A252284 -id:A252284 - cp's OEIS frontend

A001464 Expansion of e.g.f. exp(-x - (1/2)*x^2).

1, -1, 0, 2, -2, -6, 16, 20, -132, -28, 1216, -936, -12440, 23672, 138048, -469456, -1601264, 9112560, 18108928, -182135008, -161934624, 3804634784, -404007680, -83297957568, 92590134208, 1906560847424, -4221314202624, -45349267830400, 159324751301248

Offset: 0

N. J. A. Sloane, J. H. Conway and Simon Plouffe

From Robert Israel, Apr 27 2017: (Start)
(-1)^n*a(n) is (the number of even involutions) - (the number of odd involutions) in the symmetric group S_n.
a(n) == (-1)^n (mod A069834(n-1)) for n >= 3.
a(n) is divisible by n-2 and by A200675(n+2). (End)

			G.f. = 1 - x + 2*x^3 - 2*x^4 - 6*x^5 + 16*x^6 + 20*x^7 - 132*x^8 + ...

Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, Dover Publications, New York, 1945, page 32.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Israel, Table of n, a(n) for n = 0..807
John Campbell, A class of symmetric difference-closed sets related to commuting involutions, Discrete Mathematics & Theoretical Computer Science, Vol 19 no. 1, 2017.
Robert Israel, Solution to Problem 91-9: Even Minus Odd Involutions in the Symmetric Group, SIAM Review 34(2)(1992), 315-317.
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.
Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A000085, A000178, A001147, A066325, A069834, A099174, A200675.

Cf. A252284, A369755, A369756.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(-x-x^2/2) ))); // G. C. Greubel, Sep 03 2023

f:= gfun:-rectoproc({a(n)=-a(n-1)-(n-1)*a(n-2), a(0)=1,a(1)=-1},a(n),remember):
map(f, [$0..100]); # Robert Israel, Apr 27 2017
a := n -> (-1)^n*2^((n-1)/2)*KummerU((1-n)/2, 3/2, 1/2): seq(simplify(a(n)), n=0..28); # Peter Luschny, Apr 30 2017

With[{nn=30},CoefficientList[Series[Exp[-x-1/2 x^2],{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Sep 16 2011 *)
a[ n_] := If[ n < 0, 0, HermiteH[ n, Sqrt[1/2]] (-Sqrt[1/2])^n]; (* Michael Somos, Jan 24 2014 *)
a[ n_] := If[ n < 0, 0, (-1)^n Sum[ (-1)^k Binomial[ n, 2 k] (2 k - 1)!!, {k, 0, n/2}]]; (* Michael Somos, Jan 24 2014 *)
Table[(-1)^(n + 1)*DifferenceRoot[Function[{y, m}, {y[1 + m] == y[m] - (n - m) y[m - 1], y[0] == 0, y[1] == 1, y[2] == 1}]][n], {n, 1, 30}] (* Benedict W. J. Irwin, Nov 03 2016 *)

Vec( serlaplace( exp( -x -(1/2)*x^2 + O(x^66) ) ) ) /* Joerg Arndt, Oct 13 2012 */

{a(n) = if( n<0, 0, (-1)^n * sum(k=0, n\2, (-1/2)^k * n! / (k! * (n - 2*k)!)))}; /* Michael Somos, Jan 24 2014 */

def A001464_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-x-x^2/2) ).egf_to_ogf().list()
A001464_list(40) # G. C. Greubel, Sep 03 2023

