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.

A190782 Triangle T(n,k), read by rows, of the coefficients of x^k in the expansion of Sum_(m=0..n) binomial(x,m) = (a(k)*x^k)/n!, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 5, 0, 1, 24, 14, 11, -2, 1, 120, 94, 5, 25, -5, 1, 720, 444, 304, -75, 55, -9, 1, 5040, 3828, 364, 1099, -350, 112, -14, 1, 40320, 25584, 15980, -4340, 3969, -1064, 210, -20, 1
Offset: 0

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Author

Mokhtar Mohamed, Dec 29 2012

Keywords

Comments

There is a strong relation between this triangle and triangle A048994 which deals with the binomial (x,n), this triangle being dealing with the summation of this binomial.
Apparently A054651 with reversed rows. - Mathew Englander, May 17 2014

Examples

			Triangle begins:
n\k     0       1       2       3       4       5       6      7     8
0       1
1       1       1
2       2       1       1
3       6       5       0        1
4      24      14      11       -2      1
5     120      94       5       25     -5       1
6     720     444     304      -75     55      -9       1
7    5040    3828     364     1099   -350     112     -14      1
8   40320   25584   15980    -4340   3969   -1064     210    -20     1
...

Links

Crossrefs

T(2*n,n) gives A347987.
Column 0-5 give A000142, A024167, A348063, A348064, A348065, A348068.
Cf. A048994, A054651, A132393.

Programs

  • Mathematica
    row[n_] := CoefficientList[ Series[ Sum[ Binomial[x, m], {m, 0, n}], {x, 0, n}], x]*n!; Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jan 04 2013 *)

Formula

T(n,k) = T(n-1,k)+ T(n-1,k-1)- T(n-2,k-1)*(n-1)+ T(n-2,k)*(n-1)^2, T(n,n)=1, T(n,0)= n! for n >= 0.
T(n,k) = T(n-1,k)*n + (A048994(n,k)), T(n,n)= 1, T(n,0)= n! for n>= 0.
E.g.f. of column k: (log(1 + x))^k/(k! * (1 - x)). - Seiichi Manyama, Sep 26 2021
T(n, k) = Sum_{i=0..n-k} Stirling1(i+k, k)*n!/(i+k)!. - Igor Victorovich Statsenko, May 27 2024

A348063 Coefficient of x^2 in expansion of n!* Sum_{k=0..n} binomial(x,k).

Original entry on oeis.org

1, 0, 11, 5, 304, 364, 15980, 34236, 1368936, 4429656, 173699712, 771653376, 30605906304, 175622947200, 7149130156800, 50800930272000, 2137822335475200, 18241636315507200, 796397873127782400, 7971407298921830400, 361615771356450508800, 4168685961862906982400, 196587429737202833817600
Offset: 2

Views

Author

Seiichi Manyama, Sep 26 2021

Keywords

Crossrefs

Column k=2 of A190782.
Cf. A000399, A028339, A054651, A347987, A348064, A348065, A348068.

Programs

  • PARI
    a(n) = n!*polcoef(sum(k=2, n, binomial(x, k)), 2);
  • PARI
    a(n) = if(n<2, 0, a(n-1)+(n-1)^2*a(n-2)+(-1)^n*(n-2)!);
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(log(1+x)^2/(2*(1-x))))
  • Python
    from sympy.abc import x
from sympy import ff, expand
def A348063(n): return sum(ff(n,n-k)*expand(ff(x,k)).coeff(x**2) for k in range(2,n+1)) # Chai Wah Wu, Sep 27 2021

Formula

a(n) = a(n-1) + (n-1)^2 * a(n-2) + (-1)^n * (n-2)!.
E.g.f.: (log(1 + x))^2/(2 * (1 - x)).
a(n) ~ n! * log(2)^2 / 2 * (1 + (-1)^n*log(n)/(log(2)^2*n)). - Vaclav Kotesovec, Sep 27 2021

A348065 Coefficient of x^4 in expansion of n!* Sum_{k=0..n} binomial(x,k).

Original entry on oeis.org

1, -5, 55, -350, 3969, -31563, 408050, -3920950, 58206676, -657328100, 11111159696, -144321864960, 2747845864464, -40364369180016, 856755330487200, -14042902728462624, 329258021171239296, -5956512800554963584, 153050034289602269952, -3028534064042216488704, 84691080748928315003904
Offset: 4

Views

Author

Seiichi Manyama, Sep 26 2021

Keywords

Crossrefs

Column k=4 of A190782.
Cf. A000482, A054651, A028341, A347987, A348063, A348064, A348068.

Programs

  • PARI
    a(n) = n!*polcoef(sum(k=4, n, binomial(x, k)), 4);
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(log(1+x)^4/(24*(1-x))))
  • Python
    from sympy.abc import x
from sympy import ff, expand
def A348065(n): return sum(ff(n,n-k)*expand(ff(x,k)).coeff(x**4) for k in range(4,n+1)) # Chai Wah Wu, Sep 27 2021

Formula

E.g.f.: (log(1 + x))^4/(24 * (1 - x)).

A348068 Coefficient of x^5 in expansion of n!* Sum_{k=0..n} binomial(x,k).

Original entry on oeis.org

1, -9, 112, -1064, 12873, -140595, 1870385, -23551110, 351042406, -5043110072, 84074954600, -1361614072000, 25218570009424, -455365645674480, 9298765013106384, -185409487083100320, 4144212593899945056, -90492302454898284864, 2199399908894486591040
Offset: 5

Views

Author

Seiichi Manyama, Sep 27 2021

Keywords

Crossrefs

Column k=5 of A190782.
Cf. A000482, A001233, A054651, A347987, A348063, A348064, A348065.

Programs

  • PARI
    a(n) = n!*polcoef(sum(k=5, n, binomial(x, k)), 5);
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(log(1+x)^5/(120*(1-x))))
  • Python
    from sympy.abc import x
from sympy import ff, expand
def A348068(n): return sum(ff(n,n-k)*expand(ff(x,k)).coeff(x**5) for k in range(5,n+1)) # Chai Wah Wu, Sep 27 2021

Formula

E.g.f.: (log(1 + x))^5/(120 * (1 - x)).
Showing 1-4 of 4 results.

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