A067994 -id:A067994 - cp's OEIS frontend

A000321 H_n(-1/2), where H_n(x) is Hermite polynomial of degree n.

1, -1, -1, 5, 1, -41, 31, 461, -895, -6481, 22591, 107029, -604031, -1964665, 17669471, 37341149, -567425279, -627491489, 19919950975, 2669742629, -759627879679, 652838174519, 31251532771999, -59976412450835, -1377594095061119, 4256461892701199, 64623242860354751

Offset: 0

N. J. A. Sloane

Binomial transform gives A067994. Inverse binomial transform gives A062267(n)*(-1)^n. - Vladimir Reshetnikov, Oct 11 2016

The congruence a(n+k) == (-1)^k*a(n) (mod k) holds for all n and k. It follows that for even k the sequence obtained by reducing a(n) modulo k is purely periodic with period a divisor of k, while for odd k the sequence obtained by reducing a(n) modulo k is purely periodic with period a divisor of 2*k. See A047974. - Peter Bala, Apr 10 2023

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 209.
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).

Seiichi Manyama, Table of n, a(n) for n = 0..732 (terms 0..200 from T. D. Noe)
Koichi, Yamamoto, An asymptotic series for the number of three-line Latin rectangles, J. Math. Soc. Japan 1, (1950). 226-241.
Index entries for sequences related to Hermite polynomials

Cf. A000186, A062267, A144141, A293604.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(-x-x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 09 2018

Table[HermiteH[n, -1/2], {n, 0, 25}] (* Vladimir Joseph Stephan Orlovsky, Jun 15 2009 *)
Table[(-2)^n HypergeometricU[-n/2, 1/2, 1/4], {n, 0, 25}] (* Benedict W. J. Irwin, Oct 17 2017 *)

N=66;  x='x+O('x^N);
egf=exp(-x-x^2);  Vec(serlaplace(egf))
/* Joerg Arndt, Mar 07 2013 */

vector(50, n, n--; sum(k=0, n/2, (-1)^(n-k)*k!*binomial(n, k)*binomial(n-k, k))) \\ Altug Alkan, Oct 22 2015

a(n) = polhermite(n, -1/2); \\ Michel Marcus, Oct 12 2016

from sympy import hermite
def a(n): return hermite(n, -1/2) # Indranil Ghosh, May 26 2017

E.g.f.: exp(-x-x^2).
a(n) = Sum_{k=0..floor(n/2)} (-1)^(n-k)*k!*C(n, k)*C(n-k, k).
a(n) = - a(n-1) - 2*(n-1)*a(n-2), a(0) = 1, a(1) = -1.
A000186(n) ~ n!^2*exp(1)^(-3)*(a(0) + a(1)/n + a(2)/(2*[n]2) + ... + a(k)/(k!*[n]_k) + ...), where [n]_k = n*(n-1)*...*(n-k + 1), [n]_0 = 1. - _Vladeta Jovovic, Apr 30 2001
a(n) = Sum_{k=0..n} (-1)^(2*n-k)*C(k,n-k)*n!/k!. - Paul Barry, Oct 08 2007, corrected by Altug Alkan, Oct 22 2015
E.g.f.: 1 - x*(1 - E(0) )/(1+x) where E(k) = 1 - (1+x)/(k+1)/(1-x/(x+1/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 18 2013
E.g.f.: -x/Q(0) where Q(k) = 1 - (1+x)/(1 - x/(x - (k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 06 2013
G.f.: 1/(x*Q(0)), where Q(k) = 1 + 1/x + 2*(k+1)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 21 2013
a(n) = (-2)^n * U(-n/2, 1/2, 1/4), where U is the confluent hypergeometric function. - Benedict W. J. Irwin, Oct 17 2017
E.g.f.: Product_{k>=1} (1 + (-x)^k)^(mu(k)/k). - Ilya Gutkovskiy, May 26 2019

Formulae and more terms from Vladeta Jovovic, Apr 30 2001

A052859 Expansion of e.g.f.: exp(exp(2x) - 2exp(x) + 1).

