A144511 -id:A144511 - 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)

A144510 Array T(n,k) (n >= 1, k >= 0) read by downwards antidiagonals: T(n,k) = total number of partitions of [1, 2, ..., i] into exactly k nonempty blocks, each of size at most n, for any i in the range n <= i <= k*n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 7, 3, 1, 1, 37, 31, 4, 1, 1, 266, 842, 121, 5, 1, 1, 2431, 45296, 18252, 456, 6, 1, 1, 27007, 4061871, 7958726, 405408, 1709, 7, 1, 1, 353522, 546809243, 7528988476, 1495388159, 9268549, 6427, 8, 1

Offset: 1

David Applegate and N. J. A. Sloane, Dec 15 2008, Jan 30 2009

			Array begins:
1, 1,    1,       1,            1,                 1,                       1, ...
1, 2,    7,      37,          266,              2431,                   27007, ...
1, 3,   31,     842,        45296,           4061871,               546809243, ...
1, 4,  121,   18252,      7958726,        7528988476,          13130817809439, ...
1, 5,  456,  405408,   1495388159,    15467641899285,      361207016885536095, ...
1, 6, 1709, 9268549, 295887993624, 34155922905682979, 10893033763705794846727, ...
...

Seiichi Manyama, Antidiagonals n = 1..50, flattened

For the transposed array see A144512.
Rows include A001515, A144416, A144508, A144509.
Columns include A048775, A144511.
A(n+1,n) gives A281901.
A(n,n) gives A308296.
Cf. A308292.

b := proc(n, i, k) local r;
option remember;
if n = i then 1;
elif i < n then 0;
elif n < 1 then 0;
else add( binomial(i-1,r)*b(n-1,i-1-r,k), r=0..k);
end if;
end proc;
T:=proc(n,k); add(b(n,i,k),i=0..(k+1)*n); end proc;
# Peter Luschny, Apr 26 2011
A144510 := proc(n, k) local m;
add(m!*coeff(expand((exp(x)*GAMMA(n+1,x)/GAMMA(n+1)-1)^k),x,m),m=k..k*n)/k! end: for row from 1 to 6 do seq(A144510(row, col), col = 0..5) od;

multinomial[n_, k_List] := n!/Times @@ (k!); t[n_, k_] := Module[{i, ik}, ik = Array[i, k]; 1/k!* Sum[multinomial[Total[ik], ik], Evaluate[Sequence @@ Thread[{ik, 1, n}]]]]; Table[t[n-k, k], {n, 1, 10}, {k, n-1, 0, -1}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

T(n,k) = (1/k!)*Sum_{i_1=1..n} Sum_{i_2=1..n} ... Sum_{i_k=1..n} multinomial(i_1+i_2+...+i_k; i_1, i_2, ..., i_k).

T(n,k) = (1/k!)*Sum_{m=k..k*n} m! [x^m](e^x Gamma(n+1,x)/Gamma(n+1)-1)^k. Here [x^m]f(x) is the coefficient of x^m in the series expansion of f(x). - Peter Luschny, Apr 26 2011

A144512 Array read by upwards antidiagonals: T(n,k) = total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2, ..., k+1, for 0 <= k <= (k+1)*n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 4, 31, 37, 1, 1, 5, 121, 842, 266, 1, 1, 6, 456, 18252, 45296, 2431, 1, 1, 7, 1709, 405408, 7958726, 4061871, 27007, 1, 1, 8, 6427, 9268549, 1495388159, 7528988476, 546809243, 353522, 1, 1, 9, 24301, 216864652, 295887993624, 15467641899285

Offset: 0

David Applegate and N. J. A. Sloane, Dec 15 2008, Dec 21 2008

			Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 7, 37, 266, 2431, 27007, 353522, 5329837, ...
1, 3, 31, 842, 45296, 4061871, 546809243, 103123135501, ...
1, 4, 121, 18252, 7958726, 7528988476, 13130817809439, ...
1, 5, 456, 405408, 1495388159, 15467641899285, 361207016885536095, ...
1, 6, 1709, 9268549, 295887993624, 34155922905682979, 10893033763705794846727, ...
...

Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

See A144510 for Maple code.
Rows include A001515, A144416, A144508, A144509, A149187.
Columns include A048775, A144511, A144662, A147984.
Transpose of array in A144510.
Main diagonal gives A281901.

b := proc(n, i, k) local r;
option remember;
if n = i then 1;
elif i < n then 0;
elif n < 1 then 0;
else add( binomial(i-1,r)*b(n-1,i-1-r,k), r=0..k);
end if;
end proc;
T:=proc(n,k); add(b(n,i,k),i=0..(k+1)*n); end proc;

multinomial[n_, k_List] := n!/Times @@ (k!); t[n_, k_] := Module[{i, ik}, ik = Array[i, k]; 1/k!* Sum[multinomial[Total[ik], ik], Evaluate[Sequence @@ Thread[{ik, 1, n}]]]]; Table[t[n-k, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

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

A144658 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} (i+j+k)!/(i!j!k!).

Original entry on oeis.org

0, 6, 222, 5052, 109512, 2432448, 55611294, 1301187912, 30992726652, 748573575780, 18283412752602, 450657126652626, 11192820097106112, 279787295335659972, 7032532241701837758, 177611430241032329568

Offset: 0

N. J. A. Sloane, Jan 30 2009

nonn

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

Cf. A144511.

Table[Sum[Sum[Sum[(i+j+k)!/i!/j!/k!,{i,1,n}],{j,1,n}],{k,1,n}],{n,1,30}]

{a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, (i+j+k)!/(i!*j!*k!))))} \\ Seiichi Manyama, May 19 2019

{a(n) = sum(i=3, 3*n, i!*polcoef(sum(j=1, n, x^j/j!)^3, i))} \\ Seiichi Manyama, May 19 2019

A147984 Column 5 of A144512.

Original entry on oeis.org

1, 2431, 4061871, 7528988476, 15467641899285, 34155922905682979, 79397199549271412737, 191739533381111401455478, 476872353039366288373555323

Offset: 0

N. J. A. Sloane, May 13 2009

nonn

Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

Cf. A144512, A048775, A144511, A144662.

A307353 a(n) = Sum_{1<=i<=j<=k<=n} (i+j+k)!/(i!j!k!).

Original entry on oeis.org

0, 6, 138, 2808, 59083, 1298797, 29538183, 688783509, 16365391557, 394523905488, 9621386549905, 236859066714283, 5876752842394018, 146774130963028054, 3686474939155802036, 93044751867415156290, 2358431594463240429469

Offset: 0

Seiichi Manyama, Apr 03 2019

nonn

Cf. A120279, A144511, A144660, A307352.

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

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

a(n) ~ 3^(3*n + 13/2) / (832*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!).

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A144510 Array T(n,k) (n >= 1, k >= 0) read by downwards antidiagonals: T(n,k) = total number of partitions of [1, 2, ..., i] into exactly k nonempty blocks, each of size at most n, for any i in the range n <= i <= k*n.

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A144512 Array read by upwards antidiagonals: T(n,k) = total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2, ..., k+1, for 0 <= k <= (k+1)*n.

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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!).

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A144658 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} (i+j+k)!/(i!*j!*k!).

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A147984 Column 5 of A144512.

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A307353 a(n) = Sum_{1<=i<=j<=k<=n} (i+j+k)!/(i!*j!*k!).

Views

Author

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Mathematica

PARI

Formula

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

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!).

A144658 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} (i+j+k)!/(i!j!k!).

A307353 a(n) = Sum_{1<=i<=j<=k<=n} (i+j+k)!/(i!j!k!).