A066667 -id:A066667 - cp's OEIS frontend

A355171 a(n) = Sum_{k=0..n} binomial(n, k + 1)k!(n + 1)!/(k + 2)!.

0, 1, 7, 50, 406, 3804, 41028, 506064, 7084656, 111690720, 1967421600, 38425449600, 825970435200, 19404363283200, 495012834489600, 13632039812966400, 403120633444300800, 12740557701389414400, 428546132879432601600, 15284163618598275072000, 576073025410937628672000

Offset: 0

Peter Luschny, Jun 22 2022

nonn

Cf. A066667, A001705, A103213.

a := n -> n*(n + 1)!*hypergeom([1, 1, 1 - n], [2, 3], -1) / 2;
seq(simplify(a(n)), n = 0..20);

a[n_] := n * (n + 1)! * HypergeometricPFQ[{1, 1, 1 - n}, {2, 3}, -1]/2; Array[a, 21, 0] (* Amiram Eldar, Jun 22 2022 *)

from math import comb, factorial
def A355171(n):
    f = factorial(n+1)
    return sum(f*comb(n,k+1)//(k+2)//(k+1) for k in range(n+1)) # Chai Wah Wu, Jun 22 2022

a(n) = n*(n + 1)!*hypergeom([1, 1, 1 - n], [2, 3], -1) / 2.
a(n) = Sum_{k=0..n} (-1)^(k+1)*k!*A066667(n, k + 1).
E.g.f.: log((1 - x) / (1 - 2*x)) / (1 - x)^2. - Mélika Tebni, Jun 23 2022
a(n) ~ 2^(n+2) * (n-1)!. - Vaclav Kotesovec, Feb 17 2024

A222468 Numerator sequence of the n-th convergent of the continued fraction 1/(1+2/(2+2/(3+2/(4+...

Original entry on oeis.org

1, 2, 8, 36, 196, 1248, 9128, 75520, 697936, 7130400, 79830272, 972224064, 12798573376, 181124475392, 2742464277632, 44241677392896, 757593444234496, 13725165351006720, 262293328557596672, 5273316901853946880

Offset: 1

Gary Detlefs and Wolfdieter Lang, Mar 21 2013

The corresponding denominator sequence is A222467(n).
a(n) = Phat(n,2) with the numerator polynomials Phat of A221913. All the given formulas follow from there and the comments given under A084950. The limit of the continued fraction (0 + K_{k>=1} (2/k))/2 = 1/(1+2/(2+2/(3+2/(4+... is (1/2)*sqrt(2)*BesselI(1,2*sqrt(2))/BesselI(0,2*sqrt(2)) = 0.5631786198117... See A222466 for more decimals.
For a combinatorial interpretation in terms of labeled Morse codes see a comment on A221913. Here each dash has label x=2, and the dots have label j if they are at position j. Labels are multiplied and all codes on [2,...,n+1] are summed.

			a(4) = 4*a(3) + 2*a(2) = 4*8 + 2*2 = 36.
Continued fraction convergent: 1/(1+2/(2+2/(3+2/4))) = 9/16 = 36/64 = a(4)/A222467(4).
Morse code a(5) = 196 from the sum of all 5 labeled codes on [2,3,4,5], one with no dash, three with one dash and one with two dashes: 5!/1 + (4*5 + 2*5 + 2*3)*2 +2^2 = 196.

G. C. Greubel, Table of n, a(n) for n = 1..445

Cf. A084950, A221913, A222467, A001040(n+1) (x=1), A058797 (x=-1).

I:=[1, 2]; [n le 2 select I[n] else n*Self(n-1) + 2*Self(n-2): n in [1..30]]; // G. C. Greubel, May 17 2018

RecurrenceTable[{a[1] == 1, a[2] == 2, a[n] == n*a[n - 1] + 2 a[n - 2]}, a[n], {n, 20}] (* Ray Chandler, Jul 30 2015 *)

m=30; v=concat([1,2], vector(m-2)); for(n=3, m, v[n]=n*v[n-1] +2*v[n-2]); v \\ G. C. Greubel, May 17 2018

