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.

A144660 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} (i+j+k)!/(i!*j!*k!).

Original entry on oeis.org

1, 16, 271, 5248, 110251, 2435200, 55621567, 1301226496, 30992872483, 748574130016, 18283414868863, 450657134765056, 11192820128307871, 279787295456009728, 7032532242167190271, 177611430242835570688, 4504491083159761986451, 114662734697313744041248
Offset: 0

Views

Author

N. J. A. Sloane, Jan 31 2009, Feb 01 2009

Keywords

Links

Crossrefs

Cf. A030662, A144661, A307318. This sum is very close to that in A144511.

Programs

  • Maple
    f:=n->add( add( add( (i+j+k)!/(i!*j!*k!), i=0..n),j=0..n),k=0..n); [seq(f(n),n=0..20)];
  • Mathematica
    Table[Sum[(i + j + k)!/(i!*j!*k!), {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 02 2019 *)
Table[Sum[(1 + k + 2*n)! * HypergeometricPFQ[{1, -1 - k - n, -n}, {-1 - k - 2*n, -k - n}, 1] / ((1 + k + n)*k!*n!^2), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 04 2019 *)
  • PARI
    {a(n) = sum(i=0, n, sum(j=0, n, sum(k=0, n, (i+j+k)!/(i!*j!*k!))))} \\ Seiichi Manyama, Apr 02 2019

Formula

From Vaclav Kotesovec, Apr 02 2019: (Start)
Recurrence: n^2*(2*n + 1)*(91*n^4 - 478*n^3 + 917*n^2 - 755*n + 222)*a(n) = 3*(2*n - 3)*(3*n - 5)*(3*n - 4)*(91*n^4 - 114*n^3 + 29*n^2 + 9*n - 3)*a(n-1) + n^2*(2*n + 1)*(91*n^4 - 478*n^3 + 917*n^2 - 755*n + 222)*a(n-2) - 3*(2*n - 3)*(3*n - 5)*(3*n - 4)*(91*n^4 - 114*n^3 + 29*n^2 + 9*n - 3)*a(n-3).
a(n) ~ 3^(3*n + 7/2) / (16*Pi*n). (End)

A308292 A(n,k) = Sum_{i_1=0..n} Sum_{i_2=0..n} ... Sum_{i_k=0..n} multinomial(i_1 + i_2 + ... + i_k; i_1, i_2, ..., i_k), square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 16, 19, 4, 1, 1, 65, 271, 69, 5, 1, 1, 326, 7365, 5248, 251, 6, 1, 1, 1957, 326011, 1107697, 110251, 923, 7, 1, 1, 13700, 21295783, 492911196, 191448941, 2435200, 3431, 8, 1, 1, 109601, 1924223799, 396643610629, 904434761801, 35899051101, 55621567, 12869, 9, 1
Offset: 0

Views

Author

Seiichi Manyama, May 19 2019

Keywords

Comments

For r > 1, row r is asymptotic to sqrt(2*Pi) * (r*n)^(r*n + 1/2) / ((r!)^n * exp(r*n-1)). - Vaclav Kotesovec, May 24 2020

Examples

			For (n,k) = (3,2), (Sum_{i=0..3} x^i/i!)^2 = (1 + x + x^2/2 + x^3/6)^2 = 1 + 2*x + 4*x^2/2 + 8*x^3/6 + 14*x^4/24 + 20*x^5/120 + 20*x^6/720. So A(3,2) = 1 + 2 + 4 + 8 + 14 + 20 + 20 = 69.
Square array begins:
   1, 1,    1,        1,             1,                   1, ...
   1, 2,    5,       16,            65,                 326, ...
   1, 3,   19,      271,          7365,              326011, ...
   1, 4,   69,     5248,       1107697,           492911196, ...
   1, 5,  251,   110251,     191448941,        904434761801, ...
   1, 6,  923,  2435200,   35899051101,    1856296498826906, ...
   1, 7, 3431, 55621567, 7101534312685, 4098746255797339511, ...

