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!)
A119673 Triangle read by rows, T(n, k) = 3*T(n-1, k-1) + T(n-1, k) for k<n and T(n, n) = 1, T(n, k) = 0, if k<0 or k>n. 2
1, 1, 1, 1, 4, 1, 1, 7, 13, 1, 1, 10, 34, 40, 1, 1, 13, 64, 142, 121, 1, 1, 16, 103, 334, 547, 364, 1, 1, 19, 151, 643, 1549, 2005, 1093, 1, 1, 22, 208, 1096, 3478, 6652, 7108, 3280, 1, 1, 25, 274, 1720, 6766, 17086, 27064, 24604, 9841, 1, 1, 28, 349, 2542, 11926, 37384, 78322, 105796, 83653, 29524, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

G. C. Greubel, Rows n = 0..100 of triangle, flattened

FORMULA

T(n,k) = R(n,k,3) where R(n,k,m) = (1-m)^(-n+k)-m^(k+1)*pochhammer(n-k, k+1)* hyper2F1([1,n+1],[k+2],m)/(k+1)!. - Peter Luschny, Jul 25 2014

EXAMPLE

Triangle begins:

  1;

  1,  1;

  1,  4,   1;

  1,  7,  13,    1;

  1, 10,  34,   40,    1;

  1, 13,  64,  142,  121,     1;

  1, 16, 103,  334,  547,   364,     1;

  1, 19, 151,  643, 1549,  2005,  1093,     1;

  1, 22, 208, 1096, 3478,  6652,  7108,  3280,    1;

  1, 25, 274, 1720, 6766, 17086, 27064, 24604, 9841, 1;

MAPLE

T := (n, k, m) -> (1-m)^(-n+k)-m^(k+1)*pochhammer(n-k, k+1)* hypergeom([1, n+1], [k+2], m)/(k+1)!; A119673 := (n, k) -> T(n, k, 3);

seq(print(seq(round(evalf(A119673(n, k))), k=0..n)), n=0..10); # Peter Luschny, Jul 25 2014

MATHEMATICA

T[_, 0]=1; T[n_, n_]=1; T[n_, k_]/; 0<k<n := T[n, k] = 3T[n-1, k-1] + T[n-1, k]; T[_, _] = 0;

Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)

PROG

(PARI) T(n, k) = if(k<0 || k>n, 0, if(k==n, 1, 3*T(n-1, k-1) +T(n-1, k)));

for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Nov 18 2019

(MAGMA)

function T(n, k)

  if k lt 0 or k gt n then return 0;

  elif k eq n then return 1;

  else return 3*T(n-1, k-1) + T(n-1, k);

  end if;

  return T;

end function;

[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 18 2019

(Sage)

@CachedFunction

def T(n, k):

    if (k<0 or k>n): return 0

    elif (k==n): return 1

    else: return 3*T(n-1, k-1) + T(n-1, k)

[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 18 2019

CROSSREFS

Cf. A003462, A014915, A081271, A119258.

Sequence in context: A209414 A193636 A232968 * A144447 A051455 A289511

Adjacent sequences:  A119670 A119671 A119672 * A119674 A119675 A119676

KEYWORD

easy,nonn,tabl

AUTHOR

Zerinvary Lajos, Jun 11 2006

EXTENSIONS

Definition clarified by Philippe Deléham, Jun 13 2006

Entry revised by N. J. A. Sloane, Jun 19 2006

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 6 21:48 EST 2019. Contains 329809 sequences. (Running on oeis4.)