A060821 -id:A060821 - cp's OEIS frontend

A171531 Irregular triangle read by rows: first row is 1, n-th row (n > 0) consists of the coefficients in the expansion of H(x;n)*(x + 1)^(n - 1)/2^floor(n/2), where H(x;n) is the Hermite polynomial of order n.

1, 0, 2, -1, -1, 2, 2, 0, -6, -12, -2, 8, 4, 3, 9, -3, -33, -32, 0, 12, 4, 0, 30, 120, 140, -40, -202, -128, 8, 32, 8, -15, -75, -60, 300, 765, 585, -142, -470, -220, 20, 40, 8, 0, -210, -1260, -2730, -1680, 2982, 6132, 3586, -744, -1860, -688, 72, 96, 16, 105, 735, 1365

Offset: 0

Roger L. Bagula, Dec 11 2009

			Triangle begins:
    1;
    0,   2;
   -1,  -1,   2,   2;
    0,  -6, -12,  -2,   8,    4;
    3,   9,  -3, -33, -32,    0,   12,    4;
    0,  30, 120, 140, -40, -202, -128,    8,   32,  8;
  -15, -75, -60, 300, 765,  585, -142, -470, -220, 20, 40, 8;
   ... reformatted. - _Franck Maminirina Ramaharo_, Oct 02 2018

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 695.

Cf. A007318, A060821, A171229, A171631.

Join[{1},Table[CoefficientList[HermiteH[n,x]*(x + 1)^(n - 1)/2^Floor[n/2], x], {n, 1, 12}]]//Flatten

Edited and new name by Franck Maminirina Ramaharo, Oct 02 2018

A176410 A symmetrical triangle of adjusted polynomial coefficients based on Hermite orthogonal polynomials.

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, -191, 2113, -191, 1, 1, 1, 1, 1, 1, 1, 1, 7681, -337919, 7681, -337919, 7681, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -430079, 47738881, -430079, 180203521, -430079, 47738881, -430079, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Roger L. Bagula, Apr 16 2010

Row sums are: {1, 2, 11, 4, 1733, 6, -652793, 8, 273960969, 10, -143712092149, ...}.

			Triangle begins as:
  1;
  1,    1;
  1,    9,       1;
  1,    1,       1,    1;
  1, -191,    2113, -191,       1;
  1,    1,       1,    1,       1,    1;
  1, 7681, -337919, 7681, -337919, 7681, 1;
  1,    1,       1,    1,       1,    1, 1, 1;

G. C. Greubel, Rows n = 0..50 of triangle, flattened

Cf. A060821.

T[n_, m_]:= CoefficientList[HermiteH[n, x], x][[m + 1]]Reverse[ CoefficientList[ HermiteH[n, x], x]][[m + 1]] - (CoefficientList[ HermiteH[n, x], x][[1]]Reverse[CoefficientList[HermiteH[n, x], x]][[1]]) + 1;
Table[T[n, m], {n, 0, 10}, {m, 0, n}]//Flatten

Edited by G. C. Greubel, Apr 26 2019

A328141 a(n) = a(n-1) - (n-2)*a(n-2), with a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 2, 0, -4, -4, 12, 32, -40, -264, 56, 2432, 1872, -24880, -47344, 276096, 938912, -3202528, -18225120, 36217856, 364270016, -323869248, -7609269568, -808015360, 166595915136, 185180268416, -3813121694848, -8442628405248, 90698535660800, 318649502602496, -2220909495899904

Offset: 0

G. C. Greubel, Oct 04 2019

sign

Former title and formula of A122033, but not the data.

G. C. Greubel, Table of n, a(n) for n = 0..800

Cf. A001464, A060821, A121966, A122033.

a:=[1,2];; for n in [3..35] do a[n]:=a[n-1]-(n-3)*a[n-2]; od; a;

