A135230 Triangle A103451 * A000012(signed) * A007318, read by rows.

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

Offset

0,4

Comments

row sums = A135231

Links

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

Formula

T(n,k) = A103451 * A000012(signed) * A007318, where 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) = (3+(-1)^n)/2 and T(n,n) = 1. - G. C. Greubel, Nov 20 2019

Example

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

Maple

T:= proc(n, k) option remember;
      if k=n then 1
    elif k=0 then (3+(-1)^n)/2
    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==n, 1, If[k==0, (3+(-1)^n)/2, 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==n, 1, if(k==0, (3+(-1)^n)/2, 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 n then return 1;
  elif k eq 0 then return (3+(-1)^n)/2;
  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==n): return 1
    elif (k==0): return (3+(-1)^n)/2
    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. A000012, A007318, A103451, A135231.

Sequence in context: A086995 A220492 A229873 * A117957 A145702 A145704

Adjacent sequences: A135227 A135228 A135229 * A135231 A135232 A135233

Keyword

nonn,tabl

Author

Gary W. Adamson, Nov 23 2007

Extensions

More terms and offset changed by G. C. Greubel, Nov 20 2019

Status

approved