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.

A307318 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} (-1)^(i+j+k) * (i+j+k)!/(i!*j!*k!).

Original entry on oeis.org

1, -2, 37, -692, 14371, -315002, 7156969, -166785320, 3960790687, -95442311582, 2326713829837, -57260397539204, 1420295354815351, -35463581316556850, 890530353765972817, -22472131364683145552, 569507678494598796631, -14487492070374441746150
Offset: 0

Views

Author

Seiichi Manyama, Apr 02 2019

Keywords

Links

Crossrefs

Cf. A120305, A144660, A307324.

Programs

  • Mathematica
    Table[Sum[(-1)^(i + j + k) * (i + j + k)!/(i!*j!*k!), {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 02 2019 *)
  • PARI
    {a(n) = sum(i=0, n, sum(j=0, n, sum(k=0, n, (-1)^(i+j+k)*(i+j+k)!/(i!*j!*k!))))}
  • PARI
    {a(n) = sum(i=0, 3*n, (-1)^i*i!*polcoef(sum(j=0, n, x^j/j!)^3, i))} \\ Seiichi Manyama, May 20 2019

Formula

From Vaclav Kotesovec, Apr 02 2019: (Start)
Recurrence: 6*(n-1)*n^2*(490*n^4 - 3948*n^3 + 11668*n^2 - 14967*n + 7027)*a(n) = - (n-1)*(74480*n^6 - 675066*n^5 + 2399756*n^4 - 4233492*n^3 + 3852029*n^2 - 1682577*n + 272160)*a(n-1) + (131320*n^7 - 1437814*n^6 + 6472114*n^5 - 15414556*n^4 + 20770423*n^3 - 15610855*n^2 + 5939868*n - 861840)*a(n-2) - (27440*n^7 - 355838*n^6 + 1853810*n^5 - 4998800*n^4 + 7460459*n^3 - 6071312*n^2 + 2439561*n - 362880)*a(n-3) - 3*(2*n - 5)*(3*n - 8)*(3*n - 7)*(490*n^4 - 1988*n^3 + 2764*n^2 - 1515*n + 270)*a(n-4).
a(n) ~ (-1)^n * 3^(3*n + 7/2) / (128*Pi*n). (End)

A144661 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} Sum_{l=0..n} (i+j+k+l)!/(i!*j!*k!*l!).

Original entry on oeis.org

1, 65, 7365, 1107697, 191448941, 35899051101, 7101534312685, 1458965717496881, 308290573348183629, 66577182435768923245, 14629025943480502591445, 3260160391173522631759533, 735119604833362632050789701, 167408468505328518543519208949, 38448088693846486556578015883325
Offset: 0

Views

Author

N. J. A. Sloane, Feb 01 2009

Keywords

Links

Crossrefs

Cf. A030662, A144660, A144662, A307324.

Programs

  • Maple
    f:=n->add( add( add( add( (i+j+k+l)!/(i!*j!*k!*l!), i=0..n),j=0..n),k=0..n),l=0..n); [seq(f(n),n=0..20)];
  • Mathematica
    Table[Sum[(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 + n)!/((1 + j + k + l)*j!*k!*l!), {j, 0, n}, {k, 0, n}, {l, 0, n}] / n!, {n, 0, 20}] (* Vaclav Kotesovec, Apr 03 2019 *)
Table[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!), {k, 0, n}, {l, 0, n}]/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, (i+j+k+l)!/(i!*j!*k!*l!)))))} \\ Seiichi Manyama, Apr 02 2019

Formula

a(n) ~ 2^(8*n + 15/2) / (81 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Apr 02 2019

A307351 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} Sum_{l=1..n} (-1)^(i+j+k+l) * (i+j+k+l)!/(4!*i!*j!*k!).

Original entry on oeis.org

0, 1, 36, 6286, 1056496, 197741887, 38987482590, 7992252465604, 1685955453442326, 363605412277403725, 79808698852014867735, 17769930438868419048744, 4003861131932651139989514, 911215485942545343663605503, 209160405405110598032066208338
Offset: 0

Views

Author

Seiichi Manyama, Apr 03 2019

Keywords

Crossrefs

Cf. A144662, A307324, A307349, A307350.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[(-1)^(i + j + k + l)*(i + j + k + l)!/(4!*i!*j!*k!*l!), {i, 1, n}], {j, 1, n}], {k, 1, n}], {l, 1, n}], {n, 0, 14}] (* Amiram Eldar, Apr 03 2019 *)
  • PARI
    {a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, sum(l=1, n, (-1)^(i+j+k+l)*(i+j+k+l)!/(24*i!*j!*k!*l!)))))}
  • PARI
    {a(n) = sum(i=4, 4*n, (-1)^i*i!*polcoef(sum(j=1, n, x^j/j!)^4, i))/24} \\ Seiichi Manyama, May 20 2019

Formula

a(n) ~ 2^(8*n + 9/2) / (1875 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Apr 04 2019

A308322 A(n,k) = Sum_{i_1=0..n} Sum_{i_2=0..n} ... Sum_{i_k=0..n} (-1)^(i_1 + i_2 + ... + i_k) * 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, 0, 1, 1, 1, 1, 1, 1, -2, 3, 0, 1, 1, 9, 37, 9, 1, 1, 1, -44, 997, -692, 31, 0, 1, 1, 265, 44121, 148041, 14371, 111, 1, 1, 1, -1854, 2882071, -66211704, 25413205, -315002, 407, 0, 1, 1, 14833, 260415373, 53414037505, 120965241901, 4744544613, 7156969, 1513, 1, 1
Offset: 0

Views

Author

Seiichi Manyama, May 20 2019

Keywords

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 = 9.
Square array begins:
   1, 1,   1,       1,            1,                  1, ...
   1, 0,   1,      -2,            9,                -44, ...
   1, 1,   3,      37,          997,              44121, ...
   1, 0,   9,    -692,       148041,          -66211704, ...
   1, 1,  31,   14371,     25413205,       120965241901, ...
   1, 0, 111, -315002,   4744544613,   -247578134832564, ...
   1, 1, 407, 7156969, 935728207597, 545591130328772081, ...

Links

Crossrefs

Columns k=0..5 give A000012, A059841, A120305, A307318, A307324, A308325.
Rows n=0..1 give A000012, A182386.
Main diagonal gives A308323.
Cf. A308292.

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.
Showing 1-4 of 4 results.

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

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