Displaying 1-10 of 11 results found.
Triangle read by rows. T(n, k) are the coefficients of the Hermite polynomial of order n, for 0 <= k <= n.
+10
30
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
COMMENTS
Exponential Riordan array [exp(-x^2), 2x]. - Paul Barry, Jan 22 2009
FORMULA
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).
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)
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
EXAMPLE
[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;
MAPLE
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
MATHEMATICA
Flatten[ Table[ CoefficientList[ HermiteH[n, x], x], {n, 0, 10}]] (* Jean-François Alcover, Jan 18 2012 *)
PROG
(PARI) for(n=0, 9, v=Vec(polhermite(n)); forstep(i=n+1, 1, -1, print1(v[i]", "))) \\ Charles R Greathouse IV, Jun 20 2012 (Python)
from sympy import hermite, Poly, symbols
x = symbols('x')
def a(n): return Poly(hermite(n, x), x).all_coeffs()[::-1]
(Python)
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))
CROSSREFS
Without initial zeros, same as A059343.
Triangle read by rows: coefficients of modified Hermite polynomials.
+10
24
1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 3, 0, 6, 0, 1, 0, 15, 0, 10, 0, 1, 15, 0, 45, 0, 15, 0, 1, 0, 105, 0, 105, 0, 21, 0, 1, 105, 0, 420, 0, 210, 0, 28, 0, 1, 0, 945, 0, 1260, 0, 378, 0, 36, 0, 1, 945, 0, 4725, 0, 3150, 0, 630, 0, 45, 0, 1, 0, 10395, 0, 17325, 0, 6930, 0, 990, 0, 55, 0, 1
COMMENTS
T(n,k) is the number of involutions of {1,2,...,n}, having k fixed points (0 <= k <= n). Example: T(4,2)=6 because we have 1243,1432,1324,4231,3214 and 2134. - Emeric Deutsch, Oct 14 2006 Riordan array [exp(x^2/2),x]. - Paul Barry, Nov 06 2008 Same as triangle of Bessel numbers of second kind, B(n,k) (see Cheon et al., 2013). - N. J. A. Sloane, Sep 03 2013 The modified Hermite polynomial h(n,x) (as in the Formula section) is the numerator of the rational function given by f(n,x) = x + (n-2)/f(n-1,x), where f(x,0) = 1. - Clark Kimberling, Oct 20 2014 Second lower diagonal T(n,n-2) equals positive triangular numbers A000217 \ {0}. - M. F. Hasler, Oct 23 2014 T(n,k) is the number of R-classes (equivalently, L-classes) in the D-class consisting of all rank k elements of the Brauer monoid of degree n.
For n < k with n == k (mod 2), T(n,k) is the rank (minimal size of a generating set) and idempotent rank (minimal size of an idempotent generating set) of the ideal consisting of all rank <= k elements of the Brauer monoid. (End)
This array provides the coefficients of a Laplace-dual sequence H(n,x) of the Dirac delta function, delta(x), and its derivatives, formed by taking the inverse Laplace transform of these modified Hermite polynomials. H(n,x) = h(n,D) delta(x) with h(n,x) as in the examples and the lowering and raising operators L = -x and R = -x + D = -x + d/dx such that L H(n,x) = n * H(n-1,x) and R H(n,x) = H(n+1,x). The e.g.f. is exp[t H(.,x)] = e^(t^2/2) e^(t D) delta(x) = e^(t^2/2) delta(x+t). - Tom Copeland, Oct 02 2016 This triangle is the reverse of that in Table 2 on p. 7 of the Artioli et al. paper and Table 6.2 on p. 234 of Licciardi's thesis, with associations to the telephone numbers. - Tom Copeland, Jun 18 2018 and Jul 08 2018 See A344678 for connections to a Heisenberg-Weyl algebra of differential operators, matching and independent edge sets of the regular n-simplices with partially labeled vertices, and telephone switchboard scenarios. - Tom Copeland, Jun 02 2021