I:=[1,2]; [n le 2 select I[n] else Self(n-1) - (n-3)*Self(n-2): n in [1..35]];

a:= proc (n) option remember;
if n < 2 then n+1
else a(n-1) - (n-2)*a(n-2)
fi;
end proc; seq(a(n), n = 0..35);

a[n_]:= a[n]= If[n<2, n+1, a[n-1]-(n-2)*a[n-2]]; Table[a[n], {n,0,35}]

my(m=35, v=concat([1,2], vector(m-2))); for(n=3, m, v[n] = v[n-1] - (n-3)*v[n-2] ); v

def a(n):
    if n<2: return n+1
    else: return a(n-1) - (n-2)*a(n-2)
[a(n) for n in (0..35)]

a(n) = a(n-1) - (n-2)*a(n-2), with a(0)=1, a(1)=2.
E.g.f.: 1 + sqrt(2*e*Pi)*( erf(1/sqrt(2)) + erf((x-1)/sqrt(2)) ), where erf(x) is the error function.
a(n) = 2*(-1)^(n-1)*A001464(n-1).
a(n) = 2*(1/sqrt(2))^(n-1) * Hermite(n-1, 1/sqrt(2)), n > 0.

A381737 Orders k of Hermite polynomials whose maximal coefficient in absolute value appears twice.

Original entry on oeis.org

8, 13, 34, 43, 76, 89, 134, 151, 208, 229, 298, 323, 404, 433, 526, 559, 664, 701, 818, 859, 988, 1033, 1174, 1223, 1376, 1429, 1594, 1651, 1828, 1889, 2078, 2143, 2344, 2413, 2626, 2699, 2924, 3001, 3238, 3319, 3568, 3653, 3914, 4003, 4276, 4369, 4654, 4751, 5048

Offset: 1

Mike Sheppard, Mar 05 2025

nonn

			H_8(x) = 1680 - 13440 x^2 + 13440 x^4 - 3584 x^6 + 256 x^8, maximum coefficient in absolute value is 13440, which appears twice. Hence 8 is a term.
H_6(x) = -120 + 720 x^2 - 480 x^4 + 64 x^6. Absolute maximum unique. Hence 6 is not a term.

Mike Sheppard, Table of n, a(n) for n = 1..70

Cf. A060821, A277281, A381524.

Flatten@Position[Table[Count[#, Max@#] &@Abs@CoefficientList[HermiteH[n, x], x], {n, 1000}], 2]

isok(k) = my(vp=apply(x->abs(x), Vec(polhermite(k))), m=vecmax(vp)); #select(x->(x==m), vp) == 2; \\ Michel Marcus, Mar 09 2025

Conjecture 1: a(n) = 2*n*(n + 2) + (n + 1)*(-1)^(n+1).
Conjecture 2: a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
Conjecture 3: G.f.: (8*x + 5*x^2 + 5*x^3 - x^4 - x^5) / ((1 - x)^3 * (1 + x)^2).
Terms < 20000 consistent with conjectures. - Jinyuan Wang, Mar 09 2025.

A067613 Triangular table of coefficients of the Hermite polynomials, divided by 2^floor(n/2).

Original entry on oeis.org

1, 0, -2, -1, 0, 2, 0, 6, 0, -4, 3, 0, -12, 0, 4, 0, -30, 0, 40, 0, -8, -15, 0, 90, 0, -60, 0, 8, 0, 210, 0, -420, 0, 168, 0, -16, 105, 0, -840, 0, 840, 0, -224, 0, 16, 0, -1890, 0, 5040, 0, -3024, 0, 576, 0, -32, -945, 0, 9450, 0, -12600, 0, 5040, 0, -720, 0, 32, 0, 20790, 0, -69300, 0, 55440, 0, -15840, 0, 1760, 0, -64

Offset: 0

Wouter Meeussen, Feb 01 2002

Series development of exp(-(c+x)^2) at x=0 gives a Hermite polynomial in c as coefficient for x^k.

Robert Israel, Table of n, a(n) for n = 0..10010(rows 0 to 140, flattened)

Cf. A060821.

S:=series(exp(-2*c*x-x^2),x,13):
seq(seq(coeff(coeff(S,x,n)*n!/2^floor(n/2),c,j),j=0..n),n=0..12); # Robert Israel, Dec 07 2018

Table[ CoefficientList[ HermiteH[ n, c ], c ](-1)^n/2^Floor[ n/2 ], {n, 0, 12} ] (* or, equivalently *) a1=CoefficientList[ Series[ Exp[ c^2 ]Exp[ -(c+x)^2 ], {x, 0, 12} ], x ]; a2=(CoefficientList[ #, c ]&/@ a1 ) Range[ 0, 12 ]! 2^-Floor[ Range[ 0, 12 ]/2 ]

row(n) = Vecrev((-1)^n*polhermite(n)/2^floor(n/2)) \\ Michel Marcus, Dec 07 2018

HermiteH[n, c](-1)^n / 2^Floor[n/2]

A139158 Triangle a(n,k) of the expansion coefficients of the Hermite polynomial 2*H(n/2,x) if n even, of H((n-1)/2,x)+H((n+1)/2,x) if n odd.

