A067655 -id:A067655 - cp's OEIS frontend

A097324 Numbers n such that A067655(n) is different from A049606(n).

14, 18, 23, 25, 29, 35, 36, 40, 41, 42, 47, 51, 53, 58, 61, 62, 63, 69, 70, 71, 73, 80, 81, 84, 86, 88, 89, 90, 91, 95, 96, 99, 100, 102, 104, 106, 107, 109, 110, 113, 117, 118, 124, 127, 128, 130, 132, 135, 137, 139, 141, 146, 147, 150, 152, 155, 156, 157, 161

Offset: 1

Ralf Stephan, Aug 11 2004

nonn

Or, denominator of 2^n/n! differs from denominator of sum(k=1,n,C(n-1,k-1)*2^k/k!).

We conjecture that the sequence is infinite, the sequence and its complement (cases where the two values are equal) equipartition N and the difference between consecutive members of this sequence never exceeds c=7.

A067654 Numerators of the coefficients in power series expansion of exp(2x/(1-x)).

Original entry on oeis.org

2, 4, 22, 38, 314, 1504, 16346, 24796, 331906, 2426404, 38445118, 163944598, 2993870654, 2647864592, 601353085846, 820420721222, 18874008233858, 20741357283932, 5782164084903302, 38314394323494974, 1059898973520975218

Offset: 1

Benoit Cloitre, Feb 03 2002

K. Knopp, Theory and application of infinite series, Dover, p. 547.
O. Perron, Über das infinitäre Verhalten der Koeffizienten einer gewissen Potenzreihe, Archiv d. Math. u. Phys. (3), Vol. 22, pp. 329-340, 1914.

Cf. A067655.

a(n) is the numerator of Sum_{i=1..n} binomial(n-1, i-1)*2^i/i!.

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.

Original entry on oeis.org

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

Roger L. Bagula, Jun 05 2007

			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;

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

Cf. A049606, A067655, A113474.

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

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 *)

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

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).

Edited by G. C. Greubel, Sep 28 2024

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A097324 Numbers n such that A067655(n) is different from A049606(n).

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A067654 Numerators of the coefficients in power series expansion of exp(2x/(1-x)).

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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.

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