A000321 -id:A000321 - cp's OEIS frontend

A060821 Triangle read by rows. T(n, k) are the coefficients of the Hermite polynomial of order n, for 0 <= k <= n.

1, 0, 2, -2, 0, 4, 0, -12, 0, 8, 12, 0, -48, 0, 16, 0, 120, 0, -160, 0, 32, -120, 0, 720, 0, -480, 0, 64, 0, -1680, 0, 3360, 0, -1344, 0, 128, 1680, 0, -13440, 0, 13440, 0, -3584, 0, 256, 0, 30240, 0, -80640, 0, 48384, 0, -9216, 0, 512, -30240, 0, 302400, 0, -403200, 0, 161280, 0, -23040, 0, 1024

Offset: 0

Vladeta Jovovic, Apr 30 2001

Exponential Riordan array [exp(-x^2), 2x]. - Paul Barry, Jan 22 2009

			[1], [0, 2], [ -2, 0, 4], [0, -12, 0, 8], [12, 0, -48, 0, 16], [0, 120, 0, -160, 0, 32], ... .
Thus H_0(x) = 1, H_1(x) = 2*x, H_2(x) = -2 + 4*x^2, H_3(x) = -12*x + 8*x^3, H_4(x) = 12 - 48*x^2 + 16*x^4, ...
Triangle starts:
     1;
     0,     2;
    -2,     0,      4;
     0,   -12,      0,      8;
    12,     0,    -48,      0,      16;
     0,   120,      0,   -160,       0,    32;
  -120,     0,    720,      0,    -480,     0,     64;
     0, -1680,      0,   3360,       0, -1344,      0,   128;
  1680,     0, -13440,      0,   13440,     0,  -3584,     0,    256;
     0, 30240,      0, -80640,       0, 48384,      0, -9216,      0, 512;
-30240,     0, 302400,      0, -403200,     0, 161280,     0, -23040,   0, 1024;

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 24, equations 24:4:1 - 24:4:8 at page 219.

T. D. Noe, 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, p. 801.
Taekyun Kim and Dae San Kim, A note on Hermite polynomials, arXiv:1602.04096 [math.NT], 2016.
Alexander Minakov, Question about integral of product of four Hermite polynomials integrated with squared weight, arXiv:1911.03942 [math.CO], 2019.
Wikipedia, Hermite polynomials.
Index entries for sequences related to Hermite polynomials.

Cf. A001814, A001816, A000321, A062267 (row sums).

Without initial zeros, same as A059343.

with(orthopoly):for n from 0 to 10 do H(n,x):od;
T := proc(n,m) if n-m >= 0 and n-m mod 2 = 0 then ((-1)^((n-m)/2))*(2^m)*n!/(m!*((n-m)/2)!) else 0 fi; end;
# Alternative:
T := proc(n,k) option remember; if k > n then 0 elif n = k then 2^n else
(T(n, k+2)*(k+2)*(k+1))/(2*(k-n)) fi end:
seq(print(seq(T(n, k), k = 0..n)), n = 0..10); # Peter Luschny, Jan 08 2023

Flatten[ Table[ CoefficientList[ HermiteH[n, x], x], {n, 0, 10}]] (* Jean-François Alcover, Jan 18 2012 *)

for(n=0,9,v=Vec(polhermite(n));forstep(i=n+1,1,-1,print1(v[i]", "))) \\ Charles R Greathouse IV, Jun 20 2012

from sympy import hermite, Poly, symbols
x = symbols('x')
def a(n): return Poly(hermite(n, x), x).all_coeffs()[::-1]
for n in range(21): print(a(n)) # Indranil Ghosh, May 26 2017

def Trow(n: int) -> list[int]:
    row: list[int] = [0] * (n + 1); row[n] = 2**n
    for k in range(n - 2, -1, -2):
        row[k] = -(row[k + 2] * (k + 2) * (k + 1)) // (2 * (n - k))
    return row  # Peter Luschny, Jan 08 2023