Original entry on oeis.org

1, 0, 2, 6, 26, 150, 962, 6846, 54266, 471750, 4439762, 44911086, 485570186, 5581383990, 67890295202, 870493380126, 11726471352986, 165475293394470, 2439632685738482, 37491028556508366, 599285435979866666, 9945441791592272790, 171062503783616702402

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.

a(n) is the number of ways to select a nonempty proper subset from each block of the set partitions of {1,2,...,n}. - Geoffrey Critzer, Jan 20 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 827
Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022 (set m=1, b=2, r=-2, d=1, s=1).
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A060311, A067994.

spec := [S,{B=Prod(C,C),C=Set(Z,1 <= card),S=Set(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
      *2*binomial(n-1, j-1)*Stirling2(j, 2), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 02 2019

nn=20; a=Exp[x]-1; Range[0,nn]! CoefficientList[Series[Exp[a^2], {x,0,nn}], x]  (* Geoffrey Critzer, Jan 20 2012 *)
Table[Sum[BellY[n, k, 2^Range[n] - 2], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)
Table[Sum[(2*k)!*StirlingS2[n, 2*k]/k!, {k, 0, n/2}], {n, 0, 25}] (* Vaclav Kotesovec, Oct 04 2022 *)

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (2*k)!*x^(2*k)/(k!*prod(j=1, 2*k, 1-j*x)))) \\ Seiichi Manyama, May 07 2022

a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 2)/k!); \\ Seiichi Manyama, May 07 2022

E.g.f.: exp(exp(x)^2-2*exp(x)+1).
Stirling transform of unsigned Hermite numbers: Sum_{k=0..n} Stirling2(n, k)*|HermiteH(k, 0)|. - Vladeta Jovovic, Sep 12 2003
From Seiichi Manyama, May 07 2022: (Start)
G.f.: Sum_{k>=0} (2*k)! * x^(2*k)/(k! * Product_{j=1..2*k} (1 - j * x)).
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * Stirling2(n,2*k)/k!. (End)
a(n) ~ 2^n * exp(1/2 - n - 2*sqrt(n/LambertW(n)) + n/LambertW(n)) * n^n / (sqrt(1 + LambertW(n)) * LambertW(n)^n). - Vaclav Kotesovec, Oct 04 2022

New name using e.g.f. from Vaclav Kotesovec, Oct 04 2022

A067147 Triangle of coefficients for expressing x^n in terms of Hermite polynomials.

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

Offset: 0

Christian G. Bower, Jan 03 2002

x^n = (1/2^n) * Sum_{k=0..n} a(n,k)*H_k(x).

These polynomials, H_n(x), are an Appell sequence, whose umbral compositional inverse sequence HI_n(x) consists of the same polynomials signed with the e.g.f. e^{-t^2} e^{xt}. Consequently, under umbral composition H_n(HI.(x)) = x^n = HI_n(H.(x)). Other differently scaled families of Hermite polynomials are A066325, A099174, and A060821. See Griffin et al. for a relation to the Catalan numbers and matrix integration. - Tom Copeland, Dec 27 2020

			Triangle begins with:
    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;

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 801. (Table 22.12)

G. C. Greubel, Rows n = 0..100 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Griffin, K. Ono, L. Rolen, and D. Zagier, Jensen polynomials for the Riemann zeta function and other sequences, arXiv:1902.07321 [math.NT], 2019.
Index entries for sequences related to Hermite polynomials

Row sums give A047974. Columns 0-2: A001813, A000407, A001814. Cf. A048854, A060821.

Cf. A060821, A066325, and A099174.

[[Round(Factorial(n)*(1+(-1)^(n+k))/(2*Factorial(k)*Gamma((n-k+2)/2))): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jun 09 2018

T := proc(n, k) (n - k)/2; `if`(%::integer, (n!/k!)/%!, 0) end:
for n from 0 to 11 do seq(T(n, k), k=0..n) od; # Peter Luschny, Jan 05 2021

Table[n!*(1+(-1)^(n+k))/(2*k!*Gamma[(n-k+2)/2]), {n,0,20}, {k,0,n}]// Flatten (* G. C. Greubel, Jun 09 2018 *)

T(n, k) = round(n!*(1+(-1)^(n+k))/(2*k! *gamma((n-k+2)/2)))
for(n=0,20, for(k=0,n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jun 09 2018

