login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A191238 Triangle T(n,k) = coefficient of x^n in expansion of (x+x^3+x^5)^k. 1
1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 3, 0, 1, 0, 3, 0, 4, 0, 1, 0, 0, 6, 0, 5, 0, 1, 0, 2, 0, 10, 0, 6, 0, 1, 0, 0, 7, 0, 15, 0, 7, 0, 1, 0, 1, 0, 16, 0, 21, 0, 8, 0, 1, 0, 0, 6, 0, 30, 0, 28, 0, 9, 0, 1, 0, 0, 0, 19, 0, 50, 0, 36, 0, 10, 0, 1, 0, 0, 3, 0, 45, 0, 77, 0, 45, 0, 11, 0, 1, 0, 0, 0, 16, 0, 90, 0, 112, 0, 55, 0, 12, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

1. Riordan Array (1,x+x^3+x^5) without first column.

2. Riordan Array (1+x^2+x^4,x+x^3+x^5) numbering triangle (0,0).

3. For the g.f. 1/(1-x-x^3-x^5) we have a(n)=sum(k=1..n, T(n,k)) (see A060961).

4. For the e.g.f. exp(1-x-x^3-x^5) we have a(n)=n!*sum(k=1..n, T(n,k)/k!) (see A191237).

5. Bell Polynomial of second kind B(n,k){1,0,6,0,120,0,0,...,0}=n!/k!*T(n,k).

For more formulas see preprints.

LINKS

Table of n, a(n) for n=1..105.

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582

Vladimir Kruchinin, Derivation of Bell Polynomials of the Second Kind, arXiv:1104.5065

Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3

FORMULA

T(n,k)=sum(j=0..k, binomial(j,((n-k-2*j)/2))*binomial(k,j)*((-1)^(n-k)+1))/2.

EXAMPLE

triangle begins:

1,

0,1,

1,0,1,

0,2,0,1,

1,0,3,0,1,

0,3,0,4,0,1,

0,0,6,0,5,0,1,

0,2,0,10,0,6,0,1,

0,0,7,0,15,0,7,0,1,

0,1,0,16,0,21,0,8,0,1

MAPLE

A191238 := proc(n, k)

    add(binomial(j, ((n-k-2*j)/2))*binomial(k, j)*((-1)^(n-k)+1), j=0..k)/2 ;

end proc: # R. J. Mathar, Dec 16 2015

PROG

(Maxima)

T(n, k):=sum(binomial(j, ((n-k-2*j)/2))*binomial(k, j)*((-1)^(n-k)+1), j, 0, k)/2;

CROSSREFS

Cf. A060961 (row sums).

Sequence in context: A234954 A321201 A180649 * A049310 A168561 A253190

Adjacent sequences:  A191235 A191236 A191237 * A191239 A191240 A191241

KEYWORD

nonn,tabl

AUTHOR

Vladimir Kruchinin, May 27 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 8 18:37 EST 2019. Contains 329865 sequences. (Running on oeis4.)