T(n, k) = ((-1)^((n-k)/2))*(2^k)*n!/(k!*((n-k)/2)!) if n-k is even and >= 0, else 0.
E.g.f.: exp(-y^2 + 2*y*x).
From Paul Barry, Aug 28 2005: (Start)
T(n, k) = n!/(k!*2^((n-k)/2)((n-k)/2)!)2^((n+k)/2)cos(Pi*(n-k)/2)(1 + (-1)^(n+k))/2;
T(n, k) = A001498((n+k)/2, (n-k)/2)*cos(Pi*(n-k)/2)2^((n+k)/2)(1 + (-1)^(n+k))/2.
(End)
Row sums: A062267. - Derek Orr, Mar 12 2015
a(n*(n+3)/2) = a(A000096(n)) = 2^n. - Derek Orr, Mar 12 2015
Recurrence for fixed n: T(n, k) = -(k+2)*(k+1)/(2*(n-k)) * T(n, k+2), starting with T(n, n) = 2^n. - Ralf Stephan, Mar 26 2016
The m-th row consecutive nonzero entries in increasing order are (-1)^(c/2)*(c+b)!/(c/2)!b!*2^b with c = m, m-2, ..., 0 and b = m-c if m is even and with c = m-1, m-3, ..., 0 with b = m-c if m is odd. For the 10th row starting at a(55) the 6 consecutive nonzero entries in order are -30240,302400,-403200,161280,-23040,1024 given by c = 10,8,6,4,2,0 and b = 0,2,4,6,8,10. - Richard Turk, Aug 20 2017

A293604 Expansion of e.g.f.: exp(x * (1 - x)).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Oct 12 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..732

Cf. A000321, A111884, A293571, A293572, A293573.

Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), A362177 (q=-3), A362176 (q=-2), this sequence (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), A293720 (q=4).

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(x-x^2) ))); // G. C. Greubel, Jul 12 2024

CoefficientList[Series[E^(x*(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 13 2017 *)

my(N=66, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-x))))

a(n) = polhermite(n, 1/2); \\ Michel Marcus, Oct 13 2017

[hermite(n, 1/2) for n in range(31)] # G. C. Greubel, Jul 12 2024

a(n) = (-1)^n * A000321(n).
a(n) = a(n-1) - 2 * (n-1) * a(n-2) for n > 1.
E.g.f.: Product_{k>=1} (1 + x^k)^(mu(k)/k). - Ilya Gutkovskiy, May 23 2019
a(n) = Hermite(n, 1/2). - G. C. Greubel, Jul 12 2024

A119275 Inverse of triangle related to Padé approximation of exp(x).

Original entry on oeis.org

1, -2, 1, 0, -6, 1, 0, 12, -12, 1, 0, 0, 60, -20, 1, 0, 0, -120, 180, -30, 1, 0, 0, 0, -840, 420, -42, 1, 0, 0, 0, 1680, -3360, 840, -56, 1, 0, 0, 0, 0, 15120, -10080, 1512, -72, 1, 0, 0, 0, 0, -30240, 75600, -25200, 2520, -90, 1, 0, 0, 0, 0, 0, -332640, 277200, -55440, 3960, -110, 1

Offset: 0

Paul Barry, May 12 2006

Inverse of A119274.
Row sums are (-1)^(n+1)*A000321(n+1).
Bell polynomials of the second kind B(n,k)(1,-2). - Vladimir Kruchinin, Mar 25 2011
Also the inverse Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+2) (A001813) giving unsigned values and adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015

			Triangle begins
1,
-2, 1,
0, -6, 1,
0, 12, -12, 1,
0, 0, 60, -20, 1,
0, 0, -120, 180, -30, 1,
0, 0, 0, -840, 420, -42, 1,
0, 0, 0, 1680, -3360, 840, -56, 1,
0, 0, 0, 0, 15120, -10080, 1512, -72, 1
Row 4: D(x^4) = (1 - x*(d/dx)^2 + x^2/2!*(d/dx)^4 - ...)(x^4) = x^4 - 12*x^3 + 12*x^2.

Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A059344 (unsigned row reverse).