Original entry on oeis.org

2, 1, 2, 0, 4, -2, 2, 4, -4, 0, 8, -2, -12, 4, 8, 0, -24, 0, 16, 12, -12, -48, 8, 16, 24, 0, -96, 0, 32, 12, 120, -48, -160, 16, 32, 0, 240, 0, -320, 0, 64, -120, 120, 720, -160, -480, 32, 64, -240, 0, 1440, 0, -960, 0, 128, -120, -1680, 720, 3360, -480, -1344, 64, 128, 0, -3360, 0, 6720, 0, -2688

Offset: 0

Roger L. Bagula, Jun 05 2008

Coefficients are ordered along increasing exponents [x^k], k=0,...,floor((n+1)/2).

Row sums are 2, 3, 4, 4, 4, -2, -8, -24, -40, -28, -16,..

			{2}, = 2
{1, 2}, = 1+2x
{0, 4}, = 4x^2
{-2, 2, 4}, = -2+2x+4x^2
{-4, 0, 8}, = -4+8x^2
{-2, -12, 4, 8},
{0, -24, 0, 16},
{12, -12, -48, 8, 16},
{24, 0, -96, 0, 32},
{12, 120, -48, -160, 16, 32},
{0, 240, 0, -320, 0, 64}.

Cf. A060821.

A060821 := proc(n,k) orthopoly[H](n,x) ; coeftayl(%,x=0,k) ; end:
A139158 := proc(n,k) if type(n,'even') then 2*A060821(n/2,k) ; else A060821((n+1)/2-1,k)+A060821((n+1)/2,k) ; fi; end: seq( seq(A139158(n,k),k=0..(n+1)/2),n=0..15) ;

Clear[p, x] p[x, 0] = 2*HermiteH[0, x]; p[x, 1] = HermiteH[0, x] + HermiteH[1, x]; p[x, 2] = 2*HermiteH[1, x]; p[x_, m_] := p[x, m] = If[Mod[m, 2] == 0, 2*HermiteH[Floor[m/2], x], HermiteH[ Floor[m/2], x] + HermiteH[Floor[m/ 2 + 1], x]];
Table[ExpandAll[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[a] Table[Apply[Plus, CoefficientList[p[x, n], x]], {n, 0, 10}]

a(2*n,k) = 2* A060821(n,k). a(2*n-1,k) = A060821(n-1,k)+A060821(n,k) .
sum_{k=0..n} a(2*n,k) = 2*A062267(n).
sum_{k=0..n} a(2*n-1,k) = A062267(n) + A062267(n-1).

Edited by the Associate Editors of the OEIS, Aug 28 2009

A181088 a(n) = A181089(2*n+1,n)/(n+2).

Original entry on oeis.org

1, -4, -40, 672, 8064, -253440, -3294720, 153753600, 2091048960, -130025226240, -1820353167360, 141707492720640, 2024392753152000, -189483161695027200, -2747505844577894400, 300609462994993152000, 4408938790593232896000

Offset: 0

Roger L. Bagula, Oct 02 2010

sign

What are the constraints on left-right symmetric triangles t(n,m) such that t(2*n,n)/(n+1) are integers?

G. C. Greubel, Table of n, a(n) for n = 0..450

Cf. A060821, A177042, A177043, A181089.

(* First program *)
p[x_, n_] = HermiteH[n, x] + ExpandAll[x^n*HermiteH[n, 1/x]];
b:= Table[CoefficientList[p[x, n], x], {n, 0, 50}];
Table[b[[2*n+2, n+1]]/(n+2), {n,0,20}]
(* Second program *)
A060821[n_, k_]:= If[EvenQ[n-k], (-1)^(Floor[(n-k)/2])*2^k*n!/(k!*(Floor[(n - k)/2]!)), 0];
a[n_]:= (A060821[2*n+1, n] + A060821[2*n+1, n+1])/(n+2);
Table[a[n], {n, 0, 25}] (* G. C. Greubel, Apr 04 2021 *)

def A060821(n,k): return (-1)^((n-k)//2)*2^k*factorial(n)/(factorial(k)*factorial( (n-k)//2)) if (n-k)%2==0 else 0
def a(n): return (A060821(2*n+1, n) + A060821(2*n+1, n+1))/(n+2)
[a(n) for n in (0..25)] # G. C. Greubel, Apr 04 2021

a(n) = (A060821(2*n+1, n) + A060821(2*n+1, n+1))/(n+2). - G. C. Greubel, Apr 04 2021

A185296 Triangle of connection constants between the falling factorials (x)(n) and (2*x)(n).

