A121966 -id:A121966 - cp's OEIS frontend

A122018 Modulo 2 recursion switch between A000898 and A121966: A000898 first.

1, 2, 6, 2, 40, 32, 464, 272, 7040, 4864, 136448, 87808, 3177472, 2123776, 86861824, 57128960, 2720112640, 1806049280, 96095928320, 63587041280, 3778819358720, 2507078533120, 163724570132480, 108568842403840, 7748467910901760

Offset: 0

Roger L. Bagula, Sep 11 2006

nonn

Robert Israel, Table of n, a(n) for n = 0..731

Cf. A000898, A121966.

f:= proc(n) option remember; if n::odd then procname(n-1)-(n-1)*procname(n-2) else 2*procname(n-1)+2*(n-1)*procname(n-2) fi end proc:
f(0):= 1: f(1):= 2:
map(f, [$0..50]); # Robert Israel, Jun 20 2018

a[0] = 1; a[1] = 2; a[n_] := a[n] = If[Mod[n, 2] == 1, a[n - 1] - (n - 1)*a[n - 2], 2*(a[n - 1] + (n - 1)*a[n - 2])] b = Table[a[n], {n, 0, 30}]

For n>=2, a(n) = a(n-1) - (n-1)*a(n-2) if n is odd, a(n) = 2*a(n-1) + 2*(n-1)*a(n-2) if n is even. - Edited by Robert Israel, Jun 20 2018

Offset changed by Robert Israel, Jun 20 2018

A062267 Row sums of (signed) triangle A060821 (Hermite polynomials).

Original entry on oeis.org

1, 2, 2, -4, -20, -8, 184, 464, -1648, -10720, 8224, 230848, 280768, -4978816, -17257600, 104891648, 727511296, -1901510144, -28538404352, 11377556480, 1107214478336, 1759326697472, -42984354695168, -163379084079104

Offset: 0

Wolfdieter Lang, Jun 19 2001

G. C. Greubel, Table of n, a(n) for n = 0..730
Index entries for sequences related to Hermite polynomials

Cf. A000898, A121966.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x*(2-x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 08 2018

A062267 := proc(n)
    HermiteH(n,1) ;
    simplify(%) ;
end proc: # R. J. Mathar, Feb 05 2013

lst={};Do[p=HermiteH[n,1];AppendTo[lst,p],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 15 2009 *)
Table[2^n HypergeometricU[-n/2, 1/2, 1], {n, 0, 23}] (* Benedict W. J. Irwin, Oct 17 2017 *)
With[{nmax=50}, CoefficientList[Series[Exp[x*(2-x)], {x,0,nmax}],x]* Range[0, nmax]!] (* G. C. Greubel, Jun 08 2018 *)

x='x+O('x^30); Vec(serlaplace(exp(-x*(x-2)))) \\ G. C. Greubel, Jun 08 2018

a(n) = polhermite(n,1); \\ Michel Marcus, Jun 09 2018

from sympy import hermite, Poly
def a(n): return sum(Poly(hermite(n, x), x).all_coeffs()) # Indranil Ghosh, May 26 2017

a(n) = Sum_{m=0..n} A060821(n, m) = H(n, 1), with the Hermite polynomials H(n, x).
E.g.f.: exp(-x*(x-2)).
a(n) = 2*(a(n - 1) - (n - 1)*a(n - 2)). - Roger L. Bagula, Sep 11 2006
a(n) = 2^n * U(-n/2, 1/2, 1), where U is the confluent hypergeometric function. - Benedict W. J. Irwin, Oct 17 2017
E.g.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^(mu(k)/k). - Ilya Gutkovskiy, May 26 2019

A122021 a(n) = a(n-2) - (n-1)*a(n-3), with a(0) = 0, a(1) = 1, a(2) = 2.

