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A067147 Triangle of coefficients for expressing x^n in terms of Hermite polynomials. 7
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

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. (Table 22.12)

LINKS

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

FORMULA

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

EXAMPLE

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;

MAPLE

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

MATHEMATICA

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 *)

PROG

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

(MAGMA) [[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

(PARI) {T(n, k) = if(k<0 || n<k || (n-k)%2, 0, n!/(k!*((n-k)/2)!))}; /* Michael Somos, Jan 15 2020 */

CROSSREFS

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

Cf. A060821, A066325, and A099174.

Sequence in context: A109187 A265089 A166357 * A112227 A166378 A249820

Adjacent sequences:  A067144 A067145 A067146 * A067148 A067149 A067150

KEYWORD

nonn,tabl

AUTHOR

Christian G. Bower, Jan 03 2002

STATUS

approved

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Last modified April 13 11:56 EDT 2021. Contains 342936 sequences. (Running on oeis4.)