A006976 Coefficients of Chebyshev T polynomials: a(n) = A053120(n+12, n), n >= 0.

1, 13, 98, 560, 2688, 11424, 44352, 160512, 549120, 1793792, 5637632, 17145856, 50692096, 146227200, 412778496, 1143078912, 3111714816, 8341487616, 22052208640, 57567870976, 148562247680, 379364311040, 959384125440

Offset: 0

Author

Simon Plouffe

Keywords

• nonn
• easy

Comments

Binomial transform of A069039. - Paul Barry, Feb 19 2003

If X_1, X_2, ..., X_n are 2-blocks of a (2n+1)-set X then, for n >= 5, a(n-5) is the number of (n+6)-subsets of X intersecting each X_i, (i = 1, 2, ..., n). - Milan Janjic, Nov 18 2007

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• 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].
• Milan Janjic, Two Enumerative Functions
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (14,-84,280,-560,672,-448,128).

Crossrefs

a(n) = A039991(n+12, 12), A053120.

Partial sums are in A002409.

Programs

GAP

List([0..25], n-> 2^(n-1)*Binomial(n+5,5)*(n+12)/6); # G. C. Greubel, Aug 27 2019

Magma

[2^(n-1)/6*Binomial(n+5,5)*(n+12) : n in [0..25]]; // Brad Clardy, Mar 10 2012

Maple

seq(2^(n-1)*binomial(n+5,5)*(n+12)/6, n=0..25); # G. C. Greubel, Aug 27 2019

Mathematica

Table[2^(n-1)*Binomial[n+5,5]*(n+12)/6, {n,0,25}] (* G. C. Greubel, Aug 27 2019 *)
LinearRecurrence[{14,-84,280,-560,672,-448,128},{1,13,98,560,2688,11424,44352},30] (* Harvey P. Dale, Sep 26 2024 *)

PARI

vector(26, n, 2^(n-2)*binomial(n+4,5)*(n+11)/6) \\ G. C. Greubel, Aug 27 2019

Sage

[2^(n-1)*binomial(n+5,5)*(n+12)/6 for n in (0..25)] # G. C. Greubel, Aug 27 2019

Formula

G.f.: (1-x)/(1-2*x)^7.

a(n) = 2^n*binomial(n+5, 5)*(n+12)/12. [See a comment in A053120 on subdiagonal sequences. - Wolfdieter Lang, Jan 03 2020]

a(n) = Sum_{k = 0..floor((n+12)/2)} C(n+12,2*k)*C(k,6). - Paul Barry, May 15 2003

E.g.f.: (1/45)*exp(2*x)*(45 + 495*x + 1125*x^2 + 900*x^3 + 300*x^4 + 42*x^5 + 2*x^6). - Stefano Spezia, Jan 03 2020

Extensions

More terms from James Sellers, Aug 21 2000

Name clarified by Wolfdieter Lang, Nov 26 2019