A129915 Irregular triangle read by rows: T(n, k) = f(n, A113474(n-1) - k), where f(n, k) = (n-1)!/2^k if (n-1)!/2^k is an integer, otherwise f(n, k) = 0.

1, 1, 1, 2, 3, 6, 3, 6, 12, 24, 15, 30, 60, 120, 45, 90, 180, 360, 720, 315, 630, 1260, 2520, 5040, 315, 630, 1260, 2520, 5040, 10080, 20160, 40320, 2835, 5670, 11340, 22680, 45360, 90720, 181440, 362880, 14175, 28350, 56700, 113400, 226800, 453600

Offset

1,4

Links

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

Formula

T(n, k) = f(n, A113474(n-1) - k), where f(n, k) = (n-1)!/2^k if (n-1)!/2^k is an integer, otherwise f(n, k) = 0, for n >= 1, 1 <= k <= A113474(n-1).

Example

Irregular triangle begins as:
    1;
    1;
    1,   2;
    3,   6;
    3,   6,   12,   24;
   15,  30,   60,  120;
   45,  90,  180,  360,  720;
  315, 630, 1260, 2520, 5040;
  315, 630, 1260, 2520, 5040, 10080, 20160, 40320;

Mathematica

A113474[n_]:= n+1 - DigitCount[n, 2, 1];
f[n_, k_]:= If[IntegerQ[(n-1)!/2^k], (n-1)!/2^k, 0];
A129915[n_, k_]:= f[n, A113474[n-1]-k];
Table[A129915[n, k], {n, 15}, {k, A113474[n-1]}]//Flatten (* modified by G. C. Greubel, Sep 28 2024 *)

Prog

(Magma)
A113474:= func< n | n+1 - Multiplicity(Intseq(n, 2), 1) >;
f:= func< n, k | IsIntegral(Factorial(n-1)/2^k) select Factorial(n-1)/2^k else 0 >;
A129915:= func< n, k | f(n, A113474(n-1) - k) >;
[A129915(n, k): k in [1..A113474(n-1)], n in [1..12]]; // G. C. Greubel, Sep 28 2024
(SageMath)
def A113474(n): return n+1 - sum((n+0).digits(2))
def f(n, k): return factorial(n-1)/2^k if (factorial(n-1)/2^k).is_integer() else 0
def A129915(n, k): return f(n, A113474(n-1) - k)
flatten([[A129915(n, k) for k in range(1, A113474(n-1)+1)] for n in range(1, 16)]) # G. C. Greubel, Sep 28 2024

Crossrefs

Cf. A049606, A067655, A113474.

Sequence in context: A245499 A323642 A193898 * A019773 A350728 A109536

Adjacent sequences: A129912 A129913 A129914 * A129916 A129917 A129918

Keyword

nonn,tabf

Author

Roger L. Bagula, Jun 05 2007

Extensions

Edited by G. C. Greubel, Sep 28 2024

Status

approved