A352981
a(n) = Sum_{k=0..floor(n/2)} k^n.
Original entry on oeis.org
1, 0, 1, 1, 17, 33, 794, 2316, 72354, 282340, 10874275, 53201625, 2438235715, 14350108521, 762963987380, 5249352196144, 317685943157892, 2502137235710736, 169842891165484965, 1506994510201252425, 113394131858832552133, 1119223325228757961465
Offset: 0
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[(&+[k^n: k in [0..Floor(n/2)]]): n in [0..40]]; // G. C. Greubel, Nov 01 2022
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a[0] = 1; a[n_] := Sum[k^n, {k, 0, Floor[n/2]}]; Array[a, 22, 0] (* Amiram Eldar, Apr 13 2022 *)
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a(n) = sum(k=0, n\2, k^n);
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my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^(2*k)/(1-k*x)))
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[sum( k^n for k in range((n//2)+1)) for n in range(41)] # G. C. Greubel, Nov 01 2022
A352984
a(n) = Sum_{k=0..floor(n/3)} k^(3*n).
Original entry on oeis.org
1, 0, 0, 1, 1, 1, 262145, 2097153, 16777217, 7625731702716, 205892205836474, 5559069156490116, 4722516577573661689554, 302235507459360068466700, 19342922532827596354169130, 28422947373397605556855075614825, 3552792907042781637051562368414979
Offset: 0
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a[0] = 1; a[n_] := Sum[k^(3*n), {k, 0, Floor[n/3]}]; Array[a, 16, 0] (* Amiram Eldar, Apr 13 2022 *)
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a(n) = sum(k=0, n\3, k^(3*n));
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^3*x)^(3*k)/(1-k^3*x)))
A357193
a(n) = n! * Sum_{k=0..floor(n/2)} k^(2*n)/k!.
Original entry on oeis.org
1, 0, 2, 6, 3096, 61560, 65248200, 4058986680, 7506140268480, 1062517243193280, 3052268000677879200, 822543740977513816800, 3395913346775619237617280, 1553795963458841732838848640, 8727392877498334693600263757440
Offset: 0
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a(n) = n!*sum(k=0, n\2, k^(2*n)/k!);
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k^2*x)^(2*k)/(k!*(1-k^2*x)))))
A352985
a(n) = Sum_{k=0..floor(n/2)} k^(2*(n-2*k)).
Original entry on oeis.org
1, 0, 1, 1, 2, 5, 18, 74, 339, 1770, 10915, 79555, 663140, 6109351, 61264436, 669862580, 8044351557, 106331744724, 1536980041573, 24028469781765, 402558463751974, 7195932984364585, 137204787854813174, 2792969599543659326, 60668198155262809815
Offset: 0
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a[0] = 1; a[n_] := Sum[k^(2*(n - 2*k)), {k, 0, Floor[n/2]}]; Array[a, 25, 0] (* Amiram Eldar, Apr 13 2022 *)
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a(n) = sum(k=0, n\2, k^(2*(n-2*k)));
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my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^(2*k)/(1-k^2*x)))
Showing 1-4 of 4 results.