Cf. A034839, A001813, A001815, A086645, A098158, A130561, A231846.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n<2,(n+1)*(-1)^n,0), 9); # Peter Luschny, Jan 27 2016

Table[(-1)^(n - k) (n - k)!*Binomial[n + 1, k + 1] Binomial[k + 1, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Oct 12 2016 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[If[#<2, (#+1) (-1)^#, 0]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

# uses[inverse_bell_matrix from A265605]
# Unsigned values and an additional first column (1,0,0, ...).
multifact_4_2 = lambda n: prod(4*k + 2 for k in (0..n-1))
inverse_bell_matrix(multifact_4_2, 9) # Peter Luschny, Dec 31 2015

T(n,k) = [k<=n]*(-1)^(n-k)*(n-k)!*C(n+1,k+1)*C(k+1,n-k).
From Peter Bala, May 07 2012: (Start)
E.g.f.: exp(x*(t-t^2)) - 1 = x*t + (-2*x+x^2)*t^2/2! + (-6*x^2+x^3)*t^3/3! + (12*x^2-12*x^3+x^4)*t^4/4! + .... Cf. A059344. Let D denote the operator sum {k >= 0} (-1)^k/k!*x^k*(d/dx)^(2*k). The n-th row polynomial R(n,x) = D(x^n) and satisfies the recurrence equation R(n+1,x) = x*R(n,x)-2*n*x*R(n-1,x). The e.g.f. equals D(exp(x*t)).
(End)
From Tom Copeland, Oct 11 2016: (Start)
With initial index n = 1 and unsigned, these are the partition row polynomials of A130561 and A231846 with c_1 = c_2 = x and c_n = 0 otherwise. The first nonzero, unsigned element of each diagonal is given by A001813 (for each row, A001815) and dividing along the corresponding diagonal by this element generates A098158 with its first column removed (cf. A034839 and A086645).
The n-th polynomial is generated by (x - 2y d/dx)^n acting on 1 and then evaluated at y = x, e.g., (x - 2y d/dx)^2 1 = (x - 2y d/dx) x = x^2 - 2y evaluated at y = x gives p_2(x) = -2x + x^2.
(End)

A334561 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(-Sum_{j=1..k} x^j).

Original entry on oeis.org

1, 1, -1, 1, -1, 1, 1, -1, -1, -1, 1, -1, -1, 5, 1, 1, -1, -1, -1, 1, -1, 1, -1, -1, -1, 25, -41, 1, 1, -1, -1, -1, 1, 19, 31, -1, 1, -1, -1, -1, 1, 139, -209, 461, 1, 1, -1, -1, -1, 1, 19, 151, -2269, -895, -1, 1, -1, -1, -1, 1, 19, 871, -1429, 2801, -6481, 1, 1, -1, -1, -1, 1, 19, 151, 1091, -19039, 68615, 22591, -1

Offset: 0

Seiichi Manyama, May 06 2020

			Square array begins:
   1,   1,    1,   1,   1,   1,   1, ...
  -1,  -1,   -1,  -1,  -1,  -1,  -1, ...
   1,  -1,   -1,  -1,  -1,  -1,  -1, ...
  -1,   5,   -1,  -1,  -1,  -1,  -1, ...
   1,   1,   25,   1,   1,   1,   1, ...
  -1, -41,   19, 139,  19,  19,  19, ...
   1,  31, -209, 151, 871, 151, 151, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=1..5 give A033999, A000321, A334562, A334564, A334565.
Main diagonal gives A293116.
Cf. A293669, A334568.

A(0,k) = 1 and A(n,k) = - (n-1)! * Sum_{j=1..min(k,n)} j*A(n-j,k)/(n-j)!.

A144141 a(n) = Hermite(n,2).

Original entry on oeis.org

1, 4, 14, 40, 76, -16, -824, -3104, -880, 46144, 200416, -121216, -4894016, -16666880, 60576896, 708980224, 1018614016, -18612911104, -109084520960, 233726715904, 5080118660096, 10971406004224, -169479359707136, -1160659303014400, 3153413334470656

Offset: 0

Vladimir Joseph Stephan Orlovsky, Sep 11 2008

sign

G. C. Greubel, Table of n, a(n) for n = 0..729

Cf. A000898, A000321, A062267.

