A062806
a(n) = Sum_{i=1..n} i*n^i.
Original entry on oeis.org
1, 10, 102, 1252, 18555, 324726, 6565468, 150652552, 3868151445, 109876543210, 3420886930626, 115835619046764, 4238148192940207, 166614350863692382, 7004068093591806840, 313512663723845890576, 14887093306681068252201, 747445606689889243214802
Offset: 1
A113630
1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6 + 8*n^7 + 9*n^8.
Original entry on oeis.org
1, 45, 4097, 83653, 757305, 4272461, 17736745, 59409477, 169826513, 429794605, 987654321, 2098573445, 4178995657, 7879732173, 14181546905, 24517448581, 40926266145, 66242446637, 104327377633, 160347899205, 241108033241
Offset: 0
a(3) = 1 + 2*3 + 3*3^2 + 4*3^3 + 5*3^4 + 6*3^5 + 7*3^6 + 8*3^7 + 9*3^8 = 83653 is prime.
a(5) = 1 + 2*5 + 3*5^2 + 4*5^3 + 5*5^4 + 6*5^5 + 7*5^6 + 8*5^7 + 9*5^8 = 4272461 is prime.
a(8) = 1 + 2*8 + 3*8^2 + 4*8^3 + 5*8^4 + 6*8^5 + 7*8^6 + 8*8^7 + 9*8^8 = 169826513 is prime.
a(23) = 1 + 2*23 + 3*23^2 + 4*23^3 + 5*23^4 + 6*23^5 + 7*23^6 + 8*23^7 + 9*23^8 = 733113789893 is prime.
- Chai Wah Wu, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
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a113630 n = sum $ zipWith (*) [1..9] $ iterate (* n) 1
-- Reinhard Zumkeller, Nov 22 2014
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[1+2*n+3*n^2+4*n^3+5*n^4+6*n^5+7*n^6+8*n^7+9*n^8: n in [0..20]]; // Vincenzo Librandi, Nov 09 2014
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CoefficientList[Series[(5 x^8 + 1548 x^7 + 31360 x^6 + 129620 x^5 + 148266 x^4 + 48316 x^3 + 3728 x^2 + 36 x + 1) / (1 - x)^9, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 09 2014 *)
With[{c=Total[Table[k n^(k-1),{k,9}]]},Table[c,{n,0,30}]] (* or *) LinearRecurrence[ {9,-36,84,-126,126,-84,36,-9,1},{1,45,4097,83653,757305,4272461,17736745,59409477,169826513},30] (* Harvey P. Dale, Jul 18 2017 *)
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vector(100,n,1 + 2*(n-1)+ 3*(n-1)^2 + 4*(n-1)^3 + 5*(n-1)^4 + 6*(n-1)^5 + 7*(n-1)^6 + 8*(n-1)^7 + 9*(n-1)^8) \\ Derek Orr, Nov 09 2014
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A113630_list, m = [1], [362880, -1229760, 1607760, -1011480, 309816, -40752, 1584, -4, 1]
for _ in range(10**3):
for i in range(8):
m[i+1]+= m[i]
A113630_list.append(m[-1]) # Chai Wah Wu, Nov 09 2014
A368534
a(n) = Sum_{k=1..n} binomial(k+1,2) * n^(n-k).
Original entry on oeis.org
0, 1, 5, 24, 146, 1215, 13431, 186816, 3130436, 61291125, 1371742105, 34522712136, 964626945558, 29621465864627, 991330604373851, 35906022352657920, 1399219698628043016, 58367293868445147657, 2594796705962971336125, 122463905297217627859000
Offset: 0
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Table[Sum[Binomial[k+1,2]n^(n-k),{k,n}],{n,0,20}] (* Harvey P. Dale, May 14 2025 *)
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a(n) = sum(k=1, n, binomial(k+1, 2)*n^(n-k));
Showing 1-3 of 3 results.
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