A031972 -id:A031972 - cp's OEIS frontend

A228275 A(n,k) = Sum_{i=1..k} n^i; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 6, 3, 0, 0, 4, 14, 12, 4, 0, 0, 5, 30, 39, 20, 5, 0, 0, 6, 62, 120, 84, 30, 6, 0, 0, 7, 126, 363, 340, 155, 42, 7, 0, 0, 8, 254, 1092, 1364, 780, 258, 56, 8, 0, 0, 9, 510, 3279, 5460, 3905, 1554, 399, 72, 9, 0

Offset: 0

Alois P. Heinz, Aug 19 2013

A(n,k) is the total sum of lengths of longest ending contiguous subsequences with the same value over all s in {1,...,n}^k:
A(4,1) = 4 = 1+1+1+1: [1], [2], [3], [4].
A(1,4) = 4: [1,1,1,1].
A(3,2) = 12 = 2+1+1+1+2+1+1+1+2: [1,1], [1,2], [1,3], [2,1], [2,2], [2,3], [3,1], [3,2], [3,3].
A(2,3) = 14 = 3+1+1+2+2+1+1+3: [1,1,1], [1,1,2], [1,2,1], [1,2,2], [2,1,1], [2,1,2], [2,2,1], [2,2,2].

			Square array A(n,k) begins:
  0, 0,  0,   0,    0,     0,      0,      0, ...
  0, 1,  2,   3,    4,     5,      6,      7, ...
  0, 2,  6,  14,   30,    62,    126,    254, ...
  0, 3, 12,  39,  120,   363,   1092,   3279, ...
  0, 4, 20,  84,  340,  1364,   5460,  21844, ...
  0, 5, 30, 155,  780,  3905,  19530,  97655, ...
  0, 6, 42, 258, 1554,  9330,  55986, 335922, ...
  0, 7, 56, 399, 2800, 19607, 137256, 960799, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000004, A001477, A002378, A027444, A027445, A152031, A228290, A228291, A228292, A228293, A228294.
Rows n=0-11 give: A000004, A001477, A000918(k+1), A029858(k+1), A080674, A104891, A105281, A104896, A052379(k-1), A052386, A105279, A105280.
Main diagonal gives A031972.
Lower diagonal gives A226238.
Cf. A228250.

A:= (n, k)-> `if`(n=1, k, (n/(n-1))*(n^k-1)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

a[0, 0] = 0; a[1, k_] := k; a[n_, k_] := n*(n^k-1)/(n-1); Table[a[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 16 2013 *)

A(1,k) = k, else A(n,k) = n/(n-1)*(n^k-1).
A(n,k) = Sum_{i=1..k} n^i.
A(n,k) = Sum_{i=1..k+1} binomial(k+1,i)*A(n-i,k)*(-1)^(i+1) for n>k, given values A(0,k), A(1,k),..., A(k,k). - Yosu Yurramendi, Sep 03 2013

A031973 a(n) = Sum_{k=0..n} n^k.

Original entry on oeis.org

1, 2, 7, 40, 341, 3906, 55987, 960800, 19173961, 435848050, 11111111111, 313842837672, 9726655034461, 328114698808274, 11966776581370171, 469172025408063616, 19676527011956855057, 878942778254232811938, 41660902667961039785743, 2088331858752553232964200

Offset: 0

N. J. A. Sloane

These are the generalized repunits of length n+1 in base n for all n >= 1: a(n) expressed in base n is 111...111 (n+1 1's): a(1) = 1^0 + 1^1 = 2 = A000042(2), a(2) = 2^0 + 2^1 + 2^2 = 7 = A000225(3), a(3) = 3^0 + 3^1 + 3^2 + 3^3 = 40 = A003462(4), etc., a(10) = 10^0 + 10^1 + 10^2 + ... + 10^9 + 10^10 = 11111111111 = A002275(11), etc. - Rick L. Shepherd, Aug 26 2004
a(n)=the total number of ordered selections of up to n objects from n types with repetitions allowed.  Thus for 2 objects a,b there are 7 possible selections: aa,bb,ab,ba,a,b, and the null set. - J. M. Bergot, Mar 26 2014
a(n)=the total number of ordered arrangements of 0,1,2..n objects, with repetitions allowed, selected from n types of objects. - J. M. Bergot, Apr 11 2014

			a(3) = 3^0 + 3^1 + 3^2 + 3^3 = 40.