{T(n,k) = if(k<0 || nMichael Somos, Jan 15 2020 */

E.g.f. (rel to x): A(x, y) = exp(x*y + x^2).
Sum_{ k>=0 } 2^k*k!*T(m, k)*T(n, k) = T(m+n, 0) = |A067994(m+n)|. - Philippe Deléham, Jul 02 2005
T(n, k) = 0 if n-k is odd; T(n, k) = n!/(k!*((n-k)/2)!) if n-k is even. - Philippe Deléham, Jul 02 2005
T(n, k) = n!/(k!*2^((n-k)/2)*((n-k)/2)!)*2^((n+k)/2)*(1+(-1)^(n+k))/2^(k+1).
T(n, k) = A001498((n+k)/2, (n-k)/2)2^((n+k)/2)(1+(-1)^(n+k))/2^(k+1). - Paul Barry, Aug 28 2005
Exponential Riordan array (e^(x^2),x). - Paul Barry, Sep 12 2006
G.f.: 1/(1-x*y-2*x^2/(1-x*y-4*x^2/(1-x*y-6*x^2/(1-x*y-8*x^2/(1-... (continued fraction). - Paul Barry, Apr 10 2009
The n-th row entries may be obtained from D^n(exp(x*t)) evaluated at x = 0, where D is the operator sqrt(1+4*x)*d/dx. - Peter Bala, Dec 07 2011
As noted in the comments this is an Appell sequence of polynomials, so the lowering and raising operators defined by L H_n(x) = n H_{n-1}(x) and R H_{n}(x) = H_{n+1}(x) are L = D_x, the derivative, and R = D_t log[e^{t^2} e^{xt}] |{t = D_x} = x + 2 D_x, and the polynomials may also be generated by e^{-D^2} x^n = H_n(x). - _Tom Copeland, Dec 27 2020

A086660 Stirling transform of Hermite numbers: Sum_{k=0..n} Stirling2(n,k) * HermiteH(k,0).

Original entry on oeis.org

1, 0, -2, -6, -2, 90, 598, 1554, -10082, -164310, -1101242, -1496286, 64767118, 965876730, 7104497398, 57428274, -856472198402, -14195316779190, -122409183339482, 25272908324034, 21770258523698158

Offset: 0

Vladeta Jovovic, Sep 12 2003

sign

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A067994, A052859.

m:=50; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(-(Exp(x)-1)^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 12 2018

Table[Sum[StirlingS2[n,k]HermiteH[k,0],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Mar 24 2013 *)
With[{nmax = 50}, CoefficientList[Series[Exp[-(Exp[x] - 1)^2], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Jul 12 2018 *)

x='x+O('x^50); Vec(serlaplace(exp(-(exp(x)-1)^2))) \\ G. C. Greubel, Jul 12 2018

E.g.f.: exp(-(exp(x)-1)^2).

A101109 Number of sets of lists (sequences) of n labeled elements with k=3 elements per list.

Original entry on oeis.org

1, 0, 0, 6, 0, 0, 360, 0, 0, 60480, 0, 0, 19958400, 0, 0, 10897286400, 0, 0, 8892185702400, 0, 0, 10137091700736000, 0, 0, 15388105201717248000, 0, 0, 30006805143348633600000, 0, 0, 73096577329197271449600000

Offset: 0

Thomas Wieder, Dec 01 2004

nonn

The (labeled) case for k=2 is A067994, the Hermite numbers. The (labeled) case for k>=1 is A000262, Number of "sets of lists".

			Let Z[i] denote the i-th labeled element. Then a(3) = 6 with the following six sets:
Set(Sequence(Z[3],Z[1],Z[2])), Set(Sequence(Z[2],Z[1],Z[3])), Set(Sequence(Z[3],Z[2],Z[1])), Set(Sequence(Z[2],Z[3],Z[1])), Set(Sequence(Z[1],Z[3],Z[2])), Set(Sequence(Z[1],Z[2],Z[3])).

Alois P. Heinz, Table of n, a(n) for n = 0..582

Cf. A000262, A067994.

A101109 := n -> n!*PIECEWISE([1/GAMMA(1/3*n+1), irem(n,3) = 0],[0, irem(n-1,3) = 0],[0, irem(n-2,3) = 0]); [ seq(n!*PIECEWISE([1/GAMMA(1/3*n+1), irem(n,3) = 0],[0, irem(n-1,3) = 0],[0, irem(n-2,3) = 0]),n=0..30) ];
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
       j!*binomial(n-1, j-1), j=`if`(n>2, 3, [][])))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 10 2016

With[{nn=30},CoefficientList[Series[Exp[x^3],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 16 2013 *)

def A101109(n) : return factorial(n)/factorial(n/3) if n%3 == 0 else 0
[A101109(n) for n in (0..30)] # Peter Luschny, Jul 12 2012

E.g.f.: exp(z^3).
a(0) = 1, a(1) = 0, a(2) = 0, (-n-3)*a(n+3)+3*a(n).
a(n) = n!/(n/3)!, if 3 divides n, 0 otherwise. - Mitch Harris, Jan 19 2006

A277378 Expansion of e.g.f. exp(2*x/(1-x))/sqrt(1-x^2).

