0,3

Coefficient of x^(2n-2) in Chebyshev polynomial T(2n) is -a(n).

Let M_n be the n X n matrix m_(i,j)=1+2*abs(i-j) then det(M_n)=(-1)^(n-1)*a(n-1) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 28 2002

Number of subsequences 00 in all words of length n+1 on the alphabet {0,1,2,3}. Example: a(2)=8 because we have 000,001,002,003,100,200,300 (the other 57=A125145(3) words of length 3 have no subsequences 00). a(n)=Sum(k*A128235(n+1,k),k=0..n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 27 2007

Let P(A) be the power set of an n-element set A. Then a(n) = the sum of the size of the symmetric difference of x and y for every subset {x,y} of P(A). - Ross La Haye (rlahaye(AT)new.rr.com), Dec 30 2007

Number of n-permutations of 5 objects u,v,w z x, with repetition allowed, containing exactly one u. Example: a(1)=8 because we have uv, vu, uw, wu, uz, zu, ux and xu. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 28 2007

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 516.

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

F. Ellermann, Illustration of binomial transforms

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 414

Eric Weisstein's World of Mathematics, Wiener Index

G.f.: x/(1-4x)^2. Convolution of powers of 4. - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)

Third binomial transform of n. E.g.f.: xexp(4x) - Paul Barry (pbarry(AT)wit.ie), Jul 22 2003

a(n)=sum(k=0, n, k*binomial(2*n, 2*k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jul 30 2003

For n>=0, a(n+1) = sum(i+j+k+l=n, binomial(2i, i)binomial(2j, j)binomial(2k, k)binomial(2l, l)). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 22 2004

a(n)=sum{k=0..n, 4^(n-k)binomial(n-k+1, k)binomial(1, (k+1)/2)(1-(-1)^k)/2} - Paul Barry (pbarry(AT)wit.ie), Oct 15 2004

seq(add((count(Composition(n)))^2, k=1..n), n=0..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 17 2006

A002697:=1/(4*z-1)**2; [Conjectured by S. Plouffe in his 1992 dissertation.]

seq(seq(binomial(i, j)*4^(i-1), j =i-1), i=0..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 03 2007

with(finance):seq(add(futurevalue( 2, 3, n), k=0..n)/2, n=-1..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 20 2008

(PARI) a(n)=if(n<0, 0, n*4^(n-1))

Cf. A000302, A002697, A027656, A083672.

Cf. A125145, A128235.

Cf. A038231.

Cf. A002699.

Adjacent sequences: A002694 A002695 A002696 this_sequence A002698 A002699 A002700

Sequence in context: A069021 A079763 A079785 this_sequence A026761 A026706 A128734

nonn

njas