cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A122248 a(n) - a(n-1) = A113474(n).

Original entry on oeis.org

0, 1, 3, 5, 9, 13, 18, 23, 31, 39, 48, 57, 68, 79, 91, 103, 119, 135, 152, 169, 188, 207, 227, 247, 270, 293, 317, 341, 367, 393, 420, 447, 479, 511, 544, 577, 612, 647, 683, 719, 758, 797, 837, 877, 919, 961, 1004, 1047, 1094, 1141, 1189
Offset: 0

Views

Author

Paul Barry, Aug 27 2006

Keywords

Comments

First differences are A113474.
The following sequences all appear to have the same parity: A003071, A029886, A061297, A092524, A093431, A102393, A104258, A122248, A128975. - Jeremy Gardiner, Dec 28 2008

Links

Crossrefs

Cf. A113474, A122247.

Programs

  • PARI
    a(n) = sum(k=1, n, sum(j=0, n, k\2^j)) - binomial(n, 2); \\ Michel Marcus, Mar 09 2023

Formula

G.f.: (1/(1-x))*Sum{k=0..infinity, x^(2^k)/((1-x)*(1-x^(2^k)))}-x^2/(1-x)^3.
a(n) = Sum_{k=1..n} Sum_{j=0..n} floor(k/2^j) - binomial(n,2).
a(n) = A122247(n)-binomial(n,2).

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

Views

Author

Roger L. Bagula, Jun 05 2007

Keywords

Examples

			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;

Links

Crossrefs

Cf. A049606, A067655, A113474.

Programs

  • 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
  • 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 *)
  • 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

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).

Extensions

Edited by G. C. Greubel, Sep 28 2024

A086117 Denominator of mean deviation of a symmetrical binomial distribution on n elements.

Original entry on oeis.org

2, 2, 4, 4, 16, 16, 32, 32, 256, 256, 512, 512, 2048, 2048, 4096, 4096, 65536, 65536, 131072, 131072, 524288, 524288, 1048576, 1048576, 8388608, 8388608, 16777216, 16777216, 67108864, 67108864, 134217728, 134217728, 4294967296, 4294967296
Offset: 1

Views

Author

Eric W. Weisstein, Jul 10 2003

Keywords

Comments

Also denominator of x(n)=Sum(x(k)*x(n-k-1):0<=kA098597(n)/a(n+2). - Reinhard Zumkeller, Feb 06 2008

Links

Crossrefs

Cf. A086116, A113474.
Twice A101926.

Programs

  • Mathematica
    Denominator[Table[If[OddQ[n], n!!/2/(n-1)!!, (n-1)!!/2/(n-2)!! ], {n, 50}]]

Formula

a(1)=2, a(n) = a(n-1)*A006519(n). - Jon Perry, Mar 31 2004
a(n) = 2^A113474(n-1). - Alan Michael Gómez Calderón, Apr 03 2025

A336940 Number of odd divisors of n!.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 6, 12, 12, 20, 30, 60, 72, 144, 216, 336, 336, 672, 864, 1728, 2160, 3200, 4800, 9600, 10560, 14784, 22176, 28224, 35280, 70560, 86400, 172800, 172800, 245760, 368640, 497664, 559872, 1119744, 1679616, 2363904, 2626560, 5253120, 6451200, 12902400, 16128000
Offset: 0

Views

Author

Gus Wiseman, Aug 23 2020

Keywords

Examples

			The a(1) = 1 through a(8) = 12 divisors:
  1  1  1  1  1   1   1    1
        3  3  3   3   3    3
              5   5   5    5
              15  9   7    7
                  15  9    9
                  45  15   15
                      21   21
                      35   35
                      45   45
                      63   63
                      105  105
                      315  315

Links

Crossrefs

A049606 gives the maximum among these divisors, with quotient A060818.
A337257 is the even version.
A000265 gives the maximum odd divisor of n.
A001227 counts odd divisors.
A183063 counts even divisors.
Cf. A000005, A001013, A001055, A006939, A113474, A124010, A253249.
Factorial numbers: A000142, A022559, A027423 (divisors), A048656, A071626, A076716 (factorizations), A325272, A325273, A325617, A336414, A336498.

Programs

  • Mathematica
    Table[Length[Select[Divisors[n!],OddQ]],{n,0,15}]
  • PARI
    a(n) = sumdiv(n!, d, d%2); \\ Michel Marcus, Aug 24 2020
  • PARI
    a(n) = numdiv(prod(k=1, n, k >> valuation(k, 2))); \\ Michel Marcus, Aug 27 2020

Formula

a(n) = A001227(n!).
a(n) = A000005(A049606(n)).
a(n) + A337257(n) = A027423(n) = A000005(n!).
From Seiichi Manyama, Aug 27 2020: (Start)
If p is odd prime, a(p) = 2 * a(p-1).
a(n) = A027423(n) / A113474(n) for n > 0. (End)

Extensions

a(36)-a(44) from Seiichi Manyama, Aug 26 2020
Showing 1-4 of 4 results.

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