A135221 Triangle A007318 + A000012(signed) - I, I = Identity matrix, read by rows.

1, 0, 1, 2, 1, 1, 0, 4, 2, 1, 2, 3, 7, 3, 1, 0, 6, 9, 11, 4, 1, 2, 5, 16, 19, 16, 5, 1, 0, 8, 20, 36, 34, 22, 6, 1, 2, 7, 29, 55, 71, 55, 29, 7, 1, 0, 10, 35, 85, 125, 127, 83, 37, 8, 1, 2, 9, 46, 119, 211, 251, 211, 119, 46, 9, 1, 0, 12, 54, 166, 329, 463, 461, 331, 164, 56, 10, 1

Offset

0,4

Comments

row sums = A051049: (1, 1, 4, 7, 16, 31, 64, ...).

Links

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

Formula

T(n,k) = A007318 + A000012(signed) - Identity matrix, where A000012(signed) = (1; -1,1; 1,-1,1; ...).

T(n,k) = (-1)^(n-k) + binomial(n,k), with T(n,n)=1. - G. C. Greubel, Nov 20 2019

Example

First few rows of the triangle:
  1;
  0, 1;
  2, 1,  1;
  0, 4,  2,  1;
  2, 3,  7,  3,  1;
  0, 6,  9, 11,  4,  1;
  2, 5, 16, 19, 16,  5,  1;
  0, 8, 20, 36, 34, 22,  6, 1;
  2, 7, 29, 55, 71, 55, 29, 7, 1;
...

Maple

seq(seq( `if`(k=n, 1, binomial(n, k) + (-1)^(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, Binomial[n, k] + (-1)^(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, binomial(n, k) + (-1)^(n-k)); \\ G. C. Greubel, Nov 20 2019
(Magma) T:= func< n, k | k eq n select 1 else Binomial(n, k) +(-1)^(n-k) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019
(Sage)
def T(n, k):
    if (k==n): return 1
    else: return binomial(n, k) + (-1)^(n-k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019
(GAP)
T:= function(n, k)
    if k=n then return 1;
    else return Binomial(n, k) + (-1)^(n-k);
    fi; end;
Flat(List([0..12], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Nov 20 2019

Crossrefs

Cf. A007318, A051049.

Sequence in context: A326757 A147787 A247288 * A318686 A214546 A255704

Adjacent sequences: A135218 A135219 A135220 * A135222 A135223 A135224

Keyword

nonn,tabl

Author

Gary W. Adamson, Nov 23 2007

Extensions

More terms added by G. C. Greubel, Nov 20 2019

Status

approved