A164899 Binomial matrix (1,10^n) read by antidiagonals.

1, 1, 10, 1, 11, 100, 1, 12, 110, 1000, 1, 13, 121, 1100, 10000, 1, 14, 133, 1210, 11000, 100000, 1, 15, 146, 1331, 12100, 110000, 1000000, 1, 16, 160, 1464, 13310, 121000, 1100000, 10000000, 1, 17, 175, 1610, 14641, 133100, 1210000, 11000000, 100000000

Offset

1,3

Links

G. C. Greubel, Antidiagonals n = 1..50, flattened

Formula

Sum_{k=1..n} T(n, k) = A094704(n).

As a triangle T(n,k) read by rows, T(n,1) = 1, T(n,n) = 10^(n-1), and T(n,k) = T(n-1, k) + T(n-2, k-1) otherwise. - Joerg Arndt, Dec 10 2016

From G. C. Greubel, Feb 10 2023: (Start)

A(n, k) = A(n-1, k) + A(n-1, k-1), with A(n, 1) = 1 and A(1, k) = 10^(k-1) (array).

T(n, k) = A(n-k+1, k) (antidiagonal triangle). (End)

Example

Matrix array, A(n, k), begins:
  1, 10, 100, 1000, ...
  1, 11, 110, 1100, ...
  1, 12, 121, 1210, ...
  1, 13, 133, 1331, ...
  1, 14, 146, 1464, ...
  1, 15, 160, 1610, ...
Antidiagonal triangle, T(n, k), begins as:
  1;
  1, 10;
  1, 11, 100;
  1, 12, 110, 1000;
  1, 13, 121, 1100, 10000;
  1, 14, 133, 1210, 11000, 100000;
  1, 15, 146, 1331, 12100, 110000, 1000000;

Mathematica

T[n_, k_]:= T[n, k]= If[k==n, 10^(n-1), If[k==1, 1, T[n-1, k] +T[n-2, k-1]]];
Table[T[n, k], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Feb 10 2023 *)

Prog

(Magma)
function T(n, k) // T = A164899
  if k eq n then return 10^(n-1);
  elif k eq 1 then return 1;
  else return T(n-1, k) + T(n-2, k-1);
  end if; return T;
end function;
[T(n, k): k in [1..n], n in [1..15]]; // G. C. Greubel, Feb 10 2023
(SageMath)
def T(n, k): # T = A164899
    if (k==n): return 10^(n-1)
    elif (k==1): return 1
    else: return T(n-1, k) + T(n-2, k-1)
flatten([[T(n, k) for k in range(1, n+1)] for n in range(1, 16)]) # G. C. Greubel, Feb 10 2023

Crossrefs

Cf. A001020, A007318, A021093, A164844.

Cf. A094704 (antidiagonal row sums).

Sequence in context: A370401 A172171 A327723 * A164844 A287015 A130311

Adjacent sequences: A164896 A164897 A164898 * A164900 A164901 A164902

Keyword

nonn,tabl

Author

Mark Dols, Aug 30 2009

Status

approved