[(&+[(-1)^k*Factorial(n)*(4)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Jul 10 2018

lst={};Do[AppendTo[lst,HermiteH[n,2]],{n,0,7^2}];lst
HermiteH[Range[0,30],2]  (* Harvey P. Dale, May 20 2012 *)

for(n=0, 50, print1(polhermite(n, 2), ", " )) \\ G. C. Greubel, Jul 10 2018

From G. C. Greubel, Jul 10 2018: (Start)
E.g.f.: exp(4*x - x^2).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*n!*4^(n-2*k)/(k!*(n-2*k)!). (End)

A145881 Triangle read by rows: T(n,k) is the number of even permutations of {1,2,...,n} with no fixed points and having k excedances (n>=1; k>=1).

Original entry on oeis.org

0, 0, 1, 1, 0, 3, 0, 1, 11, 11, 1, 0, 25, 80, 25, 0, 1, 57, 407, 407, 57, 1, 0, 119, 1680, 3815, 1680, 119, 0, 1, 247, 6211, 26917, 26917, 6211, 247, 1, 0, 501, 21432, 160053, 303504, 160053, 21432, 501, 0, 1, 1013, 70775, 852347, 2747009, 2747009, 852347, 70775

Offset: 1

Emeric Deutsch, Nov 06 2008

Row n has n-1 entries (n>=2).
Sum of entries in row n = A000321(n).
Sum_{k=1..n-1} k*T(n,k) = A145887(n) (n>=2).

			T(4,2)=3 because the even derangements of {1,2,3,4} are 3412, 2143 and 4321.
Triangle starts:
  0;
  0;
  1,  1;
  0,  3,  0;
  1, 11, 11,  1;
  0, 25, 80, 25,  0;

R. Mantaci and F. Rakotondrajao, Exceedingly deranging!, Advances in Appl. Math., 30 (2003), 177-188.

Cf. A000321, A145880, A145886, A145887.

G:=((1-t)*exp(-t*z)/(1-t*exp((1-t)*z))-(t*exp(-z)-exp(-t*z))/(1-t))*1/2: Gser:=simplify(series(G,z=0,15)): for n to 11 do P[n]:=sort(expand(factorial(n)*coeff(Gser,z,n))) end do: 0; for n to 11 do seq(coeff(P[n],t,j),j=1..n-1) end do; # yields sequence in triangular form

E.g.f.: ((1-t)*exp(-tz)/(1-t*exp((1-t)z)) - (t*exp(-z)-exp(-tz))/(1-t))/2.

Formula corrected by Jon E. Schoenfield, Jul 21 2017 at the request of the author

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

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

A308461 Expansion of e.g.f. exp(x + 2 * Sum_{k>=2} x^(2^k)/2^k).

Original entry on oeis.org

1, 1, 1, 1, 13, 61, 181, 421, 15961, 137593, 682921, 2498761, 77344741, 927575221, 6402167773, 31881065581, 4104839160241, 68050288734961, 609856397747281, 3857727706737553, 222655237411428541, 4351842324095032621, 47276537013742616581, 361153046139022585141

Offset: 0

Ilya Gutkovskiy, May 28 2019

nonn

Cf. A000321, A005388, A008683, A308392.

nmax = 23; CoefficientList[Series[Exp[x + 2 Sum[x^(2^k)/2^k, {k, 2, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Product[(1 + (-x)^k)^((-1)^k MoebiusMu[k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: Product_{k>=1} (1 + (-x)^k)^((-1)^k*mu(k)/k).

E.g.f.: exp(-x*(1 + x))*g(x)^2, where g(x) = e.g.f. of A005388.

cp's OEIS Frontend

A060821 Triangle read by rows. T(n, k) are the coefficients of the Hermite polynomial of order n, for 0 <= k <= n.

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A293604 Expansion of e.g.f.: exp(x * (1 - x)).

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A119275 Inverse of triangle related to Padé approximation of exp(x).

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A334561 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(-Sum_{j=1..k} x^j).

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A144141 a(n) = Hermite(n,2).

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A145881 Triangle read by rows: T(n,k) is the number of even permutations of {1,2,...,n} with no fixed points and having k excedances (n>=1; k>=1).

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

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