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))))
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))))
A349863
Expansion of Sum_{k>=0} k^2 * x^k/(1 + k^2 * x).
Original entry on oeis.org
0, 1, 3, -6, -2, 243, -2031, 3796, 187212, -3860139, 36467311, 284357502, -21796446486, 538332144295, -5605176351651, -182065102478856, 12963817679287960, -422751776737348503, 5483284328996107803, 327213964461103956802, -30082452646697648945898
Offset: 0
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a[n_] := Sum[If[k == n - k == 0, 1, (-k^2)^(n - k)] * k^2, {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Dec 03 2021 *)
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a(n, s=2, 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^2*x^k/(1+k^2*x))))
Showing 1-4 of 4 results.