A307318 -id:A307318 - cp's OEIS frontend

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

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

Offset: 0

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

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..100
Vidunas, Raimundas Counting derangements and Nash equilibria Ann. Comb. 21, No. 1, 131-152 (2017).

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

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)];

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 *)

{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

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)

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

Seiichi Manyama, Apr 02 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..400
Vaclav Kotesovec, Recurrence (of order 5)

Cf. A120305, A144661, A307318.

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 *)

{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!)))))}

{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

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

A307350 a(n) = -Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} (-1)^(i+j+k) * (i+j+k)!/(3!i!j!*k!).

Original entry on oeis.org

0, 1, -5, 120, -2380, 52556, -1192625, 27798310, -660128942, 15907062666, -387785597485, 9543399745815, -236715891871160, 5910596888393926, -148421725618783545, 3745355227481531010, -94917946415633366050, 2414582011729590475886

Offset: 0

Seiichi Manyama, Apr 03 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..100

Cf. A144511, A307318, A307349, A307351.

Table[-Sum[Sum[Sum[(-1)^(i + j + k)*(i + j + k)!/(3!*i!*j!*k!), {i, 1, n}], {j, 1, n}], {k, 1, n}], {n, 0, 17}] (* Amiram Eldar, Apr 03 2019 *)

{a(n) = -sum(i=1, n, sum(j=1, n, sum(k=1, n, (-1)^(i+j+k)*(i+j+k)!/(6*i!*j!*k!))))}

{a(n) = -sum(i=3, 3*n, (-1)^i*i!*polcoef(sum(j=1, n, x^j/j!)^3, i))/6} \\ Seiichi Manyama, May 20 2019

a(n) ~ -(-1)^n * 3^(3*n + 5/2) / (256*Pi*n). - 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

Seiichi Manyama, May 20 2019

			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, ...

Seiichi Manyama, Antidiagonals n = 0..50, flattened

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

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.

A307358 a(n) = Sum_{0<=i<=j<=k<=n} (-1)^(i+j+k) * (i+j+k)!/(i!j!k!).

Original entry on oeis.org

1, -4, 72, -1345, 27886, -610558, 13861334, -322838475, 7663363513, -184598740512, 4498935186891, -110693299767349, 2745124008220296, -68532209858173364, 1720678086867077832, -43415209670536390089, 1100146390869600888470

Offset: 0

Seiichi Manyama, Apr 04 2019

sign

Cf. A144660, A307318, A307352, A307354.

Table[Sum[Sum[Sum[(-1)^(i+j+k) * (i+j+k)!/(i!*j!*k!), {i, 0, j}], {j, 0, k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 04 2019 *)

{a(n) = sum(i=0, n, sum(j=i, n, sum(k=j, n, (-1)^(i+j+k)*(i+j+k)!/(i!*j!*k!))))}

a(n) ~ (-1)^n * 3^(3*n + 13/2) / (1792*Pi*n). - Vaclav Kotesovec, Apr 04 2019

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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!).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A307350 a(n) = -Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} (-1)^(i+j+k) * (i+j+k)!/(3!*i!*j!*k!).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A307358 a(n) = Sum_{0<=i<=j<=k<=n} (-1)^(i+j+k) * (i+j+k)!/(i!*j!*k!).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

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

A307350 a(n) = -Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} (-1)^(i+j+k) * (i+j+k)!/(3!i!j!*k!).

A307358 a(n) = Sum_{0<=i<=j<=k<=n} (-1)^(i+j+k) * (i+j+k)!/(i!j!k!).