A135228 Triangle A000012(signed) * A007318 * A103451, read by rows.

1, 1, 1, 3, 1, 1, 5, 2, 2, 1, 11, 2, 4, 3, 1, 21, 3, 6, 7, 4, 1, 43, 3, 9, 13, 11, 5, 1, 85, 4, 12, 22, 24, 16, 6, 1, 171, 4, 16, 34, 46, 40, 22, 7, 1, 341, 5, 20, 50, 80, 86, 62, 29, 8, 1, 683, 5, 25, 70, 130, 166, 148, 91, 37, 9, 1, 1365, 6, 30, 95, 200, 296, 314, 239, 128, 46, 10, 1

Offset

0,4

Comments

Row sums = A000975: (1, 2, 5, 10, 21, 42, 85, 170, ...).

Left border = A001045: (1, 1, 3, 5, 11, 21, 43, ...).

Links

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

Formula

T(n,k) = A000012(signed) * A007318 * A103451 as infinite lower triangular matrices. A000012(signed) = (1; -1,1; 1,-1,1; ...).

T(n,k) = Sum_{j=0..floor((n-1)/2)} binomial(n-2*j-1, k-1), with T(n,0) =

(2^(n+1) - (-1)^(n+1))/3 (Jacobsthal_{n+1}).- G. C. Greubel, Nov 20 2019

Example

First few rows of the triangle are:
    1;
    1, 1;
    3, 1,  1;
    5, 2,  2,  1;
   11, 2,  4,  3,  1;
   21, 3,  6,  7,  4,  1;
   43, 3,  9, 13, 11,  5,  1;
   85, 4, 12, 22, 24, 16,  6, 1;
  171, 4, 16, 34, 46, 40, 22, 7, 1;
...

Maple

T:= proc(n, k) option remember;
      if k=0 then (2^(n+1) +(-1)^n)/3
    else add(binomial(n-2*j-1, k-1), j=0..floor((n-1)/2))
      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==0, (2^(n+1) +(-1)^n)/3, Sum[Binomial[n-1-2*j, k-1], {j, 0, Floor[(n-1)/2]}]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

Prog

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

Crossrefs

Cf. A000975, A001045, A007318, A103451.

Sequence in context: A014475 A358612 A320436 * A021325 A174609 A280533

Adjacent sequences: A135225 A135226 A135227 * A135229 A135230 A135231

Keyword

nonn,tabl

Author

Gary W. Adamson, Nov 23 2007

Extensions

Offset changed by G. C. Greubel, Nov 20 2019

Status

approved