A172452 -id:A172452 - cp's OEIS frontend

A172453 Triangle T(n, k) = round( A172452(n)/(A172452(k)*A172452(n-k)) ), read by rows.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 6, 6, 3, 1, 1, 4, 12, 12, 12, 4, 1, 1, 4, 16, 24, 24, 16, 4, 1, 1, 4, 16, 32, 48, 32, 16, 4, 1, 1, 5, 20, 40, 80, 80, 40, 20, 5, 1, 1, 6, 30, 60, 120, 160, 120, 60, 30, 6, 1

Offset: 0

Roger L. Bagula, Feb 03 2010

The original definition of this sequence did not produce an integer valued triangular sequence. The application of the "round" function was the method chosen to formulate an integer sequence. - G. C. Greubel, Apr 27 2021

			Triangle begins as:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,  1;
  1, 2,  4,  2,   1;
  1, 3,  6,  6,   3,   1;
  1, 4, 12, 12,  12,   4,   1;
  1, 4, 16, 24,  24,  16,   4,  1;
  1, 4, 16, 32,  48,  32,  16,  4,  1;
  1, 5, 20, 40,  80,  80,  40, 20,  5, 1;
  1, 6, 30, 60, 120, 160, 120, 60, 30, 6, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A004001, A172452.

f[n_]:= f[n]= If[n<3, Fibonacci[n], f[f[n-1]] + f[n-f[n-1]]]; (* f=A004001 *)
c[n_]:= Product[f[j], {j,n}]; (* c=A172452 *)
T[n_, k_]:= Round[c[n]/(c[k]*c[n-k])];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Apr 27 2021 *)

@CachedFunction
def b(n): return fibonacci(n) if (n<3) else b(b(n-1)) + b(n-b(n-1)) # b=A004001
def c(n): return product(b(j) for j in (1..n)) # c=A172452
def T(n,k): return round(c(n)/(c(k)*c(n-k)))
[[T(n,k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Apr 27 2021

T(n, k) = round( A172452(n)/(A172452(k)*A172452(n-k)) ).

T(n, k) = round( c(n)/(c(k)*c(n-k)) ) where c(n) = Product_{j=1..n} A004001(j) with c(0) = 1.

Definition changed to give integral terms and edited by G. C. Greubel, Apr 27 2021

A172970 Triangle T(n, k) = A172452(n) - A172452(k) - A172452(n-k), read by rows.

Original entry on oeis.org

-1, -1, -1, -1, -1, -1, -1, 0, 0, -1, -1, 1, 2, 1, -1, -1, 7, 9, 9, 7, -1, -1, 35, 43, 44, 43, 35, -1, -1, 143, 179, 186, 186, 179, 143, -1, -1, 575, 719, 754, 760, 754, 719, 575, -1, -1, 3071, 3647, 3790, 3824, 3824, 3790, 3647, 3071, -1, -1, 19199, 22271, 22846, 22988, 23016, 22988, 22846, 22271, 19199, -1

Offset: 0

Roger L. Bagula, Feb 06 2010

Row sums are: {-1, -2, -3, -2, 2, 30, 198, 1014, 4854, 28662, 197622, ...}.

			Triangle begins as:
  -1;
  -1,    -1;
  -1,    -1,    -1;
  -1,     0,     0,    -1;
  -1,     1,     2,     1,    -1;
  -1,     7,     9,     9,     7,    -1;
  -1,    35,    43,    44,    43,    35,    -1;
  -1,   143,   179,   186,   186,   179,   143,    -1;
  -1,   575,   719,   754,   760,   754,   719,   575,    -1;
  -1,  3071,  3647,  3790,  3824,  3824,  3790,  3647,  3071,    -1;
  -1, 19199, 22271, 22846, 22988, 23016, 22988, 22846, 22271, 19199, -1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A004001, A172452, A172453.

f[n_]:= f[n]= If[n<3, Fibonacci[n], f[f[n-1]] + f[n-f[n-1]]]; (* f=A004001 *)
c[n_]:= Product[f[j], {j,n}]; (* c=A172452 *)
T[n_, k_]:= c[n] - c[k] - c[n-k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Apr 27 2021 *)

@CachedFunction
def b(n): return fibonacci(n) if (n<3) else b(b(n-1)) + b(n-b(n-1)) # b=A004001
def c(n): return product(b(j) for j in (1..n)) # c=A172452
def T(n,k): return c(n) - c(k) - c(n-k)
[[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 27 2021

T(n, k) = A172452(n) - A172452(k) - A172452(n-k).

T(n, k) = c(n) - c(k) - c(n-k) where c(n) = Product_{j=1..n} A004001(j).

Edited by G. C. Greubel, Apr 27 2021

cp's OEIS Frontend

A172453 Triangle T(n, k) = round( A172452(n)/(A172452(k)*A172452(n-k)) ), read by rows.

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