FORMULA
h(k, x) = (-I/sqrt(2))^k * H(k, I*x/sqrt(2)), H(n, x) the Hermite polynomials ( A060821, A059343). T(n,k) = n!/(2^((n-k)/2)*((n-k)/2)!k!) if n-k >= 0 is even; 0 otherwise. - Emeric Deutsch, Oct 14 2006 G.f.: 1/(1-x*y-x^2/(1-x*y-2*x^2/(1-x*y-3*x^2/(1-x*y-4*x^2/(1-... (continued fraction). - Paul Barry, Apr 10 2009 Recurrence: T(0,0)=1, T(0,k)=0 for k>0 and for n >= 1 T(n,k) = T(n-1,k-1) + (k+1)*T(n-1,k+1). - Peter Luschny, Oct 06 2012 The row polynomials P(n,x) = (a. + x)^n, umbrally evaluated with (a.)^n = a_n = aerated A001147, are an Appell sequence with dP(n,x)/dx = n * P(n-1,x). The umbral compositional inverses (cf. A001147) of these polynomials are given by the same polynomials signed, A066325. - Tom Copeland, Nov 15 2014 The odd rows are (2x^2)^n x n! L(n,-1/(2x^2),1/2), and the even, (2x^2)^n n! L(n,-1/(2x^2),-1/2) in sequence with n= 0,1,2,... and L(n,x,a) = Sum_{k=0..n} binomial(n+a,k+a) (-x)^k/k!, the associated Laguerre polynomial of order a. The odd rows are related to A130757, and the even to A176230 and A176231. Other versions of this entry are A122848, A049403, A096713 and A104556, and reversed A100861, A144299, A111924. With each non-vanishing diagonal divided by its initial element A001147(n), this array becomes reversed, aerated A034839. Create four shift and stretch matrices S1,S2,S3, and S4 with all elements zero except S1(2n,n) = 1 for n >= 1, S2(n,2n) = 1 for n >= 0, S3(2n+1,n) = 1 for n >= 1, and S4(n,2n+1) = 1 for n >= 0. Then this entry's lower triangular matrix is T = Id + S1 * ( A176230-Id) * S2 + S3 * (unsigned A130757-Id) * S4 with Id the identity matrix. The sandwiched matrices have infinitesimal generators with the nonvanishing subdiagonals A000384(n>0) and A014105(n>0). As an Appell sequence, the lowering and raising operators are L = D and R = x + dlog(exp(D^2/2))/dD = x + D, where D = d/dx, L h(n,x) = n h(n-1,x), and R h(n,x) = h(n+1,x), so R^n 1 = h(n,x). The fundamental moment sequence has the e.g.f. e^(t^2/2) with coefficients a(n) = aerated A001147, i.e., h(n,x) = (a. + x)^n, as noted above. The raising operator R as a matrix acting on o.g.f.s (formal power series) is the transpose of the production matrix P below, i.e., (1,x,x^2,...)(P^T)^n (1,0,0,...)^T = h(n,x). For characterization as a Riordan array and associations to combinatorial structures, see the Barry link and the Yang and Qiao reference. For relations to projective modules, see the Sazdanovic link.
(End)
From the Appell formalism, e^(D^2/2) x^n = h_n(x), the n-th row polynomial listed below, and e^(-D^2/2) x^n = u_n(x), the n-th row polynomial of A066325. Then R = e^(D^2/2) * x * e^(-D^2/2) is another representation of the raising operator, implied by the umbral compositional inverse relation h_n(u.(x)) = x^n. - Tom Copeland, Oct 02 2016 h_n(x) = p_n(x-1), where p_n(x) are the polynomials of A111062, related to the telephone numbers A000085. - Tom Copeland, Jun 26 2018 In the power basis x^n, the matrix infinitesimal generator M = A132440^2/2, when acting on a row vector for an o.g.f., is the matrix representation for the differential operator D^2/2. e^{M} gives the coefficients of the Hermite polynomials of this entry.