Recurrence: a(n) = n*a(n-1) + 2*a(n-2), a(-1) = 1/2, a(0) = 0, n >= 1.
As a sum: a(n) = Sum_{m=0..floor(n/2)} a(n-m,m)*2^m, n >= 1, with a(n,m) = binomial(n-1-m,m)*(n-m)!/(m+1)! = |A066667(n,m)| (Laguerre coefficients, parameter alpha = 1).
Explicit form: a(n) = 2*(w/2)^n*(BesselI(1,w)*BesselK(n+1,w) + BesselK(1,w)*BesselI(n+1,w)*(-1)^(n+1)), n >= 1, where w := -2*sqrt(2).
E.g.f.: Pi*(BesselJ(1, 2*i*sqrt(2)*sqrt(1-z))* BesselY(1, 2*i*sqrt(2)) - BesselY(1, (2*i)*sqrt(2)*sqrt(1-z))*BesselJ(1, 2*i*sqrt(2)))/sqrt(1-z) with Bessel functions and the imaginary unit i = sqrt(-1). Phat(0,x) = 0.
Asymptotics: lim_{n -> infinity} a(n)/n! = BesselI(1,2*sqrt(2)) /sqrt(2) = 2.3948330992734...

A222470 Numerator sequence of the n-th convergent of the continued fraction 1/(1-2/(2-2/(3-2/(4-...

Original entry on oeis.org

1, 2, 4, 12, 52, 288, 1912, 14720, 128656, 1257120, 13571008, 160337856, 2057250112, 28480825856, 423097887616, 6712604550144, 113268081577216, 2025400259289600, 38256068763347968, 761070574748380160

Offset: 1

Gary Detlefs and Wolfdieter Lang, Mar 23 2013

The corresponding denominator sequence is A222469(n).
a(n) = Phat(n,-2) with the numerator polynomials Phat of A221913. All the given formulas follow from there and the comments given under A084950. The limit of the continued fraction (0 + K_{k=1..oo} (-2/k))/(-2) = 1/(1-2/(2-2/(3-2/(4-... is (1/2)*sqrt(2)*BesselJ(1,2*sqrt(2))/BesselJ(0,2*sqrt(2)) = -1.43974932187023280... (see A222471).
For a combinatorial interpretation in terms of labeled Morse codes see a comment on A221913. Here each dash has label x=-2, and the dots have label j if they are at position j. Labels are multiplied and all codes on [2,...,n+1] are summed.

			a(4) = 4*a(3) - 2*a(2) = 4*4 - 2*2 = 12.
Continued fraction convergent: 1/(1-2/(2-2/(3-2/4))) = -3/2 = 12/(-8) = a(4)/A222469(4).
Morse code a(5) = 52 from the sum of all 5 labeled codes on [2,3,4,5], one with no dash, three with one dash and one with two dashes: 5!/1 + (4*5 + 2*5 + 2*3)*(-2) +(-2)^2 = 52.

Cf. A084950, A221913, A222467, A001040(n+1) (x=1), A058798 (x=-1), A222468 (x=2).

Rest[RecurrenceTable[{a[-1]==-(1/2),a[0]==0,a[n]==n*a[n-1]-2a[n-2]},a,{n,20}]] (* Harvey P. Dale, Oct 24 2015 *)

Recurrence: a(n) = n*a(n-1) - 2*a(n-2), a(-1) = -1/2, a(0) = 0, n >= 1.
As a sum: a(n) = Sum_{m=0..floor(n/2)} a(n-m,m)*(-2)^m, n >= 1, with a(n,m) = binomial(n,m)*(n+1)!/(m+1)! = |A066667(n,m)| (Laguerre coefficients, parameter alpha = 1).
Explicit form: a(n) = Pi*(z/2)^n*(BesselY(1,z)*BesselJ(n+1,z) - BesselJ(1,z)*BesselY(n+1,z)) with z = 2*sqrt(2).
E.g.f.: Pi*(BesselJ(1, -x*sqrt(1-z))*BesselY(1, -x) - BesselY(1, -x*sqrt(1-z))*BesselJ(1, -x))/sqrt(1-z) with x = 2*sqrt(x). Here Phat(0,x) = 0.
Asymptotics: lim_{n->oo} a(n)/n! = BesselJ(1,2*sqrt(2))/(sqrt(2)) = 0.2829799868805...

A215216 Coefficient triangle of the Hermite-Bell polynomials for power -2.

