A351340
a(n) = Sum_{k=0..n} n^k * k^(n-k).
Original entry on oeis.org
1, 1, 6, 48, 516, 6955, 112686, 2132634, 46167560, 1125116901, 30481672610, 908760877244, 29565986232396, 1042354163621927, 39584173937284438, 1610922147768721590, 69940319175066857488, 3226793787576474492657, 157649292247463953189578
Offset: 0
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a[0] = 1; a[n_] := Sum[n^k * k^(n - k), {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Feb 08 2022 *)
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a(n) = sum(k=0, n, n^k*k^(n-k));
A101753
Numbers m such that Sum_{k=0..m} m^k is prime.
Original entry on oeis.org
1, 2, 6, 126, 8598
Offset: 1
Mark Hudson (mrmarkhudson(AT)hotmail.com), Dec 15 2004
The number 6 is in this sequence because 6^0 + 6^1 + 6^2 + 6^3 + 6^4 + 6^5 + 6^6 = (6^7 - 1)/5 = 55987 is prime.
Cf.
A088856 (primes p such that cyclotomic(p,p-1) is prime).
A349961
a(n) = Sum_{k=0..n} (2*n)^k.
Original entry on oeis.org
1, 3, 21, 259, 4681, 111111, 3257437, 113522235, 4581298449, 210027483919, 10778947368421, 612142982430915, 38108188628928601, 2580398988131886039, 188802050194014479853, 14843696896551724137931, 1247923426698972051309601, 111713733654631566667971615
Offset: 0
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a[0] = 1; a[n_] := Sum[(2*n)^k, {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Dec 07 2021 *)
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a(n) = sum(k=0, n, (2*n)^k);
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a(n) = ((2*n)^(n+1)-1)/(2*n-1);
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);
A364870
Array read by ascending antidiagonals: A(n, k) = (n + k)^n, with k >= 0.
Original entry on oeis.org
1, 1, 1, 4, 2, 1, 27, 9, 3, 1, 256, 64, 16, 4, 1, 3125, 625, 125, 25, 5, 1, 46656, 7776, 1296, 216, 36, 6, 1, 823543, 117649, 16807, 2401, 343, 49, 7, 1, 16777216, 2097152, 262144, 32768, 4096, 512, 64, 8, 1, 387420489, 43046721, 4782969, 531441, 59049, 6561, 729, 81, 9, 1
Offset: 0
The array begins:
1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, ...
4, 9, 16, 25, 36, 49, ...
27, 64, 125, 216, 343, 512, ...
256, 625, 1296, 2401, 4096, 6561, ...
3125, 7776, 16807, 32768, 59049, 100000, ...
...
Cf.
A000012 (n=0),
A000169,
A000272,
A000312 (k=0),
A007830 (k=3),
A008785 (k=4),
A008786 (k=5),
A008787 (k=6),
A031973 (antidiagonal sums),
A052746 (2nd superdiagonal),
A052750,
A062971 (main diagonal),
A079901 (read by descending antidiagonals),
A085527 (1st superdiagonal),
A085528 (1st subdiagonal),
A085532,
A099753.
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A[n_,k_]:=(n+k)^n; Join[{1},Table[A[n-k,k],{n,9},{k,0,n}]]//Flatten
Comments