A349852
Expansion of Sum_{k>=0} k * x^k/(1 + k * x).
Original entry on oeis.org
0, 1, 1, 0, 2, 1, -5, 20, -28, -47, 525, -2056, 3902, 9633, -129033, 664364, -1837904, -2388687, 67004697, -478198544, 1994889946, -1669470783, -56929813933, 615188040196, -3794477505572, 12028579019537, 50780206473221, -1172949397924184, 10766410530764118
Offset: 0
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a[n_] := -Sum[(-k)^(n - k + 1), {k, 0, n}]; Array[a, 29, 0] (* Amiram Eldar, Dec 02 2021 *)
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a(n, s=1, t=1) = sum(k=0, n, (-k^t)^(n-k)*k^s);
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my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k*x^k/(1+k*x))))
A349859
Expansion of Sum_{k>=0} k * x^k/(1 + k^2 * x).
Original entry on oeis.org
0, 1, 1, -4, 8, 57, -771, 4168, 22504, -943255, 13349105, -48905364, -3284972544, 117005181841, -2071095992867, -2153841382168, 1791029813641424, -79511121545541327, 1867021096818903489, 8850156160819959100, -3592582141732300745000
Offset: 0
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a[n_] := Sum[If[k == n - k == 0, 1, (-k^2)^(n-k)] * k, {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Dec 03 2021 *)
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a(n, s=1, t=2) = sum(k=0, n, (-k^t)^(n-k)*k^s);
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my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k*x^k/(1+k^2*x))))
A349861
Expansion of Sum_{k>=0} k * x^k/(1 + k^4 * x).
Original entry on oeis.org
0, 1, 1, -28, 272, 10473, -1204227, 61879504, 5542428184, -2801375692615, 597270865802225, -6353098642040604, -85053828910331125224, 62048537484671306803057, -23357096658814809538526243, -10072546328972154349642665952
Offset: 0
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a[n_] := Sum[If[k == n - k == 0, 1, (-k^4)^(n-k)] * k, {k, 0, n}]; Array[a, 16, 0] (* Amiram Eldar, Dec 03 2021 *)
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a(n, s=1, t=4) = sum(k=0, n, (-k^t)^(n-k)*k^s);
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my(N=20, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k*x^k/(1+k^4*x))))
Showing 1-3 of 3 results.