Original entry on oeis.org

0, 1, 2, 1, -1, -7, -6, -1, 43, 47, 52, -383, -465, -1007, 4514, 5503, 19619, -66721, -73932, -419863, 1193767, 1058777, 10010890, -25204097, -14340981, -265465457, 615761444, 107400049, 7783328783, -17133920383, 4668727362

Offset: 0

Roger L. Bagula, Sep 12 2006

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000898, A121966, A062267.

a:= function(n)
    if n<3 then return n;
    else return a(n-2) - (n-1)*a(n-3);
    fi;
  end;
List([0..30], n-> a(n) ); # G. C. Greubel, Oct 06 2019

I:=[0,1,2]; [n le 3 select I[n] else Self(n-2) - (n-2)*Self(n-3): n in [1..30]]; // G. C. Greubel, Oct 06 2019

a:= proc(n) option remember;
if n < 3 then n
else a(n-2)-(n-1)*a(n-3)
fi;
end proc;
seq(a(n), n = 0..30); # G. C. Greubel, Oct 06 2019

a[0]=0; a[1]=1; a[2]=2; a[n_]:= a[n]= a[n-2] - (n-1)*a[n-3]; Table[a[n], {n, 0, 30}]
RecurrenceTable[{a[0]==0,a[1]==1,a[2]==2,a[n]==a[n-2]-(n-1)a[n-3]},a,{n,30}] (* Harvey P. Dale, Apr 29 2022 *)

my(m=30, v=concat([0,1,2], vector(m-3))); for(n=4, m, v[n] = v[n-2] - (n-2)*v[n-3]); v \\ G. C. Greubel, Oct 06 2019

def a(n):
    if (n<3): return n
    else: return a(n-2) - (n-1)* a(n-3)
[a(n) for n in (0..30)] # G. C. Greubel, Oct 06 2019

Edited by N. J. A. Sloane, Sep 12 2006

Offset changed by G. C. Greubel, Oct 06 2019

A122022 a(n) = a(n-1) - (n-1)*a(n-4), with a(0) = 0, a(1) = 1, a(2) = 2, a(3) = 1.

Original entry on oeis.org

0, 1, 2, 1, 1, -3, -13, -19, -26, -2, 115, 305, 591, 615, -880, -5150, -14015, -23855, -8895, 83805, 350090, 827190, 1013985, -829725, -8881795, -28734355, -54083980, -32511130, 207297335, 1011859275, 2580294695, 3555628595, -2870588790, -35250085590

Offset: 0

Roger L. Bagula, Sep 12 2006

sign

Harvey P. Dale, Table of n, a(n) for n = 0..999 [Offset adapted by _Georg Fischer_, Jun 06 2021]

Cf. A000898, A062267, A121966, A122050.

a:= function(n)
    if n<3 then return n;
    elif n=3 then return 1;
    else return a(n-1) - (n-1)*a(n-4);
    fi;
  end;
List([1..30], n-> a(n) ); # G. C. Greubel, Oct 06 2019

I:=[0,1,2,1]; [n le 4 select I[n] else Self(n-1) - (n-2)*Self(n-4): n in [1..30]]; // G. C. Greubel, Oct 06 2019

a:= proc(n) option remember;
      if n<3 then n
    elif n=3 then 1
    else a(n-1) - (n-1)*a(n-4)
      fi;
end: seq(a(n), n=0..30); # G. C. Greubel, Oct 06 2019

a[0]=0; a[1]=1; a[2]=2; a[3]=1; a[n_]:= a[n]= a[n-1] - (n-1)*a[n-4]; Table[a[n], {n, 0, 30}]
RecurrenceTable[{a[0]==0,a[1]==1,a[2]==2,a[3]==1,a[n]==a[n-1]- (n-1) a[n-4]},a,{n,0,30}] (* Harvey P. Dale, Nov 28 2014 *)

my(m=30, v=concat([0,1,2,1], vector(m-4))); for(n=5, m, v[n] = v[n-1] - (n-2)*v[n-4]); v \\ G. C. Greubel, Oct 06 2019

@CachedFunction
def a(n):
    if (n<3): return n
    elif (n==3): return 1
    else: return a(n-1) - (n-1)*a(n-4)
[a(n) for n in (0..30)] # G. C. Greubel, Oct 06 2019

Edited by N. J. A. Sloane, Sep 12 2006

Offset corrected by Georg Fischer, Jun 06 2021

A122033 a(n) = 2a(n-1) - 2(n-2)*a(n-2), with a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 4, 4, -8, -40, -16, 368, 928, -3296, -21440, 16448, 461696, 561536, -9957632, -34515200, 209783296, 1455022592, -3803020288, -57076808704, 22755112960, 2214428956672, 3518653394944, -85968709390336, -326758168158208, 3301044295639040, 22286480662872064

Offset: 0

Roger L. Bagula, Sep 13 2006

G. C. Greubel, Table of n, a(n) for n = 0..725

Cf. A000898, A121966, A328141.

