A286032
a(n) = a(n-1) - n*a(n-2); a(0) = a(1) = 1.
Original entry on oeis.org
1, 1, -1, -4, 0, 20, 20, -120, -280, 800, 3600, -5200, -48400, 19200, 696800, 408800, -10740000, -17689600, 175630400, 511732800, -3000875200, -13747264000, 52271990400, 368459062400, -886068707200, -10097545267200, 12940241120000, 285573963334400
Offset: 0
-
a := proc(n) option remember;
if n <= 1 then 1 else a(n-1) - n*a(n-2) fi end:
seq(a(n), n = 0..27);
a_list := proc(len) 1 - sqrt(Pi/2)*exp(-((x-1)^2)/2)*(x-1)*
(erfi((x-1)/sqrt(2)) + erfi(1/sqrt(2))); series(%, x, len+2):
seq(n!*simplify(coeff(%,x,n)),n=0..len-1) end: a_list(27);
-
l={1, 1}; Do[AppendTo[l, l[[-1]] - n*l[[-2]]], {n, 2, 30}]; l (* Indranil Ghosh, May 01 2017 *)
RecurrenceTable[{a[0]==a[1]==1,a[n]==a[n-1]-n a[n-2]},a,{n,40}] (* Harvey P. Dale, Jun 20 2021 *)
-
l=[1, 1]
a=b=1
i=2
while i<=30:
l.append(b - i*a)
b=l[-1]
a=l[-2]
i+=1
print(l) # Indranil Ghosh, May 01 2017
A136645
Triangle of coefficients of a Pascal sum of recursive orthogonal Hermite polynomials given in Hochstadt's book: P(x, n) = x*P(x, n - 1) - n*P(x, n - 2); p2(x,n)=Sum[Binomial[n,m],{m,0,n}].
Original entry on oeis.org
1, 1, 1, -1, 2, 1, -5, -2, 3, 1, -3, -16, -3, 4, 1, 21, -12, -35, -4, 5, 1, 43, 104, -33, -64, -5, 6, 1, -97, 246, 315, -74, -105, -6, 7, 1, -455, -656, 859, 752, -145, -160, -7, 8, 1, 361, -3402, -2565, 2340, 1551, -258, -231, -8, 9, 1, 4951, 3196, -14805, -7608, 5445, 2892, -427, -320, -9, 10, 1
Offset: 1
{1},
{1, 1},
{-1, 2, 1},
{-5, -2, 3, 1},
{-3, -16, -3, 4, 1},
{21, -12, -35, -4, 5, 1},
{43, 104, -33, -64, -5, 6, 1},
{-97, 246, 315, -74, -105, -6, 7,1},
{-455, -656, 859, 752, -145, -160, -7, 8, 1},
{361, -3402, -2565, 2340, 1551, -258, -231, -8, 9, 1},
{4951, 3196, -14805, -7608, 5445, 2892, -427, -320, -9, 10, 1}
- Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, pp. 8, 42-43.
-
P[x, 0] = 1; P[x, 1] = x; P[x, -1] = 0; P[x, -2] = -1; P[x_, n_] := P[x, n] = x*P[x, n - 1] - n*P[x, n - 2]; P2[x_, n_] := P2[x, n] = Sum[Binomial[n, m]*P[x, m], {m, 0, n}]; Table[ExpandAll[P2[x, n]], {n, 0, 10}]; a = Table[CoefficientList[P2[x, n], x], {n, 0, 10}]; Flatten[a]
A137437
Triangular sequence from expansion coefficients of asymptotic Hermite Polynomial from Roman: p(x,t)= (1 + t)^(-x)*Exp[x*(t - t^2/2)].
Original entry on oeis.org
1, 0, 0, 0, -2, 0, 6, 0, -24, 0, 120, 40, 0, -720, -420, 0, 5040, 3948, 0, -40320, -38304, -2240, 0, 362880, 396576, 50400
Offset: 1
{1},
{0},
{0},
{0, -2},
{0, 6},
{0, -24},
{0, 120, 40},
{0, -720, -420},
{0, 5040, 3948},
{0, -40320, -38304, -2240},
{0, 362880, 396576, 50400}
- Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 130.
-
p[t_] := (1 + t)^(-x)*Exp[x*(t - t^2/2)];
Table[ ExpandAll[n!SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}];
a = Table[n!* CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}];
Flatten[{{1}, {0}, {0}, {0, -2}, {0, 6}, {0, -24}, {0, 120, 40}, {0, -720, -420}, {0, 5040, 3948}, {0, -40320, -38304, -2240}, {0, 362880, 396576, 50400}}]
Comments