb 09 2024

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n==0, 1, (2*(n-1)*(n-2) - 1)*T[n-1,k] -((n-1)*(n-3))^2*T[n-2,k] +T[n-1,k-1]]];(*T=A129462*)
A129466[n_]:= T[n+3, 3];
Table[A129466[n], {n,0,40}] (* G. C. Greubel, Feb 09 2024 *)

@CachedFunction
def T(n, k): # T = A129462
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return (2*(n-1)*(n-2)-1)*T(n-1, k) - ((n-1)*(n-3))^2*T(n-2, k) + T(n-1, k-1)
def A129466(n): return T(n+3,3)
[A129466(n) for n in range(41)] # G. C. Greubel, Feb 09 2024

a(n) = A129462(n+3,3), n >= 0.

A129463 Row sums of triangle A129462 (v=2 member of a certain family).

Original entry on oeis.org

1, 0, -1, -4, -39, -680, -18425, -713820, -37390255, -2543067280, -217799766225, -22928327328500, -2909576503498775, -437960283393276600, -77145678498655849225, -15720035935018890359500, -3668950667796545284209375, -972327797466833893742228000

Offset: 0

Wolfdieter Lang, May 04 2007

See A129462 for the M. Bruschi et al. reference.

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A129458 (row sums v=1 member), A129462.

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n==0, 1, (2*(n-1)*(n-2) - 1)*T[n-1,k] -((n-1)*(n-3))^2*T[n-2,k] +T[n-1,k-1] ]]; (* T=A129462 *)
A129463[n_]:= A129463[n]= Sum[T[n,k], {k,0,n}];
Table[A129463[n], {n,0,40}] (* G. C. Greubel, Feb 08 2024 *)

@CachedFunction
def T(n,k): # T = A129462
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return (2*(n-1)*(n-2)-1)*T(n-1,k) - ((n-1)*(n-3))^2*T(n-2,k) + T(n-1,k-1)
def A129463(n): return sum(T(n,k) for k in range(n+1))
[A129463(n) for n in range(41)] # G. C. Greubel, Feb 08 2024

a(n) = Sum_{k=0..n} A129462(n,k), n >= 0.
From Vaclav Kotesovec, Aug 24 2016: (Start)
a(n) = 2*(n-2)*(n-1)*a(n-1) - (n-3)^2*(n-1)^2*a(n-2).
a(n) ~ c * n^(2*n+(sqrt(5)-3)/2) / exp(2*n), where c = -2.3203776630375605070105975273368548459...
(End)

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

Wolfdieter Lang, May 04 2007

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

			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).

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
G. E. Andrews, W. Gawronski, and L. L. Littlejohn, The Legendre-Stirling Numbers, Discrete Mathematics, Volume 311, Issue 14, 28 July 2011, Pages 1255-1272.
M. Bruschi, F. Calogero and R. Droghei, Proof of certain Diophantine conjectures and identification of remarkable classes of orthogonal polynomials, J. Physics A, 40(2007), pp. 3815-3829.
José L. Cereceda, A refinement of Lang's formula for the sum of powers of integers, arXiv:2301.02141 [math.NT], 2023.
José L. Cereceda, Sums of powers of integers and the sequence A304330, arXiv:2405.05268 [math.GM], 2024. See p. 14.
Mark W. Coffey and Matthew C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.
Wolfdieter Lang, First ten rows and more.

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.

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

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 *)

@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

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

A129065 Coefficients of the v=1 member of a family of certain orthogonal polynomials.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 12, 10, 1, 0, 144, 156, 28, 1, 0, 2880, 3696, 908, 60, 1, 0, 86400, 125280, 37896, 3508, 110, 1, 0, 3628800, 5780160, 2036592, 236472, 10528, 182, 1, 0, 203212800, 349090560, 138517632, 19022736, 1074176, 26600, 280, 1

