A111595 Triangle of coefficients of square of Hermite polynomials divided by 2^n with argument sqrt(x/2).

1, 0, 1, 1, -2, 1, 0, 9, -6, 1, 9, -36, 42, -12, 1, 0, 225, -300, 130, -20, 1, 225, -1350, 2475, -1380, 315, -30, 1, 0, 11025, -22050, 15435, -4620, 651, -42, 1, 11025, -88200, 220500, -182280, 67830, -12600, 1204, -56, 1, 0, 893025, -2381400, 2302020, -1020600, 235494, -29736, 2052, -72

Offset: 0

Author

Wolfdieter Lang, Aug 23 2005

Keywords

• sign
• easy
• tabl

Comments

This is a Sheffer triangle (lower triangular exponential convolution matrix). For Sheffer row polynomials see the S. Roman reference and explanations under A048854.

In the umbral notation of the S. Roman reference this would be called Sheffer for ((sqrt(1-2*t))/(1-t), t/(1-t)).

The associated Sheffer triangle is A111596.

Matrix logarithm equals A112239. - Paul D. Hanna, Aug 29 2005

The row polynomials (1/2^n)* H(n,sqrt(x/2))^2, with the Hermite polynomials H(n,x), have e.g.f. (1/sqrt(1-y^2))*exp(x*y/(1+y)).

The row polynomials s(n,x):=sum(a(n,m)*x^m,m=0..n), together with the associated row polynomials p(n,x) of A111596, satisfy the exponential (or binomial) convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), n>=0.

The unsigned column sequences are: A111601, A111602, A111777-A111784, for m=1..10.

Examples

			The triangle a(n, m) begins:
n\m       0         1         2          3         4         5       6       7     8    9  10 ...
0:        1
1:        0         1
2:        1        -2         1
3:        0         9        -6          1
4:        9       -36        42        -12         1
5:        0       225      -300        130       -20         1
6:      225     -1350      2475      -1380       315       -30       1
7:        0     11025    -22050      15435     -4620       651     -42       1
8:    11025    -88200    220500    -182280     67830    -12600    1204     -56     1
9:        0    893025  -2381400    2302020  -1020600    235494  -29736    2052   -72    1
10:  893025  -8930250  28279125  -30958200  15961050  -4396140  689850  -63000  3285  -90   1
-------------------------------------------------------------------------------------------------
		

References

• R. P. Boas and R. C. Buck, Polynomial Expansions of Analytic Functions, Springer, 1958, p. 41
• S. Roman, The Umbral Calculus, Academic Press, New York, 1984, p. 128.

Links

• G. C. Greubel, Rows n=0..100 of triangle, flattened

Crossrefs

Row sums: A111882. Unsigned row sums: A111883.

Cf. A112239 (matrix log).

Programs

Mathematica

row[n_] := CoefficientList[ 1/2^n*HermiteH[n, Sqrt[x/2]]^2, x]; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jul 17 2013 *)

Python

from sympy import hermite, Poly, sqrt, symbols
x = symbols('x')
def a(n): return Poly(1/2**n*hermite(n, sqrt(x/2))**2, x).all_coeffs()[::-1]
for n in range(11): print(a(n)) # Indranil Ghosh, May 26 2017

Formula

E.g.f. for column m>=0: (1/sqrt(1-x^2))*((x/(1+x))^m)/m!.

a(n, m)=((-1)^(n-m))*(n!/m!)*sum(binomial(2*k, k)*binomial(n-2*k-1, m-1)/(4^k), k=0..floor((n-m)/2)), n>=m>=1. a(2*k, 0)= ((2*k)!/(k!*2^k))^2 = A001818(k), a(2*k+1) = 0, k>=0. a(n, m)=0 if n

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