Original entry on oeis.org

1, 2, 9, 50, 361, 3042, 29929, 331298, 4100625, 55777922, 828691369, 13316140818, 230256982201, 4257449540450, 83834039024649, 1750225301567618, 38614608429012001, 897325298084953602, 21904718673762721225, 560258287738117292018, 14981472258320814527241

Offset: 0

Vladimir Reshetnikov, Oct 11 2016

nonn

Eric Weisstein's World of Mathematics, Hermite Polynomial.
Wikipedia, Hermite polynomials.

Cf. A000321, A000898, A059343, A062267, A067994, A277280, A277281.

Table[Abs[HermiteH[n, I]]^2/2^n, {n, 0, 20}]
With[{nn=20},CoefficientList[Series[Exp[2x/(1-x)]/Sqrt[1-x^2],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jan 27 2023 *)

E.g.f.: exp(2*x/(1-x))/sqrt(1-x^2).
a(n) = |H_n(i)|^2 / 2^n = H_n(i) * H_n(-i) / 2^n, where H_n(x) is n-th Hermite polynomial, i = sqrt(-1).
D-finite with recurrence: (n+2)*(a(n) + n*a(n-1)) = a(n+1) + n*(n-1)^2*a(n-2).
a(n) ~ n^n / (2 * exp(1 - 2*sqrt(2*n) + n)) * (1 + 2*sqrt(2)/(3*sqrt(n))). - Vaclav Kotesovec, Oct 27 2021

A308346 Expansion of e.g.f. 1/(1 - x)^log(1 - x).

Original entry on oeis.org

1, 0, -2, -6, -10, 20, 352, 2772, 18132, 104400, 469608, 238920, -35811048, -730972944, -11436661728, -164609993520, -2294024595312, -31488879303552, -426338226719904, -5626751283423072, -70000948158061728, -745703905072996800, -4142683990211677440, 110386551348875714880

Offset: 0

Ilya Gutkovskiy, May 21 2019

Robert Israel, Table of n, a(n) for n = 0..453

Cf. A000254, A008405, A009199, A052517, A067994, A086660, A087761.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (1-x)^Log(1/(1-x)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 21 2019

E:= 1/(1-x)^log(1-x):
S:= series(E,x,31):
seq(coeff(S,x,j)*j!,j=0..30); # Robert Israel, May 22 2019

nmax = 23; CoefficientList[Series[1/(1 - x)^Log[1 - x], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] HermiteH[k, 0], {k, 0, n}], {n, 0, 23}]
a[n_] := a[n] = -2 Sum[(k - 1)! HarmonicNumber[k - 1] Binomial[n - 1, k - 1] a[n - k], {k, 2, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

a(n) = sum(k=0, n, abs(stirling(n, k, 1))*polhermite(k, 0)); \\ Michel Marcus, May 21 2019

m = 30; T = taylor((1-x)^log(1/(1-x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 21 2019

a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A067994(k).

A349280 Irregular triangle read by rows: T(n,k) is the number of arrangements of n labeled children with exactly k rounds; n >= 2, 1 <= k <= floor(n/2).

Original entry on oeis.org

2, 3, 8, 12, 30, 60, 144, 330, 120, 840, 2100, 1260, 5760, 15344, 11760, 1680, 45360, 127008, 113400, 30240, 403200, 1176120, 1169280, 428400, 30240, 3991680, 12054240, 13000680, 5821200, 831600, 43545600, 135508032, 155923680, 80415720, 16632000, 665280

Offset: 2

Steven Finch, Nov 13 2021

A round means the same as a directed ring or circle.

			Triangle starts:
[2]     2;
[3]     3;
[4]     8,     12;
[5]    30,     60;
[6]   144,    330,    120;
[7]   840,   2100,   1260;
[8]  5760,  15344,  11760,  1680;
[9] 45360, 127008, 113400, 30240;
...
For n = 4, there are 8 ways to make one round and 12 ways to make two rounds.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)

Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.