Nathaniel Johnston, Table of n, a(n) for n = 0..100

Cf. A000042 (unary representations), A000225 (2^n-1: binary repunits shown in decimal), A003462 ((3^n-1)/2: ternary repunits shown in decimal), A002275 ((10^n-1)/9: decimal repunits).

Cf. A104878.

[&+[n^k: k in [0..n]]: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

a:= proc(n) local c, i; c:=1; for i to n do c:= c*n+1 od; c end:
seq(a(n), n=0..20); # Alois P. Heinz, Aug 15 2013

Join[{1},Table[Total[n^Range[0,n]],{n,20}]] (* Harvey P. Dale, Nov 13 2011 *)

a(n)=(n^(n+1)-1)/(n-1) \\ Charles R Greathouse IV, Mar 26 2014

[lucas_number1(n,n,n-1) for n in range(1, 19)] # Zerinvary Lajos, May 16 2009

a(n) = (n^(n+1)-1)/(n-1) = (A007778(n)-1)/(n-1) = A023037(n)+A000312(n) = A031972(n)+1. - Henry Bottomley, Apr 04 2003
a(n) = A125118(n,n-2) for n>2. - Reinhard Zumkeller, Nov 21 2006
a(n) = [x^n] 1/((1 - x)*(1 - n*x)). - Ilya Gutkovskiy, Oct 04 2017
a(n) = A104878(2n,n). - Alois P. Heinz, May 04 2021

A228273 T(n,k) is the number of s in {1,...,n}^n having longest ending contiguous subsequence with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 18, 6, 3, 0, 192, 48, 12, 4, 0, 2500, 500, 100, 20, 5, 0, 38880, 6480, 1080, 180, 30, 6, 0, 705894, 100842, 14406, 2058, 294, 42, 7, 0, 14680064, 1835008, 229376, 28672, 3584, 448, 56, 8, 0, 344373768, 38263752, 4251528, 472392, 52488, 5832, 648, 72, 9

Offset: 0

Alois P. Heinz, Aug 19 2013

			T(0,0) = 1: [].
T(1,1) = 1: [1].
T(2,1) = 2: [1,2], [2,1].
T(2,2) = 2: [1,1], [2,2].
T(3,1) = 18: [1,1,2], [1,1,3], [1,2,1], [1,2,3], [1,3,1], [1,3,2], [2,1,2], [2,1,3], [2,2,1], [2,2,3], [2,3,1], [2,3,2], [3,1,2], [3,1,3], [3,2,1], [3,2,3], [3,3,1], [3,3,2].
T(3,2) = 6: [1,2,2], [1,3,3], [2,1,1], [2,3,3], [3,1,1], [3,2,2].
T(3,3) = 3: [1,1,1], [2,2,2], [3,3,3].
Triangle T(n,k) begins:
  1;
  0,        1;
  0,        2,       2;
  0,       18,       6,      3;
  0,      192,      48,     12,     4;
  0,     2500,     500,    100,    20,    5;
  0,    38880,    6480,   1080,   180,   30,   6;
  0,   705894,  100842,  14406,  2058,  294,  42,  7;
  0, 14680064, 1835008, 229376, 28672, 3584, 448, 56,  8;

Alois P. Heinz, Rows n = 0..140, flattened

Row sums give: A000312.
Columns k=0-4 give: A000007, A066274(n) = 2*A081131(n) for n>1, A053506(n) for n>2, A055865(n-1) = A085389(n-1) for n>3, A085390(n-1) for n>4.
Main diagonal gives: A028310.
Lower diagonals include (offsets may differ): A002378, A045991, A085537, A085538, A085539.
Cf. A228154, A228617.