The only nonvanishing subdiagonal of M, the second subdiagonal (1,3,6,10,...), gives, aside from the initial 0, the triangular numbers A000217, the number of edges of the n-dimensional simplices with (n+1) vertices. The perfect matchings of these simplices are the aerated odd double factorials A001147 noted above, the moments for the Hermite polynomials. The polynomials are also generated from A036040 with x[1] = x, x[2] = 1, and the other indeterminates equal to zero. (End)
EXAMPLE
h(0,x) = 1
h(1,x) = x
h(2,x) = x^2 + 1
h(3,x) = x^3 + 3*x
h(4,x) = x^4 + 6*x^2 + 3
h(5,x) = x^5 + 10*x^3 + 15*x
h(6,x) = x^6 + 15*x^4 + 45*x^2 + 15
Triangle begins
1,
0, 1,
1, 0, 1,
0, 3, 0, 1,
3, 0, 6, 0, 1,
0, 15, 0, 10, 0, 1,
15, 0, 45, 0, 15, 0, 1
Production array starts
0, 1,
1, 0, 1,
0, 2, 0, 1,
0, 0, 3, 0, 1,
0, 0, 0, 4, 0, 1,
0, 0, 0, 0, 5, 0, 1 (End)
MAPLE
T:=proc(n, k) if n-k mod 2 = 0 then n!/2^((n-k)/2)/((n-k)/2)!/k! else 0 fi end: for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form; Emeric Deutsch, Oct 14 2006
MATHEMATICA
nn=10; a=y x+x^2/2!; Range[0, nn]!CoefficientList[Series[Exp[a], {x, 0, nn}], {x, y}]//Grid (* Geoffrey Critzer, May 08 2012 *) H[0, x_] = 1; H[1, x_] := x; H[n_, x_] := H[n, x] = x*H[n-1, x]-(n-1)* H[n-2, x]; Table[CoefficientList[H[n, x], x], {n, 0, 11}] // Flatten // Abs (* Jean-François Alcover, May 23 2016 *) T[ n_, k_] := If[ n < 0, 0, Coefficient[HermiteH[n, x I/Sqrt[2]] (Sqrt[1/2]/I)^n, x, k]]; (* Michael Somos, May 10 2019 *)
PROG
(Sage)
M = matrix(ZZ, dim, dim)
for n in (0..dim-1): M[n, n] = 1
for n in (1..dim-1):
for k in (0..n-1):
M[n, k] = M[n-1, k-1]+(k+1)*M[n-1, k+1]
return M
(PARI) T(n, k)=if(k<=n && k==Mod(n, 2), n!/k!/(k=(n-k)/2)!>>k) \\ M. F. Hasler, Oct 23 2014 (Python)
import sympy
from sympy import Poly
from sympy.abc import x, y
def H(n, x): return 1 if n==0 else x if n==1 else x*H(n - 1, x) - (n - 1)*H(n - 2, x)
def a(n): return [abs(cf) for cf in Poly(H(n, x), x).all_coeffs()[::-1]]
(Python)
def Trow(n: int) -> list[int]:
row: list[int] = [0] * (n + 1); row[n] = 1
for k in range(n - 2, -1, -2):
row[k] = (row[k + 2] * (k + 2) * (k + 1)) // (n - k)
CROSSREFS
Row sums (polynomial values at x=1) are A000085. Cf. A000384, A014105, A034839, A049403, A096713, A100861, A104556, A122848, A130757, A176230, A176231.
Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.
+10
9
1, 1, 1, 2, 1, 6, 1, 12, 12, 1, 20, 60, 1, 30, 180, 120, 1, 42, 420, 840, 1, 56, 840, 3360, 1680, 1, 72, 1512, 10080, 15120, 1, 90, 2520, 25200, 75600, 30240, 1, 110, 3960, 55440, 277200, 332640, 1, 132, 5940, 110880, 831600, 1995840, 665280, 1, 156
REFERENCES
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.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 50.
LINKS
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].
EXAMPLE
Triangle begins
1;
1;
1, 2;
1, 6;
1, 12, 12;
1, 20, 60;
1, 30, 180, 120;
1, 42, 420, 840;
1, 56, 840, 3360, 1680;
1, 72, 1512, 10080, 15120;
x^2 = 1/2^2*(Hermite(2,x)+2*Hermite(0,x)); x^3 = 1/2^3*(Hermite(3,x)+6*Hermite(1,x)); x^4 = 1/2^4*(Hermite(4,x)+12*Hermite(2,x)+12*Hermite(0,x)); x^5 = 1/2^5*(Hermite(5,x)+20*Hermite(3,x)+60*Hermite(1,x)); x^6 = 1/2^6*(Hermite(6,x)+30*Hermite(4,x)+180*Hermite(2,x)+120*Hermite(0,x)). - Vladeta Jovovic, Feb 21 2003 1 = H(0); 2x = H(1); 4x^2 = H(2)+2H(0); 8x^3 = H(3)+6H(1); etc. where H(k)=Hermite(k,x).
MATHEMATICA
Flatten[Table[n!/(k! * (n-2k)!), {n, 0, 13}, {k, 0, Floor[n/2]}]]
(* Second program: *)
row[n_] := Table[h[k], {k, n, Mod[n, 2], -2}] /. SolveAlways[2^n*x^n == Sum[h[k]*HermiteH[k, x], {k, Mod[n, 2], n, 2}], x] // First; Table[ row[n], {n, 0, 13}] // Flatten (* Jean-François Alcover, Jan 05 2016 *)
PROG
(PARI) for(n=0, 25, for(k=0, floor(n/2), print1(n!/(k!*(n-2*k)!), ", "))) \\ G. C. Greubel, Jan 07 2017
Maximal coefficient in Hermite polynomial of order n.
