A176242 Triangle read by rows, T(n, 1) = 1 and T(n,k) = q^k*T(n-1, k) + T(n-1, k-1) for 2 <= k <= n, n >= 1, with q=2.

1, 1, 1, 1, 5, 1, 1, 21, 13, 1, 1, 85, 125, 29, 1, 1, 341, 1085, 589, 61, 1, 1, 1365, 9021, 10509, 2541, 125, 1, 1, 5461, 73533, 177165, 91821, 10541, 253, 1, 1, 21845, 593725, 2908173, 3115437, 766445, 42925, 509, 1, 1, 87381, 4771645, 47124493, 102602157, 52167917, 6260845, 173229, 1021, 1

Offset

1,5

Comments

Row sums are: {1, 2, 7, 36, 241, 2078, 23563, 358776, 7449061, 213188690, ...}.

References

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

Links

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Formula

T(n,k) = T(n-1, k-1) + q^k*T(n-1, k), with q=2.

Example

Triangle begins as:
  1;
  1,     1;
  1,     5,      1;
  1,    21,     13,       1;
  1,    85,    125,      29,       1;
  1,   341,   1085,     589,      61,      1;
  1,  1365,   9021,   10509,    2541,    125,     1;
  1,  5461,  73533,  177165,   91821,  10541,   253,   1;
  1, 21845, 593725, 2908173, 3115437, 766445, 42925, 509, 1;

Maple

T:= proc(n, k) option remember;
    q:=2;
      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:=2; 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=2); 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:=2;
  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=2;
    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. this sequence (q=2), A176243 (q=3), A176244 (q=4).

Sequence in context: A047909 A171243 A111577 * A036969 A080249 A333143

Adjacent sequences: A176239 A176240 A176241 * A176243 A176244 A176245

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Apr 12 2010

Extensions

Edited by G. C. Greubel, Nov 22 2019

Definition clarified by Georg Fischer, Nov 12 2021

Status

approved