A307349
a(n) = Sum_{i=1..n} Sum_{j=1..n} (-1)^(i+j) * (i+j)!/(2!*i!*j!).
Original entry on oeis.org
0, 1, 1, 5, 15, 56, 203, 757, 2839, 10736, 40821, 155948, 598065, 2301118, 8878591, 34340085, 133100055, 516851528, 2010358061, 7831136920, 30546063745, 119291436738, 466379022561, 1825168170620, 7149316835465, 28027993191706, 109965636641173
Offset: 0
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Table[Sum[Sum[(-1)^(i + j)*(i + j)!/(2*i!*j!), {i, 1, n}], {j, 1, n}], {n, 0, 30}] (* Vaclav Kotesovec, Apr 03 2019 *)
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{a(n) = sum(i=1, n, sum(j=1, n, (-1)^(i+j)*(i+j)!/(2*i!*j!)))}
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{a(n) = sum(i=2, 2*n, (-1)^i*i!*polcoef(sum(j=1, n, x^j/j!)^2, i))/2} \\ Seiichi Manyama, May 20 2019
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
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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 *)
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{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!))))}
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{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
Showing 1-3 of 3 results.