+10
4
1, 2, 4, 8, 16, 120, 720, 3360, 13440, 48384, 302400, 2217600, 13305600, 69189120, 322882560, 2421619200, 19372953600, 131736084480, 790416506880, 4290832465920, 40226554368000, 337903056691200, 2477955749068800, 16283709208166400, 113985964457164800
EXAMPLE
For n = 5, H_5(x) = 32*x^5 - 160*x^3 + 120*x. The maximal coefficient is 120 (we take signs into account, so -160 < 120), hence a(5) = 120.
MATHEMATICA
Table[Max@CoefficientList[HermiteH[n, x], x], {n, 0, 25}]
PROG
(PARI) a(n) = vecmax(Vec(polhermite(n))); \\ Michel Marcus, Oct 09 2016 (Python)
from sympy import hermite, Poly
def a(n): return max(Poly(hermite(n, x), x).coeffs()) # Indranil Ghosh, May 26 2017
Maximal coefficient (ignoring signs) in Hermite polynomial of order n.
+10
4
1, 2, 4, 12, 48, 160, 720, 3360, 13440, 80640, 403200, 2217600, 13305600, 69189120, 484323840, 2905943040, 19372953600, 131736084480, 846874828800, 6436248698880, 42908324659200, 337903056691200, 2477955749068800, 18997660742860800, 151981285942886400
EXAMPLE
For n = 5, H_5(x) = 32*x^5 - 160*x^3 + 120*x. The maximal coefficient (ignoring signs) is 160, so a(5) = 160.
MATHEMATICA
Table[Max@Abs@CoefficientList[HermiteH[n, x], x], {n, 0, 25}]
PROG
(PARI) a(n) = vecmax(apply(x->abs(x), Vec(polhermite(n)))); \\ Michel Marcus, Oct 09 2016 (Python)
from sympy import hermite, Poly
def a(n): return max(map(abs, Poly(hermite(n, x), x).coeffs())) # Indranil Ghosh, May 26 2017
The 3rd Hermite Polynomial evaluated at n: H_3(n) = 8*n^3 - 12*n.
+10
3
0, -4, 40, 180, 464, 940, 1656, 2660, 4000, 5724, 7880, 10516, 13680, 17420, 21784, 26820, 32576, 39100, 46440, 54644, 63760, 73836, 84920, 97060, 110304, 124700, 140296, 157140, 175280, 194764, 215640, 237956, 261760, 287100, 314024, 342580
FORMULA
a(n) = 8*n^3 - 12*n.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: -4*x*(1-14*x+x^2)/(x-1)^4.
MATHEMATICA
CoefficientList[Series[-4*x*(1-14*x+x^2)/(x-1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 05 2012 *) LinearRecurrence[{4, -6, 4, -1}, {0, -4, 40, 180}, 40] (* Harvey P. Dale, Aug 14 2014 *)
PROG
(Python)
from sympy import hermite
The 4th Hermite Polynomial evaluated at n: H_4(n) = 16n^4 - 48n^2 + 12.
+10
2
12, -20, 76, 876, 3340, 8812, 19020, 36076, 62476, 101100, 155212, 228460, 324876, 448876, 605260, 799212, 1036300, 1322476, 1664076, 2067820, 2540812, 3090540, 3724876, 4452076, 5280780, 6220012, 7279180, 8468076, 9796876, 11276140
FORMULA
a(n) = 16*n^4 - 48*n^2 + 12.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: 4*(-3 +20*x -74*x^2 -44*x^3 +5*x^4)/(x-1)^5.
H_(m+1)(x) = 2*x*H_m(x) - 2*m*H_(m-1)(x), with H_0(x)=1, H_1(x)=2x.
MATHEMATICA
LinearRecurrence[{5, -10, 10, -5, 1}, {12, -20, 76, 876, 3340}, 40] (* Harvey P. Dale, Jul 03 2019 *)
PROG
(Python)
from sympy import hermite
Expansion of e.g.f. exp(2*x/(1-x))/sqrt(1-x^2).
+10
2
1, 2, 9, 50, 361, 3042, 29929, 331298, 4100625, 55777922, 828691369, 13316140818, 230256982201, 4257449540450, 83834039024649, 1750225301567618, 38614608429012001, 897325298084953602, 21904718673762721225, 560258287738117292018, 14981472258320814527241
FORMULA
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
MATHEMATICA
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(x/(1-x^2))/sqrt(1-x^2).
+10
1
1, 1, 2, 10, 40, 296, 1936, 17872, 164480, 1820800, 21442816, 279255296, 3967316992, 59837670400, 988024924160, 17009993230336, 318566665977856, 6177885274406912, 129053377688043520, 2786107670662021120, 64136976817284448256, 1525720008470138454016
COMMENTS
Is this the same as A227545 (at least for n>=1)?