Original entry on oeis.org

1, 2, 4, -6, 8, -36, 24, 16, -144, 300, -120, 32, -480, 2040, -2640, 720, 64, -1440, 10320, -27720, 25200, -5040, 128, -4032, 43680, -199920, 383040, -262080, 40320, 256, -10752, 163968, -1142400, 3764880, -5503680, 2963520, -362880

Offset: 0

Roman Witula, Aug 06 2012

The Hermite-Bell polynomials for negative powers H(n;-r;x), n=0,1,..., r=1,2,..., and x \in C\{0} are defined by the following relation: H(n;-r;x) = x^((r+1)*n)*exp(1/x^r)*(d^n exp(-1/x^r)/dx^n). These polynomials form the natural generalization of the concept of so-called generalized Hermite-Bell polynomials given for positive integers powers by D. Dominici (see also the R. B. Paris paper). We obtain the following recurrence formula:
H(n+1;-r;x) = (r-(r+1)*n*x^r)*H(n;-r;x) + x^(r+1)*(dH(n;-r;x)/dx). In the sequel we deduce the following special ones: H(0;-r;x)=1, H(1;-r;x)=r, H(2;-r;x)=r^2 - r*(r+1)*x^r, H(3;-r;x)=r^3 - 3*r^2*(r+1)*x^r + r*(r+1)*(r+2)*x^(2*r), H(4;-r;x)=r^4 - 6*r^3*(r+1)*x^r + r^2*(r+1)*(7*r+11)*x^(2*r) - r*(r+1)*(r+2)*(r+3)*x^(3*r) - the general formulas are given in Witula et al.'s paper.
There is a connection between H(n;-1;x) and the Laguerre polynomials L(n;x;a=1), see A066667 for details.

			Let us put W(n;x):=H(n;-2;sqrt(x)). Then we have W(0;x)=1, W(1;x)=2, W(2;x)=4-6*x, W(3;x)=8-36*x+24*x^2, W(4;x)=16-144*x+300*x^2-120*x^3, W(5;x)=32-480*x+2040*x^2-2640*x^3+720*x^4.
1;
2;
4,      -6;
8,     -36,    24;
16,   -144,   300,    -120;
32,   -480,  2040,   -2640,    720;
64,  -1440, 10320,  -27720,  25200,   -5040;
128, -4032, 43680, -199920, 383040, -262080, 40320;

R. Witula, E. Hetmaniok, D. Slota, The Hermite-Bell polynomials for negative powers, (submitted, 2012)

D. Dominici, Asymptotic analysis of generalized Hermite polynomials, arXiv:math/0606324 [math.CA], 2006
D. Dominici, Asymptotic analysis of generalized Hermite polynomials, Analysis 28 (2008), 239-261.
R. B. Paris, The asymptotics of the generalised Hermite-Bell polynomials, J. Comput. Appl. Math. 232 (2009), 216-226.

Cf. A066667.

H := proc(n,r,x)
    local e,d ;
    e := exp(-1/x^r) ;
    for d from 1 to n do
        e := diff(e,x) ;
    end do:
    x^((r+1)*n)*exp(1/x^r)*e ;
    expand(%) ;
end proc:
A215216 := proc(n,k)
    subs(x=sqrt(x),H(n,2,x)) ;
    coeftayl( %,x=0,k) ;
end proc:
seq(seq( A215216(n,k),k=0..max(0,n-1)),n=0..6) ; # R. J. Mathar, Aug 07 2012

H[n_, r_, x_] := Module[{e, d }, e = Exp[-1/x^r]; For[d = 1, d <= n, d++, e = D[e, x]]; x^((r + 1)*n)*Exp[1/x^r]*e // Expand];
A215216[n_, k_] := H[n, 2, x] /. x -> Sqrt[x] // SeriesCoefficient[#, {x, 0, k}]&;
Table[A215216[n, k], {n, 0, 8}, {k, 0, Max[0, n - 1]}] // Flatten (* Jean-François Alcover, Nov 24 2017, after R. J. Mathar *)

H(n+1;-2;x) = (2-3*n*x^2)*H(n;-2;x) + x^3*(dH(n;-2;x)/dx), with H(1;-2;x)=2.

A104558 Triangle, read by rows, equal to the matrix inverse of A104557 and related to Laguerre polynomials.

