|
| |
|
|
A111595
|
|
Triangle of coefficients of square of Hermite polynomials divided by 2^n with argument sqrt(x/2).
|
|
16
| |
|
|
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
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,5
|
|
|
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 (pauldhanna(AT)juno.com), 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.
|
|
|
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
| W. Lang, First 10 rows.
|
|
|
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<m.
|
|
|
EXAMPLE
| Triangle begins:
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; ...
|
|
|
CROSSREFS
| Row sums: A111882. Unsigned row sums: A111883.
Cf. A112239 (matrix log).
Sequence in context: A201897 A137452 A158335 * A021478 A115563 A185285
Adjacent sequences: A111592 A111593 A111594 * A111596 A111597 A111598
|
|
|
KEYWORD
| sign,easy,tabl
|
|
|
AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 23 2005
|
| |
|
|