Row sums give A066166 (Stanley's children's game).
Column 1 gives A001048.
Right border element of row n is A001813(n/2) = |A067994(n)| for even n.
Cf. A008279, A265609, A331327.

ser := series((1 - x)^(-x*t), x, 20): xcoeff := n -> coeff(ser, x, n):
T := (n, k) -> n!*coeff(xcoeff(n), t, k):
seq(seq(T(n, k), k = 1..iquo(n,2)), n = 2..12); # Peter Luschny, Nov 13 2021
# second Maple program:
A349280 := (n,k) -> binomial(n,k)*k!*abs(Stirling1(n-k,k)):
seq(print(seq(A349280(n,k), k=1..iquo(n,2))), n=2..12); # Mélika Tebni, May 03 2023

f[k_, n_] := n! SeriesCoefficient[(1 - x)^(-x t), {x, 0, n}, {t, 0, k}]
Table[f[k, n], {n, 2, 12}, {k, 1, Floor[n/2]}]

G.f.: (1 - x)^(-x*t).
T(n, k) = binomial(n, k)*k!*|Stirling1(n-k, k)|. - Mélika Tebni, May 03 2023
The above formula can also be written as T(n, k) = A008279(n, k)*A331327(n, k) or as T(n, k) = A265609(n + 1, k)*A331327(n, k). - Peter Luschny, May 03 2023

A277379 E.g.f.: exp(x/(1-x^2))/sqrt(1-x^2).

Original entry on oeis.org

1, 1, 2, 10, 40, 296, 1936, 17872, 164480, 1820800, 21442816, 279255296, 3967316992, 59837670400, 988024924160, 17009993230336, 318566665977856, 6177885274406912, 129053377688043520, 2786107670662021120, 64136976817284448256, 1525720008470138454016

Offset: 0

Vladimir Reshetnikov, Oct 11 2016

nonn

Is this the same as A227545 (at least for n>=1)?

Eric Weisstein's World of Mathematics, Hermite Polynomial.
Wikipedia, Hermite polynomials.

Cf. A000321, A000898, A059343, A062267, A067994, A227545, A277280, A277281, A277378.

Table[Abs[HermiteH[n, (1 + I)/2]]^2/2^n, {n, 0, 20}]

a(n) = |H_n((1+i)/2)|^2 / 2^n = H_n((1+i)/2) * H_n((1-i)/2) / 2^n, where H_n(x) is n-th Hermite polynomial, i = sqrt(-1).
D-finite with recurrence: (n+1)*(n+2)*(a(n) - n^2*a(n-1)) + (2*n^2+7*n+6)*a(n+1) + a(n+2) = a(n+3).
a(n) ~ n^n * exp(sqrt(2*n)-n) / 2. - Vaclav Kotesovec, Oct 14 2016

A382139 Number of matchings of [2n] with no coupled arcs.

Original entry on oeis.org

1, 1, 1, 9, 81, 705, 7665, 100905, 1524705, 26022465, 496042785, 10445342985, 240779831985, 6030718158465, 163087008669585, 4735950860666025, 146987669673669825, 4855606200012593025, 170101350767940617025, 6298861062893921346825, 245834199405298416337425

Offset: 0

R. J. Mathar, Mar 17 2025

nonn

Sergi Elizalde and Emeric Deutsch, The degree of asymmetry of a sequence, Enum. Combinat. Applic. 2 (2022) no 1 #S2R7, eq (17) at r=1, s=2.

Cf. A047974, A053871, A382134, A067994, A001147.

g:= exp(-x^2)/sqrt(1-2*x) ;
seq( coeftayl(g,x=0,n)*n!,n=0..10) ;

E.g.f: exp(-x^2)/sqrt(1-2*x).
Exponential convolution of A067994 and A001147.
D-finite with recurrence a(n) +(-2*n+1)*a(n-1) +2*(n-1)*a(n-2) -4*(n-1)*(n-2)*a(n-3)=0.

cp's OEIS Frontend

A000321 H_n(-1/2), where H_n(x) is Hermite polynomial of degree n.

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A052859 Expansion of e.g.f.: exp(exp(2*x) - 2*exp(x) + 1).

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A067147 Triangle of coefficients for expressing x^n in terms of Hermite polynomials.

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A086660 Stirling transform of Hermite numbers: Sum_{k=0..n} Stirling2(n,k) * HermiteH(k,0).

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A101109 Number of sets of lists (sequences) of n labeled elements with k=3 elements per list.

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A277378 Expansion of e.g.f. exp(2*x/(1-x))/sqrt(1-x^2).

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A052859 Expansion of e.g.f.: exp(exp(2x) - 2exp(x) + 1).