A164915 Inverse of binomial matrix (10^n,1) A164899. (See A164899 for companion sequence.)

1, 1, 10, 1, 9, 100, 1, 8, 90, 1000, 1, 7, 81, 900, 10000, 1, 6, 73, 810, 9000, 100000, 1, 5, 66, 729, 8100, 90000, 1000000, 1, 4, 60, 656, 7290, 81000, 900000, 10000000, 1, 3, 55, 590, 6561, 72900, 810000, 9000000, 100000000

Offset

1,3

Comments

Alternate sum and difference of diagonal integers generates A164913.

Links

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

Formula

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) = T(n-1, k) - T(n-2, k-1), with T(n, 1) = 1 and T(n, n) = 10^(n-1) (antidiagonal triangle).

Sum_{k=1..n} T(n, k) = (1/273)*(3*10^(n+1) - 15*A057079(n+1) - 12*A128834(n)).

Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (1/109)*(4*Fibonacci(n) + 5*LucasL(n) + (-10)^(n+1)). (End)

Example

Matrix array, A(n, k), begins:
  1, 10, 100, 1000, 10000, 100000, ...
  1,  9,  90,  900,  9000,  90000, ...
  1,  8,  81,  810,  8100,  81000, ...
  1,  7,  73,  729,  7290,  72900, ...
  1,  6,  66,  656,  6561,  65610, ...
  1,  5,  60,  590,  5905,  59049, ...
  1,  4,  55,  530,  5315,  53144, ...
Antidiagonal triangle, T(n, k), begins as:
  1;
  1, 10;
  1,  9, 100;
  1,  8,  90, 1000;
  1,  7,  81,  900, 10000;
  1,  6,  73,  810,  9000, 100000;
  1,  5,  66,  729,  8100,  90000, 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 = A164915
  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 = A164915
    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. A000032, A000045, A001019, A007318, A057079.

Cf. A128834, A164881, A164899, A164913.

Sequence in context: A337819 A046148 A231933 * A010691 A288697 A343132

Adjacent sequences: A164912 A164913 A164914 * A164916 A164917 A164918

Keyword

sign,tabl

Author

Mark Dols, Aug 31 2009

Status

approved