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))))
A349855
Expansion of Sum_{k>=0} k^4 * x^k/(1 + k * x).
Original entry on oeis.org
0, 1, 15, 50, 76, 203, 335, -84, 2696, -3011, -8433, 130606, -662348, 1840439, 2391823, -67000872, 478203152, -1994884455, 1669477263, 56929821514, -615188031396, 3794477515715, -12028579007921, -50780206459996, 1172949397939160, -10766410530747243
Offset: 0
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a[n_] := Sum[(-k)^(n - k + 4), {k, 0, n}]; Array[a, 26, 0] (* Amiram Eldar, Dec 02 2021 *)
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a(n, s=4, 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^4*x^k/(1+k*x))))
A349853
Expansion of Sum_{k>=0} k^2 * x^k/(1 + k * x).
Original entry on oeis.org
0, 1, 3, 2, 4, 11, -13, 36, 56, -515, 2067, -3890, -9620, 129047, -664349, 1837920, 2388704, -67004679, 478198563, -1994889926, 1669470804, 56929813955, -615188040173, 3794477505596, -12028579019512, -50780206473195, 1172949397924211, -10766410530764090
Offset: 0
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a[n_] := Sum[(-k)^(n - k + 2), {k, 0, n}]; Array[a, 28, 0] (* Amiram Eldar, Dec 02 2021 *)
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a(n, s=2, 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^2*x^k/(1+k*x))))
A349878
Expansion of Sum_{k>=0} k^3 * x^k/(1 - k * x).
Original entry on oeis.org
0, 1, 9, 44, 178, 689, 2723, 11304, 49772, 232657, 1151781, 6018628, 33087022, 190780001, 1150653679, 7241710656, 47454745496, 323154695841, 2282779990113, 16700904488284, 126356632389834, 987303454928465, 7957133905608283, 66071772829246808
Offset: 0
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Table[Sum[k^(n - k + 3), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 04 2021 *)
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a(n, s=3, t=1) = sum(k=0, n, k^(t*(n-k)+s));
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my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k^3*x^k/(1-k*x))))
Showing 1-4 of 4 results.