T:= (n, k)-> `if`(n=0 and k=0, 1, `if`(k<1 or k>n, 0,
             `if`(k=n, n, (n-1)*n^(n-k)))):
seq(seq(T(n,k), k=0..n), n=0..12);

f[0,0]=1;
f[n_,k_]:=Which[1<=k<=n-1,n^(n-k)*(n-1),k<1,0,k==n,n,k>n,0];
Table[Table[f[n,k],{k,0,n}],{n,0,10}]//Grid (* Geoffrey Critzer, May 19 2014 *)

T(0,0) = 1, else T(n,k) = 0 for k<1 or k>n, else T(n,n) = n, else T(n,k) = (n-1)*n^(n-k).
Sum_{k=0..n}   T(n,k) = A000312(n).
Sum_{k=0..n} k*T(n,k) = A031972(n).

A068475 a(n) = Sum_{m=0..n} m*n^(m-1).

Original entry on oeis.org

0, 1, 5, 34, 313, 3711, 54121, 937924, 18831569, 429794605, 10987654321, 310989720966, 9652968253897, 326011399456939, 11901025061692313, 466937872906120456, 19594541482740368161, 875711370981239308953, 41524755927216069067489, 2082225625247428808306410

Offset: 0

Francois Jooste (phukraut(AT)hotmail.com), Mar 10 2002

The closed form comes from taking the derivative of the closed form of A031972, for which each term of this sequence is a derivative. - Jonas Whidden, Oct 18 2011

			a(2) = Sum_{m = 1..2} m*2^(m-1) = 1 + 2*2 = 5.

Reinhard Zumkeller, Table of n, a(n) for n = 0..250

Derivative sequence of A031972.

Cf. A023037, A062805, A062806, A368534.

a068475 n = sum $ zipWith (*) [1..n] $ iterate (* n) 1
-- Reinhard Zumkeller, Nov 22 2014

[0] cat [(&+[m*n^(m-1): m in [0..n]]): n in [1..30]]; // G. C. Greubel, Oct 13 2018

Maple
```
a := n->sum(m*n^(m-1),m=1..n);
```

Join[{0}, Table[Sum[m*n^(m-1), {m,0,n}], {n,1,30}]] (* G. C. Greubel, Oct 13 2018 *)

for(n=0,30, print1(if(n==0, 0, sum(m=0,n, m*n^(m-1))), ", ")) \\ G. C. Greubel, Oct 13 2018

a(1) = 1. For n > 1, a(n) = ((n-1)*(n+1)*n^n - n^(n+1) + 1)/(n-1)^2. - Jonas Whidden, Oct 18 2011
a(n) = A062806(n) / n for n>=1. - Reinhard Zumkeller, Nov 22 2014
a(n) = [x^(n-1)] 1/((1 - x)*(1 - n*x)^2). - Peter Bala, Feb 12 2024

A117667 a(n) = n^n-n^(n-1)-n^(n-2)-n^(n-3)-...-n^3-n^2-n.

Original entry on oeis.org

1, 2, 15, 172, 2345, 37326, 686287, 14380472, 338992929, 8888888890, 256780503551, 8105545862052, 277635514376233, 10257237069745862, 406615755353655135, 17216961135462248176, 775537745518440716417

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 11 2006

nonn

			a(3) = 3^3-3^2-3 = 27-9-3 = 15.

