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.

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A066325 Coefficients of unitary Hermite polynomials He_n(x).

Original entry on oeis.org

1, 0, 1, -1, 0, 1, 0, -3, 0, 1, 3, 0, -6, 0, 1, 0, 15, 0, -10, 0, 1, -15, 0, 45, 0, -15, 0, 1, 0, -105, 0, 105, 0, -21, 0, 1, 105, 0, -420, 0, 210, 0, -28, 0, 1, 0, 945, 0, -1260, 0, 378, 0, -36, 0, 1, -945, 0, 4725, 0, -3150, 0, 630, 0, -45, 0, 1, 0, -10395, 0, 17325, 0, -6930, 0, 990, 0, -55, 0, 1
Offset: 0

Views

Author

Christian G. Bower, Dec 14 2001

Keywords

Comments

Also number of involutions on n labeled elements with k fixed points times (-1)^(number of 2-cycles).
Also called normalized Hermite polynomials.
He_n(x) := H_n(x/sqrt(2)) / sqrt(2)^n, with the coefficients of H_n(x) given in A060821. See the Maple program. - Wolfdieter Lang, Jan 13 2020

Examples

			Triangle begins:
    1;
    0,     1;
   -1,     0,   1;
    0,    -3,   0,    1;
    3,     0,  -6,    0,   1;
    0,    15,   0,  -10,   0,   1;
  -15,     0,  45,    0, -15,   0,  1;
    0,  -105,   0,  105,   0, -21,  0, 1;
  ...

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pp. 89,94 (2.3.41,54).

Links

Crossrefs

Row sums: A001464 (with different signs).
Row sums of absolute values: A000085.
Absolute values are given in A099174.
Cf. A159834, A001147, A060821 (Hermite H_n(x)).

Programs

  • Maple
    Q:= [seq(orthopoly[H](n,x/sqrt(2))/2^(n/2), n=0..20)]:
seq(seq(coeff(Q[n+1],x,k),k=0..n),n=0..20); # Robert Israel, Jan 01 2016
# Alternative:
T := proc(n,k) option remember; if k > n then 0 elif n = k then 1 else
(T(n, k+2)*(k+2)*(k+1))/(k-n) fi end:
seq(print(seq(T(n, k), k = 0..n)), n = 0..10); # Peter Luschny, Jan 08 2023
  • Mathematica
    H[0, x_] = 1; H[1, x_] := x; H[n_, x_] := H[n, x] = x*H[n-1, x] - (n-1)*H[n-2, x] // Expand; Table[CoefficientList[H[n, x], x], {n, 0, 11}] // Flatten (* Jean-François Alcover, May 11 2015 *)
  • PARI
    for(n=0, 12, for(k=0,n, print1(if(Mod(n-k,2)==0, (-2)^((k-n)/2)*n!/(k!*((n-k)/2)!), 0), ", "))) \\ G. C. Greubel, Nov 23 2018
  • Python
    from sympy import Poly
from sympy.abc import x
def H(n, x): return 1 if n==0 else x if n==1 else x*H(n - 1, x) - (n - 1)*H(n - 2, x)
def a(n): return Poly(H(n, x), x).all_coeffs()[::-1]
for n in range(21): print(a(n)) # Indranil Ghosh, May 26 2017
  • Sage
    def A066325_row(n):
    T = [0]*(n+1)
    if n==1: return [1]
    for m in (1..n-1):
        a,b,c = 1,0,0
        for k in range(m,-1,-1):
            r = a - (k+1)*c
            if k < m : T[k+2] = u;
            a,b,c = T[k-1],a,b
            u = r
        T[1] = u;
    return T[1:]
for n in (1..11): A066325_row(n)  # Peter Luschny, Nov 01 2012
  • Sage
    # uses[riordan_array from A256893]
riordan_array(exp(-x^2/2), x, 8, True) # Peter Luschny, Nov 23 2018

Formula

T(n, k) = (-2)^((k-n)/2)*n!/(k!*((n-k)/2)!) for n-k even, 0 otherwise.
E.g.f. of row polynomials {He_n(y)}: A(x, y) = exp(x*y - x^2/2).
The umbral compositional inverses (cf. A001147) of the polynomials He(n,x) are given by the same polynomials unsigned, A099174. - Tom Copeland, Nov 15 2014
Exp(-D^2/2) x^n = He_n(x) = p_n(x+1) with D = d/dx and p_n(x), the row polynomials of A159834. These are an Appell sequence of polynomials with lowering and raising operators L = D and R = x - D. - Tom Copeland, Jun 26 2018

A111062 Triangle T(n, k) = binomial(n, k) * A000085(n-k), 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 6, 3, 1, 10, 16, 12, 4, 1, 26, 50, 40, 20, 5, 1, 76, 156, 150, 80, 30, 6, 1, 232, 532, 546, 350, 140, 42, 7, 1, 764, 1856, 2128, 1456, 700, 224, 56, 8, 1, 2620, 6876, 8352, 6384, 3276, 1260, 336, 72, 9, 1, 9496, 26200, 34380, 27840, 15960, 6552, 2100, 480, 90, 10, 1
Offset: 0

