A135233 Triangle A007318 * A193554, read by rows.

1, 2, 1, 5, 3, 1, 14, 7, 5, 1, 41, 15, 17, 7, 1, 122, 31, 49, 31, 9, 1, 365, 63, 129, 111, 49, 11, 1, 1094, 127, 321, 351, 209, 71, 13, 1, 3281, 255, 769, 1023, 769, 351, 97, 15, 1, 9842, 511, 1793, 2815, 2561, 1471, 545, 127, 17, 1

Offset

0,2

Comments

Row sums = 3^n.

Left column = A007051: (1, 2, 5, 14, 41, 122, ...).

Links

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

Formula

Binomial transform of A193554, as infinite lower triangular matrices.

T(n,k) = Sum_{j=0..n-k} (-1)^(n-k+j)*binomial(n,j)*2^j, with T(n,n) = 1, and T(n,0) = (3^n + 1)/2. - G. C. Greubel, Nov 20 2019

Example

First few rows of the triangle:
   1;
   2,  1;
   5,  3,  1;
  14,  7,  5,  1;
  41, 15, 17,  7,  1;
...

Maple

T:= proc(n, k) option remember;
      if k=n then 1
    elif k=0 then (3^n_1)/2
    else add((-1)^(n-k+j)*binomial(n, j)*2^j, j=0..n-k)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019

Mathematica

T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==0, (3^n+1)/2, Sum [(-1)^(n-k+i)* Binomial[n, i]*2^i, {i, 0, n-k}]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

Prog

(PARI) T(n, k) = if(k==n, 1, if(k==0, (3^n+1)/2, sum(j=0, n-k, (-1)^(n-k+j)*binomial(n, j)*2^j) )); \\ G. C. Greubel, Nov 20 2019
(Magma)
function T(n, k)
  if k eq n then return 1;
  elif k eq 0 then return (3^n+1)/2;
  else return (&+[(-1)^(n-k+j)*2^j*Binomial(n, j): j in [0..n-k]]);
  end if; return T; end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019
(Sage)
@CachedFunction
def T(n, k):
    if (k==n): return 1
    elif (k==0): return (3^n+1)/2
    else: return sum((-1)^(n-k+j)*2^j*binomial(n, j) for j in (0..n-k))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019

Crossrefs

Cf. A007051, A007318, A118801, A119258, A193554.

Sequence in context: A162997 A112339 A132808 * A125170 A054445 A105848

Adjacent sequences: A135230 A135231 A135232 * A135234 A135235 A135236

Keyword

nonn,tabl

Author

Gary W. Adamson, Nov 23 2007

Extensions

Definition corrected by N. J. A. Sloane, Jul 30 2011

Status

approved