Christian N. K. Anderson, Table of n, a(n) for n = 1..100

Cf. A000312 (n^n), A023037 (1+n+n^2+...n^(n-1)),

A001923, A031972.

a:=n->n^n-sum(n^j,j=1..n-1): seq(a(n),n=1..19); # Emeric Deutsch, Apr 16 2006

s[n_] := Sum[n^i, {i, 1, n - 1}]; Table[n^n - s[n], {n, 17}] (* Carlos Eduardo Olivieri, Apr 14 2015 *)
f[n_] := ((n - 2) n^n + n)/(n - 1); f[1] = 1; Array[f, 18] (* Robert G. Wilson v, Apr 15 2015 *)

a(n) = A000312(n) - A023037(n) + 1. - Michel Marcus, Apr 14 2015

A191690(n)+1. - Robert G. Wilson v, Apr 16 2015

A332652 a(n) = Sum_{k=1..n} n^(k/gcd(n, k)).

Original entry on oeis.org

1, 4, 15, 76, 785, 7836, 137263, 2130976, 47895489, 1010012140, 28531167071, 743044702104, 25239592216033, 797785008119932, 31147773583464735, 1157442765678719056, 51702516367896047777, 2185932446984222457444, 109912203092239643840239, 5255987282125826560192520

Offset: 1

Ilya Gutkovskiy, Feb 18 2020

nonn

Cf. A031972, A056665, A057661, A226561, A228640, A332620, A332621, A332653, A332655.

[&+[n^(k div Gcd(n,k)):k in [1..n]]:n in [1..21]]; // Marius A. Burtea, Feb 18 2020

Table[Sum[n^(k/GCD[n, k]), {k, 1, n}], {n, 1, 20}]
Table[Sum[Sum[If[GCD[k, d] == 1, n^k, 0], {k, 1, d}], {d, Divisors[n]}], {n, 1, 20}]

a(n) = Sum_{k=1..n} n^(lcm(n, k)/n).
a(n) = Sum_{d|n} Sum_{k=1..d, gcd(k, d) = 1} n^k.
a(n) = n * A332653(n).

A229896 Sizes of logical groups of the same integer in A229895.

Original entry on oeis.org

1, 1, 4, 1, 5, 27, 1, 7, 37, 256, 1, 9, 61, 369, 3125, 1, 11, 91, 671, 4651, 46656, 1, 13, 127, 1105, 9031, 70993, 823543, 1, 15, 169, 1695, 15961, 144495, 1273609, 16777216, 1, 17, 217, 2465, 26281, 269297, 2685817, 26269505, 387420489, 1, 19, 271, 3439

Offset: 1

Carl R. White, Oct 03 2013

The two ones at the start of the parent sequence represent parent and child 1-tuples in the grandparent sequence [(1) and (2) respectively], hence this sequence also starts with 1, 1 rather than 2, which would otherwise be a more sensible way to describe the pair of ones.
All other elements are effectively run-lengths of strings of the same integer in A229895.
The first occurrence of an integer, n, in the parent sequence, is the first of a run of n^n elements of value n. For later occurrences, the run length is n^k-(n-1)^k where k is the size of the k-tuple in the grandparent sequence, A229873.
The elements can be arranged into a triangle thus:
.... 1
.... 1, 4
.... 1, 5, 27
.... 1, 7, 37, 256
.... 1, 9, 61, 369, 3125
.... etc.
where the n-th line is:
.... n^1-(n-1)^1, n^2-(n-1)^2, ..., n^(k-1)-(n-1)^(k-1), n^n; 1 <= k < n
The first terms, for sufficiently large n simplifying as:
.... 1, 2n-1, 3n^2-3n+1, 4n^3-6n^2+4n-1, etc.
Row sums are first differences of A031972, and thus the cumulative sum of rows at the end of each row is A031972 itself, i.e., n*(n^n - 1)/(n-1).

Carl R. White, Rows 1..44 for triangle, flattened

Cf. A229873, A229895, A031972.

T := proc (n, k) if k < n then n^k-(n-1)^k elif k = n then n^n else end if end proc: for n to 12 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jan 30 2017

/* GNU bc */ for(n=1;n<=10;n++)for(p=1;p<=n;p++){if(p==n){t=n^n}else{t=n^p-(n-1)^p};print t,","};print "...\n"

A088055 a(n) = n!*n^n - ((n^(n+1)-1)/(n-1) - 1) for n>1 with a(1)=0.