Original entry on oeis.org

1, 0, 2, 0, 2, 4, 0, 0, 12, 8, 0, 0, 12, 48, 16, 0, 0, 0, 120, 160, 32, 0, 0, 0, 120, 720, 480, 64, 0, 0, 0, 0, 1680, 3360, 1344, 128, 0, 0, 0, 0, 1680, 13440, 13440, 3584, 256, 0, 0, 0, 0, 0, 30240, 80640, 48384, 9216, 512

Offset: 0

Peter Bala, Feb 20 2011

The falling factorial polynomials (x)_n := x*(x-1)*...*(x-n+1), n = 0,1,2,..., form a basis for the space of polynomials. Hence the polynomial (2*x)_n may be expressed as a linear combination of x_0, x_1,...,x_n; the coefficients in the expansion form the n-th row of the table. Some examples are given below.
This triangle is connected to two families of orthogonal polynomials, the Hermite polynomials H(n,x) A060821, and the Bessel polynomials y(n,x)A001498. The first few Hermite polynomials are
... H(0,x) = 1
... H(1,x) = 2*x
... H(2,x) = -2+4*x^2
... H(3,x) = -12*x+8*x^3
... H(4,x) = 12-48*x^2+16*x^4.
The unsigned coefficients of H(n,x) give the nonzero entries of the n-th row of the triangle.
The Bessel polynomials y(n,x) begin
... y(0,x) = 1
... y(1,x) = 1+x
... y(2,x) = 1+3*x+3*x^2
... y(3,x) = 1+6*x+15*x^2+15*x^3.
The entries in the n-th column of this triangle are the coefficients of the scaled Bessel polynomials 2^n*y(n,x).
Also the Bell transform of g(n) = 2 if n<2 else 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Triangle begins
n\k|...0.....1.....2.....3.....4.....5.....6
============================================
0..|...1
1..|...0.....2
2..|...0.....2.....4
3..|...0.....0....12.....8
4..|...0.....0....12....48....16
5..|...0.....0.....0...120...160....32
6..|...0.....0.....0...120...720...480....64
..
Row 3:
(2*x)_3 = (2*x)*(2*x-1)*(2*x-2) = 8*x*(x-1)*(x-2) + 12*x*(x-1).
Row 4:
(2*x)_4 = (2*x)*(2*x-1)*(2*x-2)*(2*x-3) = 16*x*(x-1)*(x-2)*(x-3) +
48*x*(x-1)*(x-2)+ 12*x*(x-1).
Examples of recurrence relation
T(4,4) = 5*T(3,4) + 2*T(3,3) = 5*0 + 2*8 = 16;
T(5,4) = 4*T(4,4) + 2*T(4,3) = 4*16 + 2*48 = 160;
T(6,4) = 3*T(5,4) + 2*T(5,3) = 3*160 + 2*120 = 720;
T(7,4) = 2*T(6,4) + 2*T(6,3) = 2*720 + 2*120 = 1680.

L. Comtet, Advanced Combinatorics, Reidel, 1974, page 158, exercise 7.

Cf. A000898 (row sums), A001498, A001515, A059343, A060821.

T := (n,k) -> (n!/k!)*binomial(k,n-k)*2^(2*k-n):
seq(seq(T(n, k), k=0..n), n=0..9);