Original entry on oeis.org

1, -1, 1, 0, -2, 1, 0, 2, -4, 1, 0, 0, 6, -6, 1, 0, 0, -6, 18, -9, 1, 0, 0, 0, -24, 36, -12, 1, 0, 0, 0, 24, -96, 72, -16, 1, 0, 0, 0, 0, 120, -240, 120, -20, 1, 0, 0, 0, 0, -120, 600, -600, 200, -25, 1, 0, 0, 0, 0, 0, -720, 1800, -1200, 300, -30, 1, 0, 0, 0, 0, 0, 720, -4320, 5400, -2400, 450, -36, 1, 0, 0, 0, 0, 0, 0, 5040, -15120, 12600, -4200, 630, -42, 1

Offset: 0

Paul D. Hanna, Mar 16 2005

Even-indexed rows are found in A066667 (generalized Laguerre polynomials). Odd-indexed rows are found in A021009 (Laguerre polynomials L_n(x)). Row sums equal A056920 (offset 1). Absolute row sums equal A056953 (offset 1).

			Rows begin:
  1;
  -1,1;
  0,-2,1;
  0,2,-4,1;
  0,0,6,-6,1;
  0,0,-6,18,-9,1;
  0,0,0,-24,36,-12,1;
  0,0,0,24,-96,72,-16,1;
  0,0,0,0,120,-240,120,-20,1;
  0,0,0,0,-120,600,-600,200,-25,1;
  ...
Unsigned columns read downwards equals rows of matrix inverse A104557 read backwards:
  1;
  1,1;
  2,2,1;
  6,6,4,1;
  24,24,18,6,1;
  120,120,96,36,9,1;
  ...

G. C. Greubel, Rows n=0..100 of triangle, flattened

Cf. A104557, A066667, A021009, A056920, A056953.

/* As triangle */ [[(-1)^(n-k)*Factorial(n-k)*Binomial(1+ Floor(n/2), k +1 -Floor((n+1)/2))*Binomial(Floor((n+1)/2), k - Floor(n/2)): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 14 2018

T[n_, k_] := (-1)^(n - k)*(n - k)!*Binomial[1 + Floor[n/2], k + 1 - Floor[(n + 1)/2]]*Binomial[Floor[(n+1)/2], k -Floor[n/2]]; Table[T[n, k], {n, 0, 20}, {k, 0, n}]//Flatten (* G. C. Greubel, May 14 2018 *)

{T(n,k)=(-1)^(n-k)*(n-k)!*binomial(1+n\2,k+1-(n+1)\2)* binomial( (n+1)\2,k-n\2)};

T(n, k) = (-1)^(n-k)*(n-k)!*C(1+[n/2], k+1-[(n+1)/2])*C([(n+1)/2], k-[n/2]).

A145118 Denominator polynomials for continued fraction generating function for n!.

Original entry on oeis.org

1, 1, 1, -1, 1, -2, 1, -4, 2, 1, -6, 6, 1, -9, 18, -6, 1, -12, 36, -24, 1, -16, 72, -96, 24, 1, -20, 120, -240, 120, 1, -25, 200, -600, 600, -120, 1, -30, 300, -1200, 1800, -720, 1, -36, 450, -2400, 5400, -4320, 720, 1, -42, 630, -4200, 12600, -15120

Offset: 0

Paul Barry, Oct 02 2008

Row sums are A056920. T(n,1) gives quarter squares A002620. T(n,2) appears to coincide with 2*A000241(n+1).

			Triangle begins:
1;
1;
1,  -1;
1,  -2;
1,  -4,   2;
1,  -6,   6;
1,  -9,  18,    -6;
1, -12,  36,   -24;
1, -16,  72,   -96,   24;
1, -20, 120,  -240,  120;
1, -25, 200,  -600,  600,  -120;
1, -30, 300, -1200, 1800,  -720;
1, -36, 450, -2400, 5400, -4320, 720;

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A021010, A008297, A066667, A105278.

T:= (n, k)-> (-1)^k* binomial(iquo(n+1, 2),k) *binomial(iquo(n, 2), k)*k!:
seq (seq (T(n, k), k=0..iquo(n, 2)), n=0..16);  # Alois P. Heinz, Dec 04 2012

T(n,k) = (-1)^k C(floor((n+1)/2),k) * C(floor(n/2),k)*k!.

A222472 Numerator sequence of the n-th convergent of the continued fraction 1/(1+3/(2+3/(3+3/(4+...