Offset: 0

Wolfdieter Lang, May 04 2007

For v >= 1 the orthogonal polynomials p(n,v,x) have v integer zeros k*(k-1), k = 1..v, for every n >= v.
Coefficients of p(n,v=1,x) (in the quoted Bruschi, et al., paper p(nu, n)(x) of eqs. (4) and (8a),(8b)) in increasing powers of x.
The v-family p(n,v,x) consists of characteristic polynomials of the tridiagonal M x M matrix V=V(M,v) with entries V_{m,n} given by v*(v-1) - (m-1)^2 - (v-m)^2 if n=m, m=1,...,M; (m-1)^2 if n=m-1, m=2,...,M; (v-m)^2 if n=m+1, m=1..M-1 and 0 else. p(n,v,x) := det(x*I_n - V(n,v) with the n-dimensional unit matrix I_n.
p(n,v=1,x) has, for every n >= 1, a zero for x=0, i.e., det(V(n,1))=0 for every n >= 1. This is obvious.
The column sequences give A000007, A010790, A129460, A129461 for m=0,1,2,3.

			Triangle begins:
  1;
  0,    1;
  0,    2,    1;
  0,   12,   10,   1;
  0,  144,  156,  28,   1;
  0, 2880, 3696, 908,  60,  1;
  ...
n=5,[0,2880,3696,908,60,1] stands for the polynomial x*(2880 + 3696*x + 908*x^2 + 60*x^3 + 1*x^4) with one zero 0 and some other four zeros.
Tridiagonal matrix V(5,1) = [[0,0,0,0,0], [1,-2,1,0,0], [0,4,-8,4,0], [0,0,9,-18,9], [0,0,0,16,-32]].

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
M. Bruschi, F. Calogero and R. Droghei, Proof of certain Diophantine conjectures and identification of remarkable classes of orthogonal polynomials, J. Physics A, 40(2007), pp. 3815-3829.
Wolfdieter Lang, First ten rows.

Columns: A000007 (m=0), A010790, (m=1), A129460 (m=2), A129461 (m=3).

Cf. A129458 (row sums), A129462 (v=2 triangle).

function T(n,k) // T = A129065
  if k lt 0 or k gt n then return 0;
  elif n eq 0 then return 1;
  else return 2*(n-1)^2*T(n-1,k) - 4*Binomial(n-1,2)^2*T(n-2,k) + T(n-1,k-1);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 07 2024

nmax = 9; T[n_, m_] := T[n, m] = (-(n-2)^2)*(n-1)^2*T[n-2, m] + T[n-1, m-1] + 2*(n-1)^2*T[n-1, m]; T[n_, m_] /; n < m = 0; T[-1, ] = 0; T[0, 0] = 1; T[, -1] = 0; Flatten[Table[T[n, m], {n, 0, nmax}, {m, 0, n}]] (* Jean-François Alcover, Sep 26 2011, after recurrence *)

@CachedFunction
def T(n,k): # T = A129065
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return 2*(n-1)^2*T(n-1,k) - 4*binomial(n-1,2)^2*T(n-2,k) + T(n-1,k-1)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 07 2024

T(n,m) = [x^m] p(n,1,x), n >= 0, with the three-term recurrence for orthogonal polynomial systems of the form 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), n >= 1; p(-1,v,x)=0 and p(0,v,x)=1. Put v=1 here.
Recurrence: T(n,m) = T(n-1,m-1) + (2*(n-1)^2 - 2*(v-1)*(n-1) - v + 1)*T(n-1,m) - ((n-1)^2*(n-1-v)^2)*T(n-2, m); T(n,m)=0 if n < m, T(-1,m):=0, T(0,0)=1, T(n,-1)=0. Put v=1 here.
Sum_{k=0..n} T(n, k) = A129458(n) (row sums).

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A129464 Second column (m=1) of triangle A129462 (v=2 member of a certain family).

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A129465 Third column (m=2) sequence of triangle A129462 (v=2 member of a certain family).

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A129466 Fourth column (m=3) sequence of triangle A129462 (v=2 member of a certain family).

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A129463 Row sums of triangle A129462 (v=2 member of a certain family).

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

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A129065 Coefficients of the v=1 member of a family of certain orthogonal polynomials.

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