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A067147 Triangle of coefficients for expressing x^n in terms of Hermite polynomials. 6
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, 0, 332640, 0, 277200, 0 (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).

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

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;

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.

Sequence in context: A109187 A265089 A166357 * A112227 A166378 A249820

Adjacent sequences:  A067144 A067145 A067146 * A067148 A067149 A067150

KEYWORD

nonn,tabl,changed

AUTHOR

Christian G. Bower, Jan 03 2002

STATUS

approved

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Last modified January 22 16:29 EST 2020. Contains 331152 sequences. (Running on oeis4.)