Links

Crossrefs

Columns k=0..4 give A000012, A000027(n+1), A030662(n+1), A144660, A144661.
Rows n=0..4 give A000012, A000522, A003011, A308294, A308295.
Main diagonal gives A274762.
Cf. A144510.

Formula

A(n,k) = Sum_{i=0..k*n} b(i) where Sum_{i=0..k*n} b(i) * x^i/i! = (Sum_{i=0..n} x^i/i!)^k.

A307324 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} Sum_{l=0..n} (-1)^(i+j+k+l) * (i+j+k+l)!/(i!*j!*k!*l!).

Original entry on oeis.org

1, 9, 997, 148041, 25413205, 4744544613, 935728207597, 191813392024137, 40462946725744501, 8726529512888314245, 1915408781755211655133, 426478330303800465141669, 96092667172064808771832957, 21869171662479233922632691261
Offset: 0

Views

Author

Seiichi Manyama, Apr 02 2019

Keywords

Links

Crossrefs

Cf. A120305, A144661, A307318.

Programs

  • Mathematica
    Table[Sum[(-1)^(i + j + k + l) * (i + j + k + l)! / (i!*j!*k!*l!), {i, 0, n}, {j, 0, n}, {k, 0, n}, {l, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 02 2019 *)
Table[Sum[((-1)^(j + k + l) * 2^(-1 - j - k - l) * ((j + k + l)! * (1 + n)! + (-1)^n * 2^(1 + j + k + l) * (1 + j + k + l + n)! Hypergeometric2F1[1, 2 + j + k + l + n, 2 + n, -1]))/(j! k! l! (1 + n)!), {j, 0, n}, {k, 0, n}, {l, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 03 2019 *)
  • PARI
    {a(n) = sum(i=0, n, sum(j=0, n, sum(k=0, n, sum(l=0, n, (-1)^(i+j+k+l)*(i+j+k+l)!/(i!*j!*k!*l!)))))}
  • PARI
    {a(n) = sum(i=0, 4*n, (-1)^i*i!*polcoef(sum(j=0, n, x^j/j!)^4, i))} \\ Seiichi Manyama, May 20 2019

Formula

a(n) ~ 2^(8*n + 15/2) / (625 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Apr 03 2019

A144662 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} Sum_{l=1..n} (i+j+k+l)!/(4!*i!*j!*k!*l!).

Original entry on oeis.org

0, 1, 266, 45296, 7958726, 1495388159, 295887993624, 60790021361170, 12845435390707724, 2774049143394729653, 609542744597785306189, 135840016223787254538508, 30629983532857972983331740, 6975352854342057056747327899, 1602003695575764851150428242804, 370631496919828403109950449644134
Offset: 0

Views

Author

N. J. A. Sloane, Feb 01 2009

Keywords

Links

Crossrefs

Column 4 of A144512. Cf. A144660, A144661.

Programs

  • Maple
    f:=n->add( add( add( add( (i+j+k+l)!/(4!*i!*j!*k!*l!), i=1..n),j=1..n),k=1..n),l=1..n); [seq(f(n),n=0..16)];
  • Mathematica
    a[n_] := Sum[(i+j+k+l)!/(4! i! j! k! l!), {i, n}, {j, n}, {k, n}, {l, n}];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Sep 05 2018 *)
Table[(Binomial[2*n + 2, n + 1] - 2*(1 + n) + Sum[(1 + k + l + 2*n)! HypergeometricPFQ[{1, -1 - k - l - n, -n}, {-1 - k - l - 2*n, -k - l - n}, 1]/((1 + k + l + n) k! l! (n!)^2) - (2*(1 + k + l + n)!)/((1 + k + l) k! l! n!), {k, 1, n}, {l, 1, n}])/24, {n, 0, 15}] (* Vaclav Kotesovec, Apr 04 2019 *)
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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