A135224 Triangle A103451 * A007318 * A000012, read by rows. T(n, k) for 0 <= k <= n.

1, 3, 1, 5, 3, 1, 9, 7, 4, 1, 17, 15, 11, 5, 1, 33, 31, 26, 16, 6, 1, 65, 63, 57, 42, 22, 7, 1, 129, 127, 120, 99, 64, 29, 8, 1, 257, 255, 247, 219, 163, 93, 37, 9, 1, 513, 511, 502, 466, 382, 256, 130, 46, 10, 1

Offset: 0

Author

Gary W. Adamson, Nov 23 2007

Keywords

• nonn
• tabl

Comments

Row sums = A132750: (1, 4, 9, 21, 49, 113, ...).

Left border = A083318: (1, 3, 5, 9, 17, 33, ...).

Examples

			First few rows of the triangle:
   1;
   3,  1;
   5,  3,  1;
   9,  7,  4,  1;
  17, 15, 11,  5,  1;
  33, 31, 26, 16,  6,  1;
  65, 63, 57, 42, 22,  7,  1;
...
		

Links

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

Crossrefs

Cf. A083318, A103451, A132750.

Programs

Magma

function T(n,k)
  if k eq 0 and n eq 0 then return 1;
  elif k eq 0 then return 2^n +1;
  else return (&+[Binomial(n, k+j): j in [0..n]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 20 2019

Maple

T:= proc(n, k) option remember;
      if k=0 and n=0 then 1
    elif k=0 then 2^n +1
    else add(binomial(n, k+j), j=0..n)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Nov 20 2019

Mathematica

T[n_, k_]:= T[n, k] = If[k==n==0, 1, If[k==0, 2^n +1, Sum[Binomial[n, k + j], {j, 0, n}]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

PARI

T(n,k) = if(k==0 && n==0, 1, if(k==0, 2^n +1, sum(j=0, n, binomial(n, k+j)) )); \\ G. C. Greubel, Nov 20 2019

Sage

def T(n, k):
    if (k==0 and n==0): return 1
    elif (k==0): return 2^n + 1
    else: return sum(binomial(n, k+j) for j in (0..n))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 20 2019

Formula

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

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

T(n, k) = binomial(n, k)*hypergeom([1, k-n], [k+1], -1) - binomial(n, k+n+1)* hypergeom([1, k+1], [k+n+2], -1) + 0^k - 0^n. - Peter Luschny, Nov 20 2019