A294453 Array read by antidiagonals: T(0,k) = A000045(k+1) for k >= 0. T(n,0) = 1 for n >= 0; thereafter T(n,k) = T(n-1,k-1)+T(n-1,k) for n, k >= 1.

1, 1, 1, 1, 2, 2, 1, 3, 3, 3, 1, 4, 5, 5, 5, 1, 5, 8, 8, 8, 8, 1, 6, 12, 13, 13, 13, 13, 1, 7, 17, 21, 21, 21, 21, 21, 1, 8, 23, 33, 34, 34, 34, 34, 34, 1, 9, 30, 50, 55, 55, 55, 55, 55, 55, 1, 10, 38, 73, 88, 89, 89, 89, 89, 89, 89, 1, 11, 47, 103, 138, 144, 144, 144, 144, 144, 144, 144, 1, 12, 57

Offset

0,5

Comments

This is another "Fibonacci array".

Links

Robert Israel, Table of n, a(n) for n = 0..10010 (antidiagonals 0 to 140, flattened)

Richard L. Ollerton and Anthony G. Shannon, Some properties of generalized Pascal squares and triangles, Fib. Q., 36 (1998), 98-109. See Table 6.

Formula

G.f. as array: 1/((1-x-x*y)*(1-y-y^2)). - Robert Israel, Nov 22 2017

T(n,m) = Sum_{k=0..m} binomial(n+m-k,k). - Vladimir Kruchinin, Jan 14 2024

Example

The array begins:
  1, 1,  2,   3,   5,   8,  13,  21,   34, ...
  1, 2,  3,   5,   8,  13,  21,  34,   55, ...
  1, 3,  5,   8,  13,  21,  34,  55,   89, ...
  1, 4,  8,  13,  21,  34,  55,  89,  144, ...
  1, 5, 12,  21,  34,  55,  89, 144,  233, ...
  1, 6, 17,  33,  55,  89, 144, 233,  377, ...
  1, 7, 23,  50,  88, 144, 233, 377,  610, ...
  1, 8, 30,  73, 138, 232, 377, 610,  987, ...
  1, 9, 38, 103, 211, 370, 609, 987, 1597, ...
  ...
The first few antidiagonals are:
  1,
  1, 1,
  1, 2,  2,
  1, 3,  3,  3,
  1, 4,  5,  5,  5,
  1, 5,  8,  8,  8,  8,
  1, 6, 12, 13, 13, 13, 13,
  1, 7, 17, 21, 21, 21, 21, 21,
  ...

Maple

A294453:= proc(n, k) option remember;
  if n = 0 then combinat:-fibonacci(k+1)
  elif k = 0 then 1
  else procname(n-1, k-1)+procname(n-1, k)
  fi
end proc:
seq(seq(A294453(m-k, k), k=0..m), m=0..10); # Robert Israel, Nov 22 2017

Mathematica

T[0, k_] := Fibonacci[k + 1];
T[_, 0] = 1;
T[n_, k_] := T[n, k] = T[n - 1, k - 1] + T[n - 1, k];
Table[T[n - k, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 16 2020 *)

Crossrefs

Cf. A000045, A001629 (sums of antidiagonals).

Sequence in context: A210798 A117501 A117915 * A097094 A210870 A104726

Adjacent sequences: A294450 A294451 A294452 * A294454 A294455 A294456

Keyword

nonn,tabl

Author

N. J. A. Sloane, Nov 22 2017

Extensions

Better definition and more terms from Robert Israel, Nov 22 2017

Status

approved