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
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, ...
...
-
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 *)
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
-
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)];
-
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 *)
-
{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
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
-
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 *)
-
{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!)))))}
-
{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
Original entry on oeis.org
1, 2431, 4061871, 7528988476, 15467641899285, 34155922905682979, 79397199549271412737, 191739533381111401455478, 476872353039366288373555323
Offset: 0
Showing 1-4 of 4 results.