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A001794 Coefficients of Chebyshev polynomials.
(Formerly M4405 N1859)
12
1, 7, 32, 120, 400, 1232, 3584, 9984, 26880, 70400, 180224, 452608, 1118208, 2723840, 6553600, 15597568, 36765696, 85917696, 199229440, 458752000, 1049624576, 2387607552, 5402263552, 12163481600, 27262976000, 60850962432 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A negated sub-diagonal of A053120.

If X_1,X_2,...,X_n are 2-blocks of a (2n+1)-set X then a(n-2) is the number of (n+3)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007

The third corrector line for transforming 2^n offset 0 with a leading 1 into the Fibonacci sequence. - Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009

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.

Jones, C. W.; Miller, J. C. P.; Conn, J. F. C.; Pankhurst, R. C.; Tables of Chebyshev polynomials. Proc. Roy. Soc. Edinburgh. Sect. A. 62, (1946). 187-203.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..500

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. H. Albert, M. D. Atkinson, R. Brignall, The enumeration of three pattern classes using monotone grid classes, E. J. Combinat. 19 (3) (2012) P20, chapter 5.4.1.

T. Hibi, N. Li, H. Ohsugi, The face vector of a half-open hypersimplex, arXiv preprint arXiv:1309.5155, 2013

Milan Janjic, Two Enumerative Functions

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

FORMULA

a(n) = 2^(n-2)*(n+1)*(n+2)*(n+6)/3.

G.f.: (1-x)/(1-2*x)^4. - Simon Plouffe in his 1992 dissertation

a(n) = Sum_{k=0..floor((n+6)/2)} C(n+6, 2*k)*C(k, 3). - Paul Barry, May 15 2003

With a leading zero, the binomial transform of A000330. - Paul Barry, Jul 19 2003

a(n) = Sum_{i=0..n+1} (Sum{k=0..i} (k^2*binomial(n+1, i))). - Jon Perry, Feb 26 2004 [corrected by Michel Marcus, Mar 25 2017]

Binomial transform of a(n) = (2*n^3 + 6*n^2 + 7*n + 3)/3 offset 0. a(3)=120. - Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009

a(n) = 2^(n-1)/3*Binomial(n+2,2)*(n+6). - Brad Clardy, Mar 08 2012

MATHEMATICA

a[n_] := 2^(n-2)*(n+1)*(n+2)*(n+6)/3; a /@ Range[0, 20] (* Giovanni Resta, Mar 25 2017 *)

PROG

(MAGMA) [2^(n-1)/3*Binomial(n+2, 2)*(n+6) : n in [0..25]]; // Brad Clardy, Mar 08 2012

(PARI) a(n) = sum(i=0, n+1, sum(k=0, i, k^2*binomial(n+1, i))); \\ Michel Marcus, Mar 25 2017

CROSSREFS

Cf. A039991 (negative of column 6).

With alternating signs, the o.g.f (with offset 1) is the inverse of the o.g.f of A065097.

Cf. A001789 (partial sums), A081279 (binomial transform), A005900 (0 followed by inverse binomial transform).

Sequence in context: A219510 A164270 A182820 * A140289 A133107 A178851

Adjacent sequences:  A001791 A001792 A001793 * A001795 A001796 A001797

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 25 20:45 EST 2018. Contains 299656 sequences. (Running on oeis4.)