a:=[1,2];; for n in [3..35] do a[n]:=2*(a[n-1]-(n-3)*a[n-2]); od; a; # G. C. Greubel, Oct 04 2019

I:=[1,2]; [n le 2 select I[n] else 2*(Self(n-1)-(n-3)*Self(n-2)): n in [1..35]]; // G. C. Greubel, Oct 04 2019

a:= proc(n) option remember;
      if n < 2 then n+1
    else 2*(a(n-1) - (n-2)*a(n-2))
      fi
    end proc:
seq(a(n), n = 0..35); # G. C. Greubel, Oct 04 2019

a[n_]:= a[n]= If[n<2, n+1, a[n-1]-(n-2)*a[n-2]]; Table[a[n], {n,0,30}] (* modified by G. C. Greubel, Oct 04 2019 *)
nxt[{n_,a_,b_}]:={n+1,b,2b-2a(n-1)}; NestList[nxt,{1,1,2},30][[;;,2]] (* Harvey P. Dale, Jan 01 2024 *)

my(m=35, v=concat([1,2], vector(m-2))); for(n=3, m, v[n] = 2*(v[n-1] - (n-3)*v[n-2] ) ); v \\ G. C. Greubel, Oct 04 2019

def a(n):
    if n<2: return n+1
    else: return 2*(a(n-1) - (n-2)*a(n-2))
[a(n) for n in (0..35)] # G. C. Greubel, Oct 04 2019

a(n) = 2*a(n-1) - 2*(n-2)*a(n-2), with a(0)=1, a(1)=2. - G. C. Greubel, Oct 04 2019
a(n) = 2*A062267(n-1) for n > 0. - Michel Marcus, Oct 05 2019
E.g.f.: 1 + exp(1)*sqrt(Pi)*( erf(1) - erf(1-x) ), where erf(x) is the error function. - G. C. Greubel, Oct 05 2019

Edited by N. J. A. Sloane, Sep 17 2006

Corrected and offset changed by G. C. Greubel, Oct 04 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.

A122031 a(n) = a(n - 1) + (n - 1)*a(n - 2).

Original entry on oeis.org

1, 2, 3, 7, 16, 44, 124, 388, 1256, 4360, 15664, 59264, 231568, 942736, 3953120, 17151424, 76448224, 350871008, 1650490816, 7966168960, 39325494464, 198648873664, 1024484257408, 5394759478016, 28957897398400

Offset: 0

Roger L. Bagula, Sep 13 2006

nonn

Equals the eigensequence of an infinite lower triangular matrix with (1, 1, 1, ...) in the main diagaonal, (1, 1, 2, 3, 4, 5, ...) in the subdiagonal and the rest zeros. - Gary W. Adamson, Apr 13 2009

G. C. Greubel, Table of n, a(n) for n = 0..795

Cf. A000898, A062267, A121966, A122021, A122022.

a[0] = 1; a[1] = 2; a[n_] := a[n] = a[n - 1] + (n - 1)*a[n - 2] Table[a[n], {n, 0, 30}]
Table[n!*SeriesCoefficient[1/2*Exp[x+x^2/2]*(2-Sqrt[2*E*Pi]*Erf[1/Sqrt[2]]+Sqrt[2*E*Pi]*Erf[(1+x)/Sqrt[2]]),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec after Paul Abbott, Dec 27 2012 *)
RecurrenceTable[{a[0]==1,a[1]==2,a[n]==a[n-1]+(n-1)a[n-2]},a,{n,30}] (* Harvey P. Dale, Feb 21 2015 *)

E.g.f.: (1/2)*exp(x + x^2/2)*(2 - sqrt(2*exp(1)*Pi)*erf(1/sqrt(2)) + sqrt(2*exp(1)*Pi)*erf((1+x)/sqrt(2))). - Paul Abbott (paul(AT) physics.uwa.edu.au)

a(n) ~ (1/sqrt(2) + sqrt(Pi)/2*exp(1/2) * (1 - erf(1/sqrt(2)))) * n^(n/2)*exp(sqrt(n) - n/2 - 1/4) * (1+7/(24*sqrt(n))). - Vaclav Kotesovec, Dec 27 2012

Edited by N. J. A. Sloane, Sep 17 2006

Offset corrected by Vaclav Kotesovec, Dec 27 2012

A122598 a(0) = 0; a(1) = 1; if n is odd then a(n) = 2a(n-1) - (n-1)a(n-2) otherwise a(n) = 2(a(n-1) - (n-2)a(n-2)).

