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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.

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A354894 a(n) is the numerator of the n-th hyperharmonic number of order n.

Original entry on oeis.org

1, 5, 47, 319, 1879, 20417, 263111, 261395, 8842385, 33464927, 166770367, 3825136961, 19081066231, 57128792093, 236266661971, 7313175618421, 14606816124167, 102126365345729, 3774664307989373, 3771059091081773, 154479849447926113, 6637417807457499259, 6632660439700528339
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 10 2022

Keywords

Examples

			1, 5/2, 47/6, 319/12, 1879/20, 20417/60, 263111/210, 261395/56, 8842385/504, ...

References

  • J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, p. 258.

Links

Crossrefs

Cf. A001008, A001700, A027612, A058806, A076174, A124837, A354895 (denominators).
Differs from A049281.

Programs

  • Mathematica
    Table[SeriesCoefficient[-Log[1 - x]/(1 - x)^n, {x, 0, n}], {n, 1, 23}] // Numerator
Table[Binomial[2 n - 1, n - 1] (HarmonicNumber[2 n - 1] - HarmonicNumber[n - 1]), {n, 1, 23}] // Numerator
  • PARI
    H(n) = sum(i=1, n, 1/i);
a(n) = numerator(binomial(2*n-1,n-1) * (H(2*n-1) - H(n-1))); \\ Michel Marcus, Jun 10 2022
  • Python
    from math import comb
from sympy import harmonic
def A354894(n): return (comb(2*n-1,n-1)*(harmonic(2*n-1)-harmonic(n-1))).p # Chai Wah Wu, Jun 18 2022

Formula

a(n) is the numerator of the coefficient of x^n in the expansion of -log(1 - x) / (1 - x)^n.
a(n) is the numerator of binomial(2*n-1,n-1) * (H(2*n-1) - H(n-1)), where H(n) is the n-th harmonic number.
a(n) / A354895(n) ~ log(2) * 2^(2*n-1) / sqrt(Pi * n).

A354895 a(n) is the denominator of the n-th hyperharmonic number of order n.

Original entry on oeis.org

1, 2, 6, 12, 20, 60, 210, 56, 504, 504, 660, 3960, 5148, 4004, 4290, 34320, 17680, 31824, 302328, 77520, 813960, 8953560, 2288132, 27457584, 49031400, 12498200, 168725700, 42948360, 10925460, 163881900, 2540169450, 645122400, 327523680, 5567902560, 1412149200
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 10 2022

Keywords

Examples

			1, 5/2, 47/6, 319/12, 1879/20, 20417/60, 263111/210, 261395/56, 8842385/504, ...

References

  • J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, p. 258.

Links

Crossrefs

Cf. A001700, A002805, A027611, A058806, A076175, A124838, A354894 (numerators).

Programs

  • Maple
    N:= 100: # for a(1)..a(N)
H:= ListTools:-PartialSums([seq(1/i,i=1..2*N-1)]):
f:= n -> denom(binomial(2*n-1,n-1)*(H[2*n-1]-H[n-1])):
f(1):= 1:
map(f, [$1..N]); # Robert Israel, Jul 10 2023
  • Mathematica
    Table[SeriesCoefficient[-Log[1 - x]/(1 - x)^n, {x, 0, n}], {n, 1, 35}] // Denominator
Table[Binomial[2 n - 1, n - 1] (HarmonicNumber[2 n - 1] - HarmonicNumber[n - 1]), {n, 1, 35}] // Denominator
  • PARI
    H(n) = sum(i=1, n, 1/i);
a(n) = denominator(binomial(2*n-1,n-1) * (H(2*n-1) - H(n-1))); \\ Michel Marcus, Jun 10 2022
  • Python
    from math import comb
from sympy import harmonic
def A354895(n): return (comb(2*n-1,n-1)*(harmonic(2*n-1)-harmonic(n-1))).q # Chai Wah Wu, Jun 18 2022

Formula

a(n) is the denominator of the coefficient of x^n in the expansion of -log(1 - x) / (1 - x)^n.
a(n) is the denominator of binomial(2*n-1,n-1) * (H(2*n-1) - H(n-1)), where H(n) is the n-th harmonic number.
A354894(n) / a(n) ~ log(2) * 2^(2*n-1) / sqrt(Pi * n).

A292717 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. -log(1 - x)/(1 - x)^k.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 5, 11, 6, 0, 1, 7, 26, 50, 24, 0, 1, 9, 47, 154, 274, 120, 0, 1, 11, 74, 342, 1044, 1764, 720, 0, 1, 13, 107, 638, 2754, 8028, 13068, 5040, 0, 1, 15, 146, 1066, 5944, 24552, 69264, 109584, 40320, 0, 1, 17, 191, 1650, 11274, 60216, 241128, 663696, 1026576, 362880
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 21 2017

Keywords

Examples

			E.g.f. of column k: A_k(x) = x/1! + (2*k + 1)*x^2/2! + (3*k^2 + 6*k + 2)*x^3/3! + 2*(2*k^3 + 9*k^2 + 11*k + 3)*x^4/4! + ...
Square array begins:
   0,    0,     0,     0,     0,      0,  ...
   1,    1,     1,     1,     1,      1,  ...
   1,    3,     5,     7,     9,     11,  ...
   2,   11,    26,    47,    74,    107,  ...
   6,   50,   154,   342,   638,   1066,  ...
  24,  274,  1044,  2754,  5944,  11274,  ...

Crossrefs

Columns k=0..11 give A104150, A000254, A001705, A001711 (with offset 1), A001716 (with offset 1), A001721 (with offset 1), A051524, A051545, A051560, A051562, A051564, A203147.
Rows n=0..3 give A000004, A000012, A005408, A080663 (with offset 0).
Main diagonal gives A058806.

Programs

  • Mathematica
    Table[Function[k, n! SeriesCoefficient[-Log[1 - x]/(1 - x)^k, {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

Formula

E.g.f. of column k: -log(1 - x)/(1 - x)^k.

A302582 a(n) = n! * [x^n] log(1 + x)/(1 - x)^n.

Original entry on oeis.org

0, 1, 3, 29, 386, 6774, 146484, 3762744, 111868560, 3777096240, 142734788640, 5967788097600, 273488036169600, 13631083378617600, 734083968523046400, 42477063883483622400, 2628184745184816384000, 173147202267665649408000, 12100888735302910523904000, 894183767796064712795136000
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 10 2018

Keywords

Crossrefs

Cf. A000254, A024167, A058806, A104150, A109792, A300489.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Log[1 + x]/(1 - x)^n, {x, 0, n}], {n, 0, 19}]
Table[n! Sum[(-1)^(k + 1) Binomial[2 n - k - 1, n - k]/k, {k, 1,  n}], {n, 0, 19}]
Join[{0}, Table[n^2 (2 (n - 1))! HypergeometricPFQ[{1, 1, 1 - n}, {2, 2 - 2 n}, -1]/n!, {n, 19}]]

Formula

a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*binomial(2*n-k-1,n-k)/k.
a(n) ~ log(3/2) * 2^(2*n - 1/2) * n^n / exp(n). - Vaclav Kotesovec, May 05 2018
Showing 1-4 of 4 results.

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