Original entry on oeis.org

1, 2, 9, 42, 237, 1548, 11547, 97020, 907821, 9369270, 105785433, 1297533006, 17185285377, 244486594296, 3718854770571, 60235136112024, 1035153878216121, 18813475216226250, 360561490742947113, 7267670240507621010

Offset: 1

Gary Detlefs and Wolfdieter Lang, Mar 09 2013

The corresponding denominator sequence is A213190.
a(n) = Phat(n,3) with the numerator polynomials Phat of A221913. All the given formulas follow from there and from the comments given under A084950. The limit of the continued fraction (0 + K_{k=1..oo} (3/k))/3 = 1/(1+3/(2+3/(3+3/(4+... is (1/3)*sqrt(3)*BesselI(1,2*sqrt(3))/BesselI(0,2*sqrt(3)) = 0.484516174987404...
For a combinatorial interpretation in terms of labeled Morse codes see a comment on A221913. Here each dash has label x=3, and the dots have label j if they are at position j. Labels are multiplied and all codes on positions [2,...,n+1] are summed.

			a(4) = 4*a(3) + 2*a(2) = 4*9 + 3*2 = 42.
Continued fraction convergent: 1/(1+3/(2+3/(3+3/4))) = 14/29 = 42/87 = a(4)/A213190(4).
Morse code: a(5) = 237 from the sum of all 5 labeled codes on [2,3,4,5], one with no dash, three with one dash and one with two dashes: 5!/1 + (4*5 + 2*5 + 2*3)*(3) +3^2 = 237.

Cf. A084950, A221913, A222467, A001040(n+1) (x=1), A058798 (x=-1).

a=vector(50); a[1]=1;a[2]=2; for(n=3, #a, a[n]=n*a[n-1]+3*a[n-2]); a \\ Altug Alkan, Apr 20 2018

Recurrence: a(n) = n*a(n-1) + 3*a(n-2), with a(-1) = 1/3, a(0) = 0, n >= 1.
As a sum: a(n) = Sum_{m=0..floor(n/2)} b(n-m,m)*3^m, n >= 1, with b(n,m) = binomial(n,m)*(n+1)!/(m+1)! = |A066667(n,m)| (Laguerre coefficients, parameter alpha =1).
Explicit form: a(n) = -2*(sqrt(3))^n*(BesselK(1, -2*sqrt(3))*BesselI(n+1, -2*sqrt(3)) + (-1)^(n+1)*BesselI(1,-2*sqrt(3))*BesselK(n+1,-2*sqrt(3))).
E.g.f.: Pi*(BesselJ(1, 2*I*sqrt(3)*sqrt(1-z))*BesselY(1, 2*I*sqrt(3)) - BesselY(1, (2*I)*sqrt(3)*sqrt(1-z))*BesselJ(1, 2*I*sqrt(3)))/sqrt(1-z). Here Phat(0,x) = 0.
Asymptotics: lim_{n->oo} a(n)/n! = BesselI(1,2*sqrt(3))/(sqrt(3)) = 3.468649618760...

A343861 Coefficient triangle of generalized Laguerre polynomials n!*L(n,n,x) (rising powers of x).

Original entry on oeis.org

1, 2, -1, 12, -8, 1, 120, -90, 18, -1, 1680, -1344, 336, -32, 1, 30240, -25200, 7200, -900, 50, -1, 665280, -570240, 178200, -26400, 1980, -72, 1, 17297280, -15135120, 5045040, -840840, 76440, -3822, 98, -1, 518918400, -461260800, 161441280, -29352960, 3057600, -188160, 6720, -128, 1

Offset: 0

Seiichi Manyama, May 01 2021

			The triangle begins:
       1;
       2,      -1;
      12,      -8,      1;
     120,     -90,     18,     -1;
    1680,   -1344,    336,    -32,    1;
   30240,  -25200,   7200,   -900,   50,  -1;
  665280, -570240, 178200, -26400, 1980, -72, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

For k=0..1 the (unsigned) columns give A001813, A092956(n-1).
Row sums (signed) give A006902, row sums (unsigned) give A082545.
Cf. A066667, A062137, A062138, A062139, A062140.

[(-1)^k*Factorial(n-k)*Binomial(n,k)*Binomial(2*n, n+k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 11 2022

T[n_, k_] := (-1)^k * (2*n)! * Binomial[n, k]/(k + n)!; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, May 11 2021 *)

T(n, k) = (-1)^k*(2*n)!*binomial(n,k)/(k+n)!;

PARI
```
row(n) = Vecrev(n!*pollaguerre(n, n));
```

def A343861(n,k): return (-1)^k*factorial(n-k)*binomial(n,k)*binomial(2*n,n+k)
flatten([[A343861(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 11 2022

T(n, k) = (-1)^k * n! * binomial(2*n,n-k)/k! = (-1)^k * (2*n)! * binomial(n,k)/(k+n)!.
T(n, 0) = A001813(n).
T(n, 1) = -A092956(n-1).
Sum_{k=0..n} T(n, k) = A006902(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A082545(n).