Original entry on oeis.org

0, 1, 2, 2, -4, -16, 0, 96, 192, -384, -3840, -3840, 69120, 184320, -1290240, -5160960, 25804800, 134184960, -557383680, -3530096640, 13005619200, 96613171200, -326998425600, -2779486617600, 8828957491200, 84365593804800, -255058771968000, -2703622982860800, 7855810176614400

Offset: 0

Roger L. Bagula, Sep 19 2006

E. S. R. Gopal, Specific Heats at Low Temperatures, Plenum Press, New York, 1966, pages 36-40.

Robert Israel, Table of n, a(n) for n = 0..804

Cf. A000898, A121966.

f:= proc(n) option remember;
if n::odd then 2*procname(n-1) - (n-1)*procname(n-2)
else 2*procname(n-1) - 2*(n-2)*procname(n-2)
fi
end proc:
f(0):= 0: f(1):= 1:
map(f, [$0..100]); # Robert Israel, Mar 15 2017

a[0] = 0; a[1] = 1; a[n_] := a[n] = If[Mod[n, 2] == 1, 2*a[n - 1] - ( n - 1)*a[n - 2], 2*(a[n - 1] - (n - 2)*a[n - 2])] b = Table[a[n], {n, 0, 30}]
nxt[{n_,a_,b_}]:={n+1,b,If[EvenQ[n],2*b-n*a,2(b-(n-1)a)]}; Transpose[ NestList[ nxt,{1,0,1},30]][[2]] (* Harvey P. Dale, Dec 15 2014 *)

a(n) = 2*a(n-1) - (n-1)*a(n-2) for n odd > 1; a(n) = 2*(a(n-1) - (n-2)*a(n-2)) for n even > 1.

Edited by N. J. A. Sloane, Oct 01 2006

A122017 a(n) = if n even then a(n - 1) - (n - 1)a(n - 2) otherwise 2(a(n - 1) + (n - 1)*a(n - 2)).

Original entry on oeis.org

1, 2, 1, 10, 7, 94, 59, 1246, 833, 21602, 14105, 460250, 305095, 11656190, 7689955, 341753230, 226403905, 11388911170, 7540044785, 425080891690, 281820040775, 17566875749150, 11648654892875, 796239842748350, 528320780212225

Offset: 0

Roger L. Bagula, Sep 11 2006

Cf. A000898, A121966.

a[0] = 1; a[1] = 2; a[n_] := a[n] = If[Mod[n, 2] == 0, a[n - 1] - (n - 1)*a[n - 2], 2*(a[n - 1] + (n - 1)*a[n - 2])] b = Table[a[n], {n, 0, 30}]
nxt[{n_,a_,b_}]:={n+1,b,If[OddQ[n],b-n*a,2(b+n*a)]}; NestList[nxt,{1,1,2},30][[All,2]] (* Harvey P. Dale, Jun 12 2020 *)

Edited by N. J. A. Sloane, Sep 13 2006

Offset corrected by Jason Yuen, Sep 15 2024

cp's OEIS Frontend

A122018 Modulo 2 recursion switch between A000898 and A121966: A000898 first.

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A062267 Row sums of (signed) triangle A060821 (Hermite polynomials).

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A122021 a(n) = a(n-2) - (n-1)*a(n-3), with a(0) = 0, a(1) = 1, a(2) = 2.

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A122022 a(n) = a(n-1) - (n-1)*a(n-4), with a(0) = 0, a(1) = 1, a(2) = 2, a(3) = 1.

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A122033 a(n) = 2*a(n-1) - 2*(n-2)*a(n-2), with a(0)=1, a(1)=2.

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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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A122033 a(n) = 2a(n-1) - 2(n-2)*a(n-2), with a(0)=1, a(1)=2.

A122598 a(0) = 0; a(1) = 1; if n is odd then a(n) = 2a(n-1) - (n-1)a(n-2) otherwise a(n) = 2(a(n-1) - (n-2)a(n-2)).

A122017 a(n) = if n even then a(n - 1) - (n - 1)a(n - 2) otherwise 2(a(n - 1) + (n - 1)*a(n - 2)).