A176244 Triangle generated by T(n,k) = q^k*T(n-1, k) + T(n-1, k-1), with q=4.

1, 1, 1, 1, 17, 1, 1, 273, 81, 1, 1, 4369, 5457, 337, 1, 1, 69905, 353617, 91729, 1361, 1, 1, 1118481, 22701393, 23836241, 1485393, 5457, 1, 1, 17895697, 1454007633, 6124779089, 1544878673, 23837265, 21841, 1, 1, 286331153, 93074384209, 1569397454417, 1588080540241, 99182316113, 381680209, 87377, 1

Offset

1,5

Comments

Row sums are: {1, 2, 19, 356, 10165, 516614, 49146967, 9165420200, 3350402793721, 2449781908163402, ...}.

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=4.

Example

Triangle starts as:
  1;
  1,        1;
  1,       17,          1;
  1,      273,         81,          1;
  1,     4369,       5457,        337,          1;
  1,    69905,     353617,      91729,       1361,        1;
  1,  1118481,   22701393,   23836241,    1485393,     5457,     1;
  1, 17895697, 1454007633, 6124779089, 1544878673, 23837265, 21841, 1;

Maple

T:= proc(n, k) option remember;
    q:=4;
      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:=4; 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 (* G. C. Greubel, Nov 22 2019 *)

Prog

(PARI) T(n, k) = my(q=4); 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:=4;
  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=4;
    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), A176243 (q=3), this sequence (q=4).

Sequence in context: A144442 A157151 A176794 * A022180 A156581 A015143

Adjacent sequences: A176241 A176242 A176243 * A176245 A176246 A176247

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 12 2010

Extensions

Edited by G. C. Greubel, Nov 22 2019

Status

approved