From Benoit Cloitre, May 01 2003: (Start)
a(n) = -h(n, -1) where h(n, x) is the Hermite polynomial h(n, x) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n, 2*k)*Product_{i=0..k} (2*i-1)*x^(n-2*k).
a(n) = (-1)^n*Sum_{k=0..floor(n/2)} (-1)^k*C(n, 2*k)*(2k-1)!!. (End)
a(n) = -a(n-1) - (n-1)*a(n-2); a(0)=1, a(1)=-1. - Matthew J. White (mattjameswhite(AT)hotmail.com), Mar 01 2006
From Sergei N. Gladkovskii, Oct 12 2012, Nov 04 2012, Apr 17 2013, Nov 13 2013: (Start)
Continued fractions:
G.f.: 1/(U(0) + x) where U(k) = 1 + x*(k+1) - x*(k+1)/(1 + x/U(k+1)).
G.f.: 1/U(0) where U(k) = 1 + x + x^2*(k+1)/U(k+1).
G.f.: 1/Q(0) where Q(k) = 1 + x*k + x/(1 - x*(k+1)/Q(k+1)).
G.f.: T(0)/(1+x) where T(k) = 1 - x^2*(k+1)/(x^2*(k+1) + (1+x)^2/T(k+1)). (End)
From Michael Somos, Jan 24 2014: (Start)
Binomial transform is [1, 0, -1, 0, 3, 0, -15, 0, 105, ...] where A001147 = [1, 1, 3, 15, 105, ...].
Hankel transform is [1, -1, -2, 12, 288, -34560, -24883200, ...] where A000178 = [1, 1, 2, 12, 288, 34560, 24883200, ...].
0 = a(n) * (-a(n+1) - a(n+2) - a(n+3)) + a(n+1) * (a(n+1) + a(n+2)) for all n in Z. (End)
a(n) = -(-1)^n*y(n,n), where y(m+1,n) = y(m,n) - (n-m)*y(m-1,n), with y(0,n)=0, y(1,n)=y(2,n)=1 for all n. - Benedict W. J. Irwin, Nov 03 2016
a(n) = (-1)^n*2^((n-1)/2)*KummerU((1-n)/2, 3/2, 1/2). - Peter Luschny, Apr 30 2017
a(n) = Sum_{k=0..n} 2^k * Stirling1(n,k) * Bell_k(-1/2), where Bell_n(x) is n-th Bell polynomial. - Seiichi Manyama, Jan 31 2024

a(12) and a(13) corrected by Simon Plouffe

A369755 Expansion of e.g.f. exp( (1 - (1+x)^4)/4 ).

Original entry on oeis.org

1, -1, -2, 2, 28, 44, -464, -3088, 1408, 135872, 726976, -2959936, -67261952, -293413888, 3054389248, 52458520064, 178569842176, -3909868400128, -60465254054912, -149165881689088, 6569005278939136, 98054837101881344, 158559568611401728, -14356527387138039808

Offset: 0

Seiichi Manyama, Jan 31 2024

sign

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A001464, A252284, A369756.

Cf. A049426.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((1-(1+x)^4)/4)))

a(0) = 1; a(n) = -(n-1)! * Sum_{k=1..min(4,n)} binomial(3,k-1) * a(n-k)/(n-k)!.
a(n) = Sum_{k=0..n} 4^k * Stirling1(n,k) * Bell_k(-1/4), where Bell_n(x) is n-th Bell polynomial.
D-finite with recurrence a(n) +a(n-1) +3*(n-1)*a(n-2) +3*(n-1)*(n-2)*a(n-3) +(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Feb 02 2024

A369756 Expansion of e.g.f. exp( (1 - (1+x)^5)/5 ).

Original entry on oeis.org

1, -1, -3, -1, 49, 255, -275, -13105, -83775, 170495, 8290045, 69257055, -111005135, -9684015745, -109196883795, -31470300625, 17728458119425, 276531029694975, 904537471692925, -44728487203650625, -1000823562359108175, -7110596979389965825

Offset: 0

Seiichi Manyama, Jan 31 2024

sign

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A001464, A252284, A369755.

Cf. A049427, A369753.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((1-(1+x)^5)/5)))

a(0) = 1; a(n) = -(n-1)! * Sum_{k=1..min(5,n)} binomial(4,k-1) * a(n-k)/(n-k)!.

a(n) = Sum_{k=0..n} 5^k * Stirling1(n,k) * Bell_k(-1/5), where Bell_n(x) is n-th Bell polynomial.

A374882 Expansion of e.g.f. exp( (1 - (1 - 9*x)^(1/3))/3 ).

Original entry on oeis.org

1, 1, 7, 109, 2665, 88981, 3768391, 193406977, 11663021329, 808092594505, 63252127883431, 5519514702282901, 531266903931402937, 55912682968563924829, 6387276499619184590695, 787104141893585220839401, 104074098535487279656795681, 14697203663694095986066104337

Offset: 0

Seiichi Manyama, Aug 02 2024

nonn

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A080893, A252284, A317996.

Cf. A375174, A375176.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1-(1-9*x)^(1/3))/3)))

a(n) = Sum_{k=0..n} (-9)^(n-k) * Stirling1(n,k) * A317996(k) = (-9)^n * Sum_{k=0..n} (1/3)^k * Stirling1(n,k) * Bell_k(-1/3), where Bell_n(x) is n-th Bell polynomial.
From Vaclav Kotesovec, Aug 02 2024: (Start)
a(n) = 18*(n-2)*a(n-1) - 9*(3*n-8)*(3*n-7)*a(n-2) + a(n-3).
a(n) ~ Gamma(1/3) * 3^(2*n - 3/2) * n^(n - 5/6) / (sqrt(2*Pi) * exp(n - 1/3)) * (1 - 2*Pi/(3^(3/2)*Gamma(1/3)^2*n^(1/3))). (End)

A249015 A binomial convolution.

