A176243 Triangle generated by T(n,k) = q^k*T(n-1, k) + T(n-1, k-1), with q=3.

1, 1, 1, 1, 10, 1, 1, 91, 37, 1, 1, 820, 1090, 118, 1, 1, 7381, 30250, 10648, 361, 1, 1, 66430, 824131, 892738, 98371, 1090, 1, 1, 597871, 22317967, 73135909, 24796891, 892981, 3277, 1, 1, 5380840, 603182980, 5946326596, 6098780422, 675780040, 8059780, 9838, 1

Offset

1,5

Comments

Row sums are: {1, 2, 12, 130, 2030, 48642, 1882762, 121744898, 13337520498, 2503662940162, ...}.

References

Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 176

Links

G. C. Greubel, Rows n = 1..75 of triangle, flattened

Formula

T(n,k) = T(n-1, k-1) + q^k*T(n-1, k), with q=3.

Example

Triangle begins as:
  1;
  1,      1;
  1,     10,        1;
  1,     91,       37,        1;
  1,    820,     1090,      118,        1;
  1,   7381,    30250,    10648,      361,     1;
  1,  66430,   824131,   892738,    98371,  1090,     1;
  1, 597871, 22317967, 73135909, 24796891, 892981, 3277, 1;

Maple

T:= proc(n, k) option remember;
    q:=3;
      if k=1 or k=n then 1
    else T(n-1, k-1) + q^k*T(n-1, k)
      fi; end:
seq(seq(T(n, k), k=1..n), n=1..12); # G. C. Greubel, Nov 22 2019

Mathematica

q:=3; T[n_, k_]:= T[n, k]= If[k==1 || k==n, 1, q^k*T[n-1, k] + T[n-1, k-1]]; Table[T[n, k], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Nov 22 2019 *)

Prog

(PARI) T(n, k) = my(q=3); if(k==1 || k==n, 1, q^k*T(n-1, k) + T(n-1, k-1)); \\ G. C. Greubel, Nov 22 2019
(Magma)
function T(n, k)
  q:=3;
  if k eq 1 or k eq n then return 1;
  else return T(n-1, k-1) + q^k*T(n-1, k);
  end if; return T; end function;
[T(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 22 2019
(Sage)
@CachedFunction
def T(n, k):
    q=3;
    if (k==1 or k==n): return 1
    else: return q^k*T(n-1, k) + T(n-1, k-1)
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Nov 22 2019

Crossrefs

Cf. A176242 (q=2), this sequence (q=3), A176244 (q=4).

Sequence in context: A166972 A364071 A160562 * A022173 A158117 A172378

Adjacent sequences: A176240 A176241 A176242 * A176244 A176245 A176246

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 12 2010

Extensions

Edited by G. C. Greubel, Nov 22 2019

Status

approved