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))))
A349854
Expansion of Sum_{k>=0} k^3 * x^k/(1 + k * x).
Original entry on oeis.org
0, 1, 7, 12, 14, 49, 13, 8, 596, -1967, 4011, 9764, -128878, 664545, -1837695, -2388448, 67004968, -478198239, 1994890287, -1669470404, -56929813514, 615188040657, -3794477505067, 12028579020088, 50780206473820, -1172949397923535, 10766410530764819, -61183127006113196
Offset: 0
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a[n_] := -Sum[(-k)^(n - k + 3), {k, 0, n}]; Array[a, 28, 0] (* Amiram Eldar, Dec 02 2021 *)
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a(n, s=3, 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^3*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))))
Showing 1-3 of 3 results.