login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A129689 A007318 * A129688. 3
1, 1, 1, 3, 2, 1, 7, 5, 3, 1, 15, 12, 8, 4, 1, 31, 27, 20, 12, 5, 1, 63, 58, 47, 32, 17, 6, 1, 127, 121, 105, 79, 49, 23, 7, 1, 255, 248, 226, 184, 128, 72, 30, 8, 1, 511, 503, 474, 410, 312, 200, 102, 38, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
Row sums = A057711: (1, 2, 6, 16, 40, 96, ...). A129690 = A129688 * A007318.
Riordan array ( (1-2*x+2*x^2)/((1-x)*(1-2*x)), x/(1-x) ). - Peter Bala, Mar 21 2018
LINKS
FORMULA
Binomial transform of A129688. A007318 * A129688 as infinite lower triangular matrices.
From Peter Bala, Mar 21 2018: (Start)
T(n,k) = C(n, n-k) + Sum_{i = 2..n} 2^(i-1)*C(n-i, n-k-i), where C(n,k) = n!/(k!*(n-k)!) for 0 <= k <= n, otherwise 0.
Exp(x) * the e.g.f. for row n = the e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(7 + 5*x + 3*x^2/2! + x^3/3!) = 7 + 12*x + 20*x^2/2! + 32*x^3/3! + 49*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1-x) ).
(End)
EXAMPLE
First few rows of the triangle are:
1;
1, 1;
3, 2, 1;
7, 5, 3, 1;
15, 12, 8, 4, 1;
31, 27, 20, 12, 5, 1;
63, 58, 47, 32, 17, 6, 1;
...
MAPLE
C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if;
end proc:
for n from 0 to 12 do
seq(C(n, n-k) + add(2^(i-1)*C(n-i, n-k-i), i = 2..n), k = 0..n)
end do; # Peter Bala, Mar 21 2018
MATHEMATICA
T[n_, k_] := Binomial[n, n-k] + Sum[2^(i-1) Binomial[n-i, n-k-i], {i, 2, n}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 10 2019 *)
PROG
(Sage) # uses[riordan_array from A256893]
# After Peter Bala.
riordan_array((1-2*x+2*x^2)/((1-x)*(1-2*x)), x/(1-x), 8) # Peter Luschny, Mar 21 2018
(GAP) Flat(List([0..12], n->List([0..n], k->Binomial(n, k)+Sum([2..n], i->2^(i-1)*Binomial(n-i, n-k-i))))); # Muniru A Asiru, Mar 22 2018
CROSSREFS
Sequence in context: A364855 A105531 A369041 * A115990 A350571 A277919
KEYWORD
nonn,tabl,easy
AUTHOR
Gary W. Adamson, Apr 28 2007
STATUS
approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 06:44 EDT 2024. Contains 371265 sequences. (Running on oeis4.)