Original entry on oeis.org

1, 1, 5, 17, 69, 339, 1677, 9321, 55137, 343659, 2285289, 15910857, 116120781, 886308147, 7033465989, 58008074409, 495792941337, 4381170220251, 39980186877537, 376025841184329, 3640077999981189, 36224841818288547, 370112212444620861, 3878334404076375657

Offset: 0

Emanuele Munarini, Oct 20 2014

nonn

Cf. A049425, A252284.

b[n_] := Sum[(n!/k!)Sum[Binomial[k,i]Binomial[k-i+2,n-2i-k]/3^i,{i,0,k}],{k,0,n}]
c[n_] := Sum[(n!/k!)(-1)^k Sum[Binomial[k,i]Binomial[k-i,n-2i-k]/3^i,{i,0,k}],{k,0,n}]
Table[If[n==0,1,0]+Sum[Binomial[n,k]b[k]c[n-k-1],{k,0,n-1}],{n,0,40}]

b(n) := sum((n!/k!)*sum(binomial(k,i)*binomial(k-i+2,n-2*i-k)/3^i,i,0,k),k,0,n);
c(n) := sum((n!/k!)*(-1)^k*sum(binomial(k,i)*binomial(k-i,n-2*i-k)/3^i,i,0,k),k,0,n);
makelist((if n=0 then 1 else 0)+sum(binomial(n,k)*b(k)*c(n-k-1),k,0,n-1),n,0,20);

a(n) = 0^0 + Sum_{k=0..n-1} binomial(n,k)*b(k)*c(n-k-1),
where the numbers b(n) = A049425(n+1) have e.g.f. (1+t)^2*exp(t+t^2+t^3/3)
and the numbers c(n) have e.g.f. exp(-(t+t^2+t^3/3)).
D-finite with recurrence: a(n+4) + n*a(n+3) - 3*(n+3)*a(n+2) - 3*(n+3)*(n+2)*a(n+1) - (n+3)*(n+2)*(n+1)*a(n) = 0.
E.g.f.: A(t) = 1+(1+t)^2*exp(t+t^2+t^3/3)*Integral_{u=0..t} exp(-(u+u^2+u^3/3)) du.
Differential equation for the e.g.f.: (1+t)*A''(t) - (2+3*t+3*t^2+t^3)*A'(t) - 3*(1+t)^2*A(t) = 0.

A331816 Irregular triangle (read by rows) of coefficients T(n,k) of polynomials p(n,x) = Sum_{k=0..2n} T(n,k) x^k = (-1)^n * e^(x^3/3) * (((d/dx)^n) e^(-x^3/3)) for n >= 0 and 0 <= k <= 2*n.

Original entry on oeis.org

1, 0, 0, 1, 0, -2, 0, 0, 1, 2, 0, 0, -6, 0, 0, 1, 0, 0, 20, 0, 0, -12, 0, 0, 1, 0, -40, 0, 0, 80, 0, 0, -20, 0, 0, 1, 40, 0, 0, -360, 0, 0, 220, 0, 0, -30, 0, 0, 1, 0, 0, 1120, 0, 0, -1680, 0, 0, 490, 0, 0, -42, 0, 0, 1, 0, -2240, 0, 0, 9520, 0, 0, -5600, 0, 0, 952, 0, 0, -56, 0, 0, 1

Offset: 0

Werner Schulte, Jan 27 2020

Let r(s;n,x) = Sum_{k=0..s*n} A(s;n,k)*x^k = (-1)^n * e^(x^(s+1)/(s+1)) * (((d/dx)^n) e^(-x^(s+1)/(s+1))) for n >= 0 and x complex and some fixed integer s >= 1. Special cases: A(1;n,k) = A066325(n,k) and A(2;n,k) is this triangle. Formula: A(s;n,k) = (Sum_{i=0..floor(k/(s+1))} (-1)^i * binomial((n+k) /(s+1),i) * binomial(n+k-(s+1)*i,n)) * (-1)^(n-(n+k)/(s+1)) * (n!) / ((s+1)^((n+k)/(s+1)) * (((n+k)/(s+1))!)) if (n+k) mod (s+1) = 0 else 0 with n >= 0 and 0 <= k <= s*n.
  Recurrence: (1) A(s;n,k) = A(s;n-1,k-s) - (k+1) * A(s;n-1,k+1),
  (2) r(s;n,x) = x^s * r(s;n-1,x) - ((d/dx) r(s;n-1,x)) for n > 0 with initial values A(s;0,0) = 1 = r(s;0,x) and A(s;n,k) = 0 if k < 0 or k > s*n or (n+k) mod (s+1) > 0;
  E.g.f.: Sum_{n>=0} r(s;n,x)*t^n/(n!) = e^((x^(s+1)-(x-t)^(s+1))/(s+1)).
This generalization is result of a long and intensive discussion with Wolfdieter Lang. For more information see A091752.