# uses[bell_matrix from A264428]
bell_matrix(lambda n: 2 if n<2 else 0, 12) # Peter Luschny, Jan 19 2016

Defining relation: 2*x*(2*x-1)*...*(2*x-n+1) = sum {k=0..n} T(n, k)*x*(x-1)*...*(x-k+1)
Explicit formula: T(n,k) = (n!/k!)*binomial(k,n-k)*2^(2*k-n). [As defined by Comtet (see reference).]
Recurrence relation: T(n,k) = (2*k-n+1)*T(n-1,k)+2*T(n-1,k-1).
E.g.f.: exp(x*(t^2+2*t)) = 1 + (2*x)*t + (2*x+4*x^2)*t^2/2! + (12*x^2+8*x^3)*t^3/3! + ...
O.g.f. for m-th diagonal (starting at main diagonal m = 0): (2*m)!/m!*x^m/(1-2*x)^(2*m+1).
The triangle is the matrix product [2^k*s(n,k)]n,k>=0 * ([s(n,k)]n,k>=0)^(-1),
where s(n,k) denotes the signed Stirling number of the first kind.
Row sums are [1,2,6,20,76,...] = A000898.
Column sums are [1,4,28,296,...] = [2^n*A001515(n)] n>=0.

A085638 Resultant of the polynomial x^n-1 and the Hermite polynomial H_n(x).

Original entry on oeis.org

2, 4, -1216, 2310400, -13094125568, 41787366322733056, 609452979875950622670848, 150142808011575068721319772160000, -7129771654003819760990676428837143372103680, 114629197516562460020595757143135575237521247385419776

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 15 2003

sign

Cf. A060821 (Hermite polynomial), A086840.

Hnmin2 = 1; Hnmin1 = 2*x; print1(polresultant(x - 1, Hnmin1), ", "); for (n = 2, 12, H = 2*x*Hnmin1 - 2*(n - 1)*Hnmin2; print1(polresultant(x^n - 1, H), ", "); Hnmin2 = Hnmin1; Hnmin1 = H); \\ David Wasserman, Feb 08 2005

a(n) = polresultant(x^n-1, polhermite(n)); \\ Michel Marcus, Apr 13 2020

More terms from David Wasserman, Feb 08 2005

A112227 A scaled Hermite triangle.

Original entry on oeis.org

1, 0, 1, -2, 0, 1, 0, -6, 0, 1, 12, 0, -12, 0, 1, 0, 60, 0, -20, 0, 1, -120, 0, 180, 0, -30, 0, 1, 0, -840, 0, 420, 0, -42, 0, 1, 1680, 0, -3360, 0, 840, 0, -56, 0, 1, 0, 15120, 0, -10080, 0, 1512, 0, -72, 0, 1, -30240, 0, 75600, 0, -25200, 0, 2520, 0, -90, 0, 1, 0, -332640, 0, 277200, 0, -55440, 0, 3960, 0, -110, 0, 1, 665280, 0

Offset: 0

Paul Barry, Aug 28 2005

Inverse of number triangle A067147. Diagonal sums are A002119.

			Triangle begins
1;
0,1;
-2,0,1;
0,-6,0,1;
12,0,-12,0,1;
0,60,0,-20,0,1;

Index entries for sequences related to Hermite polynomials

(* The function RiordanArray is defined in A256893. *)
rows = 12;
R = RiordanArray[E^(-#^2)&, #&, rows, True];
R // Flatten

Number triangle T(n, k)=A060821(n, k)/2^k; T(n, k)=n!/(k!*2^((n-k)/2)((n-k)/2)!)*cos(pi*(n-k)/2)*2^((n+k)/2)(1+(-1)^(n+k))/2^(k+1) T(n, k)=A001498((n+k)/2, (n-k)/2)*cos(pi(n-k)/2)*2^((n+k)/2)(1+(-1)^(n+k))/2^(k+1);

Exponential Riordan array (e^(-x^2),x). - Paul Barry, Sep 12 2006

cp's OEIS Frontend

A171531 Irregular triangle read by rows: first row is 1, n-th row (n > 0) consists of the coefficients in the expansion of H(x;n)*(x + 1)^(n - 1)/2^floor(n/2), where H(x;n) is the Hermite polynomial of order n.

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A176410 A symmetrical triangle of adjusted polynomial coefficients based on Hermite orthogonal polynomials.

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A328141 a(n) = a(n-1) - (n-2)*a(n-2), with a(0)=1, a(1)=2.

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A381737 Orders k of Hermite polynomials whose maximal coefficient in absolute value appears twice.

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A067613 Triangular table of coefficients of the Hermite polynomials, divided by 2^floor(n/2).

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A139158 Triangle a(n,k) of the expansion coefficients of the Hermite polynomial 2*H(n/2,x) if n even, of H((n-1)/2,x)+H((n+1)/2,x) if n odd.

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A181088 a(n) = A181089(2*n+1,n)/(n+2).

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