A215269 Coefficient triangle of the modified Hermite-Bell polynomials for power -2.

Original entry on oeis.org

1, 1, 2, -3, 2, -9, 6, 4, -36, 75, -30, 4, -60, 255, -330, 90, 8, -180, 1290, -3465, 3150, -630, 8, -252, 2730, -12495, 23940, -16380, 2520, 16, -672, 10248, -71400, 235305, -343980, 185220, -22680, 16, -864

Offset: 0

Roman Witula, Aug 07 2012

The modified Hermite-Bell polynomials for power -2 are defined by the formula H(n;-2;sqrt(x))*2^(-floor(n/2)-(1-(-1)^n)/2), where H(n;-2;x) denotes the n-th Hermite-Bell polynomial - see A215216 for the definition and details.

			If we set V(n;x):= H(n;-2;sqrt(x))*2^(-floor(n/2)-(1-(-1)^n)/2) then we obtain V(0;x)=V(1;x)=1, V(2;x)=2-3*x, V(3;x)=2-9*x+6*x^2, V(4;x)=4-36*x+75*x^2-30*x^3, V(5;x)=4-60*x+255*x^2-330*x^3+90*x^4.
1;
1;
2,-3;
2,-9,6;
4,-36,75,-30;
4,-60,255,-330,90;
8,-180,1290,-3465,3150,-630;
8,-252,2730,-12495,23940,-16380,2520;
16,-672,10248,-71400,235305,-343980,185220,-22680;
16,-864,17640,-173880,877905,-2226420,2593080,-1134000,113400;

R. Witula, E. Hetmaniok, D. Slota, The Hermite-Bell polynomials for negative powers, (submitted, 2012)

Cf. A215216, A066667.

H := proc(n,r,x)
    local e,d ;
    e := exp(-1/x^r) ;
    for d from 1 to n do
        e := diff(e,x) ;
    end do:
    x^((r+1)*n)*exp(1/x^r)*e ;
    expand(%) ;
end proc;
A215269 := proc(n,k)
    subs(x=sqrt(x),H(n,2,x))*2^(-floor(n/2)-(1-(-1)^n)/2) ;
    coeftayl( %,x=0,k) ;
end proc:
seq(seq( A215269(n,k),k=0..max(0,n-1)),n=0..12) ; # R. J. Mathar, Aug 07 2012

A287272 a(n) is the number of zeros of the Laguerre L(n, x) polynomial in the open interval (-1, +1).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 0

Michel Lagneau, May 22 2017

nonn

The Laguerre polynomials are given by the sum: L(n,x) = Sum_{k=0..n} ((-1)^k)/k!*binomial(n,k)*x^k.
The first few Laguerre polynomials are:
L(0,x) =   1,
L(1,x) =  -x + 1,
L(2,x) = ( x^2 - 4*x + 2)/2,
L(3,x) = (-x^3 + 9*x^2 - 18*x + 6)/6,
L(4,x) = ( x^4 - 16*x^3 + 72*x^2 - 96*x + 24)/24,
L(5,x) = (-x^5 + 25*x^4 - 200*x^3 + 600*x^2 - 600*x + 120)/120.
The number of occurrences a(n) = 0, 1, 2,.. is given by the sequence {2, 6, 11, 16, 21, ...}.

			a(3) = 1 because the zeros of L(3,x) = (-x^3 + 9*x^2 - 18*x + 6)/6 are x1=.4157745568..., x2=2.294280360... and x3=6.289945083... with the root x1 in the open interval (-1, +1). Hence, a(3) = 1.

Eric Weisstein's World of Mathematics, The World of Mathematics: Laguerre Polynomial

Cf. A066667.

for n from 0 to 90 do:it:=0:
y:=[fsolve(expand(LaguerreL(n,x)),x,real)]:for m from 1 to nops(y) do:if abs(y[m])<1 then it:=it+1:else fi:od: printf(`%d, `, it):od:

cp's OEIS Frontend

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