A308356
A(n,k) = (1/k!) * Sum_{i_1=1..n} Sum_{i_2=1..n} ... Sum_{i_k=1..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, 0, 1, 0, -1, 1, 0, 1, 0, 1, 0, -1, 1, -1, 1, 0, 1, 5, 5, 0, 1, 0, -1, 36, -120, 15, -1, 1, 0, 1, 329, 6286, 2380, 56, 0, 1, 0, -1, 3655, -557991, 1056496, -52556, 203, -1, 1, 0, 1, 47844, 74741031, 1006985994, 197741887, 1192625, 757, 0, 1
Offset: 0
For (n,k) = (3,2), (1/2) * (Sum_{i=1..3} x^i/i!)^2 = (1/2) * (x + x^2/2 + x^3/6)^2 = (-x)^2/2 + (-3)*(-x)^3/6 + 7*(-x)^4/24 + (-10)*(-x)^5/120 + 10*(-x)^6/720. So A(3,2) = 1 - 3 + 7 - 10 + 10 = 5.
Square array begins:
1, 0, 0, 0, 0, 0, ...
1, -1, 1, -1, 1, -1, ...
1, 0, 1, 5, 36, 329, ...
1, -1, 5, -120, 6286, -557991, ...
1, 0, 15, 2380, 1056496, 1006985994, ...
1, -1, 56, -52556, 197741887, -2063348839223, ...
1, 0, 203, 1192625, 38987482590, 4546553764660831, ...
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
-
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
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
A309117
Number of perfect matchings on a triangular lattice of width 4 and length n.
Original entry on oeis.org
1, 1, 5, 15, 56, 203, 749, 2777, 10293, 38240, 141997, 527593, 1960029, 7282483, 27057400, 100531559, 373522965, 1387822193, 5156442953, 19158736256, 71184183353, 264484479633, 982690786037, 3651182836279, 13565952140920, 50404229548515, 187276671274621
Offset: 0
Showing 1-4 of 4 results.