A014918 -id:A014918 - cp's OEIS frontend

A014917 a(1)=1, a(n) = n*5^(n-1) + a(n-1).

1, 11, 86, 586, 3711, 22461, 131836, 756836, 4272461, 23803711, 131225586, 717163086, 3890991211, 20980834961, 112533569336, 600814819336, 3194808959961, 16927719116211, 89406967163086, 470876693725586, 2473592758178711, 12964010238647461, 67800283432006836, 353902578353881836

Offset: 1

Olivier Gérard

From Gary Detlefs, Aug 31 2021 (Start)
This is the x=5 member of the x-family of sequences with member a(x, n) = x^n*Sum_{k=1..n} S(x, k), with S(x, k) = Sum_{j=1..k} 1/x^j.
S(x, k) = (x^k - 1)/((x-1)*x^k) = (1/x^k)*Sum_{j=0..k-1} x^j, and a(x,n) = ((n*(x-1) - 1)*x^n + 1)/(x-1)^2.
The sequence {x^k*S(x, k)} with recurrence signature (x+1, -x) leads to sequence {a(x, n)} with recurrence signature (2*x+1, -x*(x+2), x^2). (End) [Rewritten by Wolfdieter Lang, Nov 30 2021]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See p. 16.
Index entries for linear recurrences with constant coefficients, signature (11,-35,25).

Cf. A000351, A003463, A014915, A014916, A014917, A014918, A014920, A014921, A014923.

I:=[1, 11]; [n le 2 select I[n] else 10*Self(n-1)-25*Self(n-2)+ 1: n in [1..30]]; // Vincenzo Librandi, Oct 23 2012

CoefficientList[Series[1/((1 - x)(1 - 5 x)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 23 2012 *)
LinearRecurrence[{11,-35,25},{1,11,86},20] (* Harvey P. Dale, May 06 2013 *)

From Vincenzo Librandi, Oct 23 2012: (Start)
a(n) = 10*a(n-1) - 25*a(n-2) + 1; a(1)=1, a(2)=11.
G.f.: x/((1-x)*(1-5*x)^2). (End)
a(n) = 11*a(n-1) - 35*a(n-2) + 25*a(n-3); a(1)=1, a(2)=11, a(3)=86. - Harvey P. Dale, May 06 2013
a(n) = 5^n*Sum_{k=1..n} (Sum_{j=1..k} 1/x^j) = ((4*n - 1)*5^n + 1)/4^2. See the general comment above, and the first formula. - Gary Detlefs, Aug 31 2021 [Edited by Wolfdieter Lang, Nov 30 2021]
E.g.f.: exp(x)*(1 + exp(4*x)*(20*x - 1))/16. - Elmo R. Oliveira, May 24 2025

A059045 Square array T(n,k) read by antidiagonals where T(0,k) = 0 and T(n,k) = 1 + 2k + 3k^2 + ... + n*k^(n-1).

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 5, 1, 0, 1, 10, 17, 7, 1, 0, 1, 15, 49, 34, 9, 1, 0, 1, 21, 129, 142, 57, 11, 1, 0, 1, 28, 321, 547, 313, 86, 13, 1, 0, 1, 36, 769, 2005, 1593, 586, 121, 15, 1, 0, 1, 45, 1793, 7108, 7737, 3711, 985, 162, 17, 1, 0, 1, 55, 4097, 24604, 36409

Offset: 0

Henry Bottomley, Dec 18 2000

			   0,   0,   0,    0,     0,      0,      0,      0,       0, ...
   1,   1,   1,    1,     1,      1,      1,      1,       1, ...
   1,   3,   5,    7,     9,     11,     13,     15,      17, ...
   1,   6,  17,   34,    57,     86,    121,    162,     209, ...
   1,  10,  49,  142,   313,    586,    985,   1534,    2257, ...
   1,  15, 129,  547,  1593,   3711,   7465,  13539,   22737, ...
   1,  21, 321, 2005,  7737,  22461,  54121, 114381,  219345, ...
   1,  28, 769, 7108, 36409, 131836, 380713, 937924, 2054353, ...

Cf. A000004, A000012, A005408, A056109, A056578, A056579 (rows 1-6).

Cf. A057427, A000217, A000337, A014915, A014916, A014917, A014918, A014920 (columns 1-7).

A059045 := proc(n,k)
    if k = 1 then
        n*(n+1) /2 ;
    else
        (1+n*k^(n+1)-k^n*(n+1))/(k-1)^2 ;
    end if;
end proc: # R. J. Mathar, Mar 29 2013

T(n,k) = n*k^(n-1)+T(n-1, k) = (n*k^(n+1)-(n+1)*k^n+1)/(k-1)^2.

cp's OEIS Frontend

A014917 a(1)=1, a(n) = n*5^(n-1) + a(n-1).

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