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

Author

Mark Dols, Aug 30 2009

Keywords

• nonn
• tabl

Examples

			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;
		

Links

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

Crossrefs

Cf. A001020, A007318, A021093, A164844.

Cf. A094704 (antidiagonal row sums).

Programs

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

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 *)

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

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)