A334190
a(n) = exp(1/2) * Sum_{k>=0} (2*k + 1)^n / ((-2)^k * k!).
Original entry on oeis.org
1, 0, -2, -4, 4, 64, 248, 48, -6512, -51200, -171296, 830400, 17870400, 144684032, 441316224, -5976726784, -119879356160, -1123892297728, -3962230563328, 70410917051392, 1686366492509184, 19578100126072832, 101728414306826240, -1258662784047370240, -42727186269262737408
Offset: 0
-
nmax = 24; CoefficientList[Series[1/(1 - x) Sum[(-x/(1 - x))^k/Product[(1 - 2 j x/(1 - x)), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 24; CoefficientList[Series[Exp[x + (1 - Exp[2 x])/2], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] * 2^k * BellB[k, -1/2], {k, 0, n}], {n, 0, 24}] (* Vaclav Kotesovec, Apr 18 2020 *)
A334192
Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = exp(1/k) * Sum_{j>=0} (k*j + 1)^n / ((-k)^j * j!).
Original entry on oeis.org
1, 1, 0, 1, 0, -1, 1, 0, -2, -1, 1, 0, -3, -4, 2, 1, 0, -4, -9, 4, 9, 1, 0, -5, -16, 0, 64, 9, 1, 0, -6, -25, -16, 189, 248, -50, 1, 0, -7, -36, -50, 384, 1377, 48, -267, 1, 0, -8, -49, -108, 625, 4416, 4374, -6512, -413, 1, 0, -9, -64, -196, 864, 10625, 26368, -26001, -51200, 2180
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 0, 0, 0, 0, 0, ...
-1, -2, -3, -4, -5, -6, ...
-1, -4, -9, -16, -25, -36, ...
2, 4, 0, -16, -50, -108, ...
9, 64, 189, 384, 625, 864, ...
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Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[(-x/(1 - x))^j/Product[(1 - k i x/(1 - x)), {i, 1, j}], {j, 0, n}], {x, 0, n}]][m - n + 1], {m, 0, 10}, {n, 0, m}] // Flatten
Table[Function[k, n! SeriesCoefficient[Exp[x + (1 - Exp[k x])/k], {x, 0, n}]][m - n + 1], {m, 0, 10}, {n, 0, m}] // Flatten
A334193
a(0) = 1; thereafter a(n) = exp(1/n) * Sum_{k>=0} (n*k + 1)^n / ((-n)^k * k!).
Original entry on oeis.org
1, 0, -2, -9, -16, 625, 21384, 571438, 13471744, 188661555, -9794500000, -1476328587789, -134710712340480, -10664210861777200, -744650964057237888, -37832162051689453125, 831929248561267474432, 725944099523076464203157, 167435684777981700601449984
Offset: 0
-
Table[SeriesCoefficient[1/(1 - x) Sum[(-x/(1 - x))^k/Product[(1 - n j x/(1 - x)), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[n! SeriesCoefficient[Exp[x + (1 - Exp[n x])/n], {x, 0, n}], {n, 1, 18}]]
Join[{1}, Table[Sum[Binomial[n, k]*n^k*BellB[k, -1/n], {k, 0, n}], {n, 1, 18}]] (* Vaclav Kotesovec, Apr 18 2020 *)
Showing 1-3 of 3 results.