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A100257 Triangle of expansions of 2^(k-1)*x^k in terms of T(n,x), in descending degrees n of T, with T the Chebyshev polynomials. 18
1, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 4, 0, 3, 1, 0, 5, 0, 10, 0, 1, 0, 6, 0, 15, 0, 10, 1, 0, 7, 0, 21, 0, 35, 0, 1, 0, 8, 0, 28, 0, 56, 0, 35, 1, 0, 9, 0, 36, 0, 84, 0, 126, 0, 1, 0, 10, 0, 45, 0, 120, 0, 210, 0, 126, 1, 0, 11, 0, 55, 0, 165, 0, 330, 0, 462, 0, 1, 0, 12, 0, 66, 0, 220, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

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. 795.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..6104

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].

H. J. Brothers, Pascal's Prism: Supplementary Material.

Index entries for sequences related to Chebyshev polynomials.

EXAMPLE

x^0 = T(0,x)

x^1 = T(1,x) + 0T(0,x)

2x^2 = T(2,x) + 0T(1,x) + 1T(0,x)

4x^3 = T(3,x) + 0T(2,x) + 3T(1,x) + 0T(0,x)

8x^4 = T(4,x) + 0T(3,x) + 4T(2,x) + 0T(1,x) + 3T(0,x)

16x^5 = T(5,x) + 0T(4,x) + 5T(3,x) + 0T(2,x) + 10T(1,x) + 0T(0,x)

MATHEMATICA

a[k_, n_] := If[k == 1, 1, If[EvenQ[n] || k < 0 || n > k, 0, If[n >= k - 1, Binomial[2*Floor[k/2], Floor[k/2]]/2, Binomial[k - 1, Floor[n/2]]]]];

Table[a[k, n], {k, 1, 13}, {n, 1, k}] // Flatten (* Jean-Fran├žois Alcover, May 04 2017, translated from PARI *)

PROG

(PARI) a(k, n)=if(k==1, 1, if(n%2==0||k<0||n>k, 0, if(n>=k-1, binomial(2*floor(k/2), floor(k/2))/2, binomial(k-1, floor(n/2)))))

CROSSREFS

Without zeros: A008311. Row sums are A011782. Cf. A092392.

Diagonals are (with interleaved zeros) twice A001700, A001791, A002054, A002694, A003516, A002696, A030053, A004310, A030054, A004311, A030055, A004312, A030056, A004313.

Sequence in context: A117178 A111527 A035695 * A100573 A049087 A178921

Adjacent sequences:  A100254 A100255 A100256 * A100258 A100259 A100260

KEYWORD

nonn,tabl

AUTHOR

Ralf Stephan, Nov 13 2004

STATUS

approved

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Last modified February 19 16:28 EST 2018. Contains 299356 sequences. (Running on oeis4.)