Original entry on oeis.org

0, 2, 123, 5804, 371095, 33536334, 4149695921, 676438175160, 140586711200271, 36287988888888890, 11388728579602327129, 4270826370748686175140, 1886009588224061851054127, 968725766842917544760889030

Offset: 1

Amarnath Murthy, Sep 20 2003

nonn

Original definition: a(n) = G(n) - A(n), where G(n) = Sum of the first n terms of a geometric progression with first term n and common ratio n. A(n) = Product of first n terms of an arithmetic progression with first term n and common difference n.

Cf. A031972, A061711.

seq(`if`(n=1, 0, n!*n^n - ((n^(n+1)-1)/(n-1) - 1)),n=1..16); # Georg Fischer, Dec 09 2022

a(n) = if (n==1, 0, n!*n^n - ((n^(n+1)-1)/(n-1) - 1)); \\ Michel Marcus, Dec 10 2022

a(n) = A061711(n) - A031972(n) for n>1 with a(1)=0.

Corrected and extended by David Wasserman, Jun 27 2005
Edited by M. F. Hasler, Feb 12 2013
Formula negated by Georg Fischer, Dec 09 2022

A264748 a(n) = Sum_{k = 1..n} (k^n - n^k).

Original entry on oeis.org

0, -1, -3, 14, 520, 11185, 239505, 5510652, 138456936, 3803230815, 113833152565, 3695302326650, 129479186068128, 4874312730972685, 196306448145080385, 8425000059348756472, 383956514250037779376, 18521535576956405481147, 942952190208348285876501

Offset: 1

Ilya Gutkovskiy, Nov 23 2015

sign

			a(1) = 1^1 - 1^1 = 0;
a(2) = 1^2 - 2^1 + 2^2 - 2^2 = -1;
a(3) = 1^3 - 3^1 + 2^3 - 3^2 + 3^3 - 3^3 = -3;
a(4) = 1^4 - 4^1 + 2^4 - 4^2 + 3^4 - 4^3 + 4^4 - 4^4 = 14;
a(5) = 1^5 - 5^1 + 2^5 - 5^2 + 3^5 - 5^3 + 4^5 - 5^4 + 5^5 - 5^5 = 520, etc.

Eric Weisstein's World of Mathematics, Riemann Zeta Function
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function

Cf. A031971, A031972.

Table[Sum[k^n - n^k, {k, 1, n}], {n, 1, 20}]
Join[{0}, Table[HarmonicNumber[n, -n] - n (n^n - 1)/(n - 1), {n, 2, 20}]]

a(n) = sum(k=1, n, k^n - n^k); \\ Altug Alkan, Nov 23 2015

a(n) =  A031971(n) - A031972(n).
a(n) = ((1 - n)*zeta(-n, n + 1) - n*(n^n - 1) + (n - 1)*zeta(-n))/(n - 1) for n>1, where zeta(s) is the Riemann zeta function and zeta(s, a) is the Hurwitz zeta function.
a(n) ~ n^n / (exp(1) - 1). - Vaclav Kotesovec, Jul 16 2025

cp's OEIS Frontend

A228275 A(n,k) = Sum_{i=1..k} n^i; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A031973 a(n) = Sum_{k=0..n} n^k.

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A228273 T(n,k) is the number of s in {1,...,n}^n having longest ending contiguous subsequence with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A068475 a(n) = Sum_{m=0..n} m*n^(m-1).

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A117667 a(n) = n^n-n^(n-1)-n^(n-2)-n^(n-3)-...-n^3-n^2-n.

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A332652 a(n) = Sum_{k=1..n} n^(k/gcd(n, k)).

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A229896 Sizes of logical groups of the same integer in A229895.

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