FORMULA
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).
MATHEMATICA
Table[Abs[HermiteH[n, (1 + I)/2]]^2/2^n, {n, 0, 20}]
Triangle of connection constants between the falling factorials (x)_(n) and (2*x)_(n).
+10
0
1, 0, 2, 0, 2, 4, 0, 0, 12, 8, 0, 0, 12, 48, 16, 0, 0, 0, 120, 160, 32, 0, 0, 0, 120, 720, 480, 64, 0, 0, 0, 0, 1680, 3360, 1344, 128, 0, 0, 0, 0, 1680, 13440, 13440, 3584, 256, 0, 0, 0, 0, 0, 30240, 80640, 48384, 9216, 512
COMMENTS
The falling factorial polynomials (x)_n := x*(x-1)*...*(x-n+1), n = 0,1,2,..., form a basis for the space of polynomials. Hence the polynomial (2*x)_n may be expressed as a linear combination of x_0, x_1,...,x_n; the coefficients in the expansion form the n-th row of the table. Some examples are given below.
This triangle is connected to two families of orthogonal polynomials, the Hermite polynomials H(n,x) A060821, and the Bessel polynomials y(n,x) A001498. The first few Hermite polynomials are ... 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.
The unsigned coefficients of H(n,x) give the nonzero entries of the n-th row of the triangle.
The Bessel polynomials y(n,x) begin
... y(0,x) = 1
... y(1,x) = 1+x
... y(2,x) = 1+3*x+3*x^2
... y(3,x) = 1+6*x+15*x^2+15*x^3.
The entries in the n-th column of this triangle are the coefficients of the scaled Bessel polynomials 2^n*y(n,x).
Also the Bell transform of g(n) = 2 if n<2 else 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, page 158, exercise 7.
FORMULA
Defining relation: 2*x*(2*x-1)*...*(2*x-n+1) = sum {k=0..n} T(n, k)*x*(x-1)*...*(x-k+1)
Explicit formula: T(n,k) = (n!/k!)*binomial(k,n-k)*2^(2*k-n). [As defined by Comtet (see reference).]
Recurrence relation: T(n,k) = (2*k-n+1)*T(n-1,k)+2*T(n-1,k-1).
E.g.f.: exp(x*(t^2+2*t)) = 1 + (2*x)*t + (2*x+4*x^2)*t^2/2! + (12*x^2+8*x^3)*t^3/3! + ...
O.g.f. for m-th diagonal (starting at main diagonal m = 0): (2*m)!/m!*x^m/(1-2*x)^(2*m+1).
The triangle is the matrix product [2^k*s(n,k)]n,k>=0 * ([s(n,k)]n,k>=0)^(-1),
where s(n,k) denotes the signed Stirling number of the first kind.
Row sums are [1,2,6,20,76,...] = A000898. Column sums are [1,4,28,296,...] = [2^n* A001515(n)] n>=0.
EXAMPLE
Triangle begins
n\k|...0.....1.....2.....3.....4.....5.....6
============================================
0..|...1
1..|...0.....2
2..|...0.....2.....4
3..|...0.....0....12.....8
4..|...0.....0....12....48....16
5..|...0.....0.....0...120...160....32
6..|...0.....0.....0...120...720...480....64
..
Row 3:
(2*x)_3 = (2*x)*(2*x-1)*(2*x-2) = 8*x*(x-1)*(x-2) + 12*x*(x-1).
Row 4:
(2*x)_4 = (2*x)*(2*x-1)*(2*x-2)*(2*x-3) = 16*x*(x-1)*(x-2)*(x-3) +
48*x*(x-1)*(x-2)+ 12*x*(x-1).
Examples of recurrence relation
T(4,4) = 5*T(3,4) + 2*T(3,3) = 5*0 + 2*8 = 16;
T(5,4) = 4*T(4,4) + 2*T(4,3) = 4*16 + 2*48 = 160;
T(6,4) = 3*T(5,4) + 2*T(5,3) = 3*160 + 2*120 = 720;
T(7,4) = 2*T(6,4) + 2*T(6,3) = 2*720 + 2*120 = 1680.
MAPLE
T := (n, k) -> (n!/k!)*binomial(k, n-k)*2^(2*k-n):
seq(seq(T(n, k), k=0..n), n=0..9);
PROG
(Sage) # uses[bell_matrix from A264428] bell_matrix(lambda n: 2 if n<2 else 0, 12) # Peter Luschny, Jan 19 2016
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