A133298
a(n) = 1 + Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} i^(j+k).
Original entry on oeis.org
2, 41, 1727, 130917, 17245160, 3546873073, 1046002784253, 417182980579609, 215861313302976046, 140463714074395109081, 112191246261394235358555, 107867952671976721983260413, 122856922623618324408724634164
Offset: 1
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List([1..20], n-> 1 + n^2 + Sum([2..n], j-> (j*(j^n-1)/(j-1))^2) ); # G. C. Greubel, Aug 02 2019
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[2] cat [1+n^2 + (&+[(j*(j^n-1)/(j-1))^2: j in [2..n]]): n in [1..20]]; // G. C. Greubel, Aug 02 2019
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Table[Sum[(i(i^n-1)/(i-1))^2, {i,2,n}] +n^2 +1,{n,20}]
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vector(20, n, 1+n^2 + sum(j=2,n, (j*(j^n-1)/(j-1))^2)) \\ G. C. Greubel, Aug 02 2019
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[1+n^2 + sum((j*(j^n-1)/(j-1))^2 for j in (2..n)) for n in (1..20)] # G. C. Greubel, Aug 02 2019
A349226
Triangle read by rows. Row n gives the coefficients of Product_{k=0..n} (x - k^k) expanded in decreasing powers of x, with row 0 = {1}.
Original entry on oeis.org
1, 1, -1, 1, -2, 1, 1, -6, 9, -4, 1, -33, 171, -247, 108, 1, -289, 8619, -44023, 63340, -27648, 1, -3413, 911744, -26978398, 137635215, -197965148, 86400000, 1, -50070, 160195328, -42565306462, 1258841772303, -6421706556188, 9236348345088, -4031078400000
Offset: 0
The triangle begins:
1;
1, -1;
1, -2, 1;
1, -6, 9, -4;
1, -33, 171, -247, 108;
1, -289, 8619, -44023, 63340, -27648;
1, -3413, 911744, -26978398, 137635215, -197965148, 86400000;
...
Row 4: x^4-33*x^3+171*x^2-247*x+108 = (x-1)*(x-1^1)*(x-2^2)*(x-3^3).
Cf.
A008276 (The Stirling numbers of the first kind in reverse order).
Cf.
A039758 (Coefficients for polynomials with roots in odd numbers).
Cf.
A355540 (Coefficients for polynomials with roots in factorials).
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T(n, k) = polcoeff(prod(m=0, n-1, (x-m^m)), n-k);
A353018
a(n) = Sum_{k=0..floor(n/3)} (n-3*k)^(n-3*k).
Original entry on oeis.org
1, 1, 4, 28, 257, 3129, 46684, 823800, 16780345, 387467173, 10000823800, 285328450956, 8916487915429, 302885107416053, 11112292154008972, 437902806868774804, 18447046958816967669, 827251374178490773149, 39346845978103406350228
Offset: 0
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a[n_] := Sum[If[3*k == n, 1, (n - 3*k)^(n - 3*k)], {k, 0, Floor[n/3]}]; Array[a, 20, 0] (* Amiram Eldar, Apr 16 2022 *)
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a(n) = sum(k=0, n\3, (n-3*k)^(n-3*k));
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k)/(1-x^3))
A068983
a(n) = Sum_{k=0..n} (k^k-k!).
Original entry on oeis.org
0, 2, 23, 255, 3260, 49196, 867699, 17604595, 404662204, 10401033404, 295672787215, 9211294233871, 312080173805324, 11423999821072140, 449316582527563515, 18896039733447227131, 846135945932355895308, 40192537618855187742732, 2018612071634068368034711
Offset: 1
a(4) = 255 because (1^1-1!)+(2^2-2!)+(3^3-3!)+(4^4-4!) = 255.
A349928
a(n) = Sum_{k=0..n} (k+n)^k.
Original entry on oeis.org
1, 3, 20, 246, 4481, 107129, 3157836, 110504876, 4473749677, 205615442135, 10574135574388, 601527803412298, 37500537926181449, 2542321872054610333, 186209553386691383388, 14653121207168215024624, 1232879877057607865696085, 110444572988776439826640683
Offset: 0
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a[0] = 1; a[n_] := Sum[(k + n)^k, {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Dec 05 2021 *)
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a(n) = sum(k=0, n, (k+n)^k);
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