			The irregular triangle T(n,k) starts:
n\k:  0     1    2    3    4     5   6     7   8   9  10   . . .      16
========================================================================
0  :  1
1  :  0     0    1
2  :  0    -2    0    0    1
3  :  2     0    0   -6    0     0   1
4  :  0     0   20    0    0   -12   0     0   1
5  :  0   -40    0    0   80     0   0   -20   0   0   1
6  : 40     0    0 -360    0     0 220     0   0 -30   0   0 1
7  :  0     0 1120    0    0 -1680   0     0 490   0   0 -42 0   0 1
8  :  0 -2240    0    0 9520     0   0 -5600   0   0 952   0 0 -56 0 0 1
etc.

Cf. A049404, A066325, A091752.

Row sums are (-1)^n*A252284(n).

T(n,k) = (-1)^k * (n!) * (Sum_{i=0..floor(k/3)} (-1)^i * binomial((n+k) /3,i) * binomial(n+k-3*i,n)) / (3^((n+k)/3) * ((n+k)/3)!) if (n+k) mod 3 = 0 else 0 with n >= 0 and 0 <= k <= 2*n.
Recurrence: (1) T(n,k) = T(n-1,k-2) - (k+1) * T(n-1,k+1),
  (2) T(n,k) = T(n-1,k-2) - 2*(n-1)*T(n-2,k-1) + (n-1)*(n-2)*T(n-3,k),
  (3) k*T(n,k) = 2*n*T(n-1,k-2) - n*(n-1)*T(n-2,k-1),
  (4) p(n,x) = x^2 * p(n-1,x) - (d/dx) p(n-1,x),
  (5) p(n,x) = x^2*p(n-1,x) - 2*(n-1)*x*p(n-2,x) + (n-1)*(n-2)*p(n-3,x),
  (6) (d/dx) p(n,x) = 2*n*x*p(n-1,x) - n*(n-1)*p(n-2,x) for n > 0 with initial values T(0,0) = 1 = p(0,x) and T(n,k) = 0 if k < 0 or k > 2*n or (n+k) mod 3 > 0.
T(n,2*n) = 1 for n >= 0.
T(3*n,0) = -T(3*n-1,1) = 2*T(3*n-2,2) = ((3*n)!)/(3^n * (n!)) for n > 0.
The polynomials p(n,x) satisfy for n >= 0 and x complex the differential equation: 0 = (((d/dx)^3) p(n,x)) - 2*x^2*(((d/dx)^2) p(n,x)) + (x^4 + 2*(n-1)*x) * ((d/dx) p(n,x)) - (2*n*x^3-(n+3)*n) * p(n,x).
E.g.f.: Sum_{n>=0} p(n,x)*t^n/(n!) = e^((x^3-(x-t)^3)/3).
((d/dx)^m) p(n,x) = Sum_{i=0..m} (-1)^i * binomial(m,i) * p(m-i,-x) * p(n+i,x) for m,n >= 0 and x complex.
T(3*n-k,k) = A091752(n+1,k+2) for 0 <= k <= 2*n.
(-1)^(n-k) * T(n,3*k-n) = A049404(n,k) for n > 0 and (n+2)/3 <= k <= n.

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A001464 Expansion of e.g.f. exp(-x - (1/2)*x^2).

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A369755 Expansion of e.g.f. exp( (1 - (1+x)^4)/4 ).

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A369756 Expansion of e.g.f. exp( (1 - (1+x)^5)/5 ).

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PARI

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A374882 Expansion of e.g.f. exp( (1 - (1 - 9*x)^(1/3))/3 ).

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PARI

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A249015 A binomial convolution.

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A331816 Irregular triangle (read by rows) of coefficients T(n,k) of polynomials p(n,x) = Sum_{k=0..2*n} T(n,k) * x^k = (-1)^n * e^(x^3/3) * (((d/dx)^n) e^(-x^3/3)) for n >= 0 and 0 <= k <= 2*n.

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A331816 Irregular triangle (read by rows) of coefficients T(n,k) of polynomials p(n,x) = Sum_{k=0..2n} T(n,k) x^k = (-1)^n * e^(x^3/3) * (((d/dx)^n) e^(-x^3/3)) for n >= 0 and 0 <= k <= 2*n.