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))))
A349860
Expansion of Sum_{k>=0} k * x^k/(1 + k^3 * x).
Original entry on oeis.org
0, 1, 1, -12, 50, 913, -35093, 557048, 15098348, -1975843727, 103722820509, -396969408196, -704487440688562, 106081534307130081, -7995154887120369801, -510404434212739199104, 340061796037700870259064, -75820293495741408235957599, 8142099722101269515604494937
Offset: 0
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a[n_] := Sum[If[k == n - k == 0, 1, (-k^3)^(n-k)] * k, {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Dec 03 2021 *)
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a(n, s=1, t=3) = 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^3*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))))
A349884
Expansion of Sum_{k>=0} (k * x)^k/(1 + k^2 * x).
Original entry on oeis.org
1, 1, 3, 12, 76, 961, 15407, 221528, 3260936, 80774113, 2462081967, 50963779604, 922244742292, 61063845514113, 2868669700179871, 2019727494212912, -47889136910252848, 461395118866593115713, 5781219348638565771423, -2108738296748190078596084
Offset: 0
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a[n_] := Sum[If[k == 2*n - k == 0, 1, (-1)^(n - k) * k^(2*n - k)], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Dec 04 2021 *)
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a(n, t=2) = sum(k=0, n, (-k^t)^(n-k)*k^k);
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1+k^2*x)))
Showing 1-4 of 4 results.