Views

Author

Philippe Deléham, Oct 07 2005

Keywords

Comments

Triangle related to A000085.
Riordan array [exp(x(2+x)/2),x]. - Paul Barry, Nov 05 2008
Array is exp(S+S^2/2) where S is A132440 the infinitesimal generator for Pascal's triangle. T(n,k) gives the number of ways to choose a subset of {1,2,...,n} of size k and then partitioning the remaining n-k elements into sets each of size 1 or 2. Cf. A122832. - Peter Bala, May 14 2012
T(n,k) is equal to the number of R-classes (equivalently, L-classes) in the D-class consisting of all rank k elements of the partial Brauer monoid of degree n. - James East, Aug 17 2015

Examples

			Rows begin:
     1;
     1,    1;
     2,    2,    1;
     4,    6,    3,    1;
    10,   16,   12,    4,    1;
    26,   50,   40,   20,    5,    1;
    76,  156,  150,   80,   30,    6,   1;
   232,  532,  546,  350,  140,   42,   7,  1;
   764, 1856, 2128, 1456,  700,  224,  56,  8, 1;
  2620, 6876, 8352, 6384, 3276, 1260, 336, 72, 9, 1;
From _Paul Barry_, Apr 23 2009: (Start)
Production matrix is:
  1, 1,
  1, 1, 1,
  0, 2, 1, 1,
  0, 0, 3, 1, 1,
  0, 0, 0, 4, 1, 1,
  0, 0, 0, 0, 5, 1, 1,
  0, 0, 0, 0, 0, 6, 1, 1,
  0, 0, 0, 0, 0, 0, 7, 1, 1,
  0, 0, 0, 0, 0, 0, 0, 8, 1, 1 (End)
From _Peter Bala_, Feb 12 2017: (Start)
The infinitesimal generator has integer entries and begins
  0
  1  0
  1  2  0
  0  3  3  0
  0  0  6  4  0
  0  0  0 10  5  0
  0  0  0  0 15  6  0
  ...
and is the generalized exponential Riordan array [x + x^2/2!,x].(End)

Links

Crossrefs

Cf. A000085, A005425 (row sums), A007318, A013989, A122832, A132440.
Cf. A099174, A133314, A159834 (inverse matrix).

Programs

  • GAP
    Flat(List([0..10],n->List([0..n],k->(Factorial(n)/Factorial(k))*Sum([0..n-k],j->Binomial(j,n-k-j)/(Factorial(j)*2^(n-k-j)))))); # Muniru A Asiru, Jun 29 2018
  • Mathematica
    a[n_] := Sum[(2 k - 1)!! Binomial[n, 2 k], {k, 0, n/2}]; Table[Binomial[n, k] a[n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 20 2015, after Michael Somos at A000085 *)
  • Sage
    def A111062_triangle(dim):
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+M[n-1,k]+(k+1)*M[n-1,k+1]
    return M
A111062_triangle(9) # Peter Luschny, Sep 19 2012

Formula

Sum_{k>=0} T(m, k)*T(n, k)*k! = T(m+n, 0) = A000085(m+n).
Sum_{k=0..n} T(n, k) = A005425(n).
Apparently satisfies T(n,m) = T(n-1,m-1) + T(n-1,m) + (m+1) * T(n-1,m+1). - Franklin T. Adams-Watters, Dec 22 2005 [corrected by Werner Schulte, Feb 12 2025]
T(n,k) = (n!/k!)*Sum_{j=0..n-k} C(j,n-k-j)/(j!*2^(n-k-j)). - Paul Barry, Nov 05 2008
G.f.: 1/(1-xy-x-x^2/(1-xy-x-2x^2/(1-xy-x-3x^2/(1-xy-x-4x^2/(1-... (continued fraction). - Paul Barry, Apr 23 2009
T(n,k) = C(n,k)*Sum_{j=0..n-k} C(n-k,j)*(n-k-j-1)!! where m!!=0 if m is even. - James East, Aug 17 2015
From Tom Copeland, Jun 26 2018: (Start)
E.g.f.: exp[t*p.(x)] = exp[t + t^2/2] e^(x*t).
These polynomials (p.(x))^n = p_n(x) are an Appell sequence with the lowering and raising operators L = D and R = x + 1 + D, with D = d/dx, such that L p_n(x) = n * p_(n-1)(x) and R p_n(x) = p_(n+1)(x), so the formalism of A133314 applies here, giving recursion relations.
The transpose of the production matrix gives a matrix representation of the raising operator R.
exp(D + D^2/2) x^n= e^(D^2/2) (1+x)^n = h_n(1+x) = p_n(x) = (a. + x)^n, with (a.)^n = a_n = A000085(n) and h_n(x) the modified Hermite polynomials of A099174.
A159834 with the e.g.f. exp[-(t + t^2/2)] e^(x*t) gives the matrix inverse for this entry with the umbral inverse polynomials q_n(x), an Appell sequence with the raising operator x - 1 - D, such that umbrally composed q_n(p.(x)) = x^n = p_n(q.(x)). (End)

Extensions

Corrected by Franklin T. Adams-Watters, Dec 22 2005
10th row added by James East, Aug 17 2015
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