A001593 -id:A001593 - cp's OEIS frontend

A001594 a(n) = 6^n + n^6.

1, 7, 100, 945, 5392, 23401, 93312, 397585, 1941760, 10609137, 61466176, 364568617, 2179768320, 13065520825, 78371693632, 470196375201, 2821126684672, 16926683582305, 101559990680640, 609359787056377

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-63,161,-245,231,-133,43,-6).

Cf. sequences of the form k^n+n^k: A001580 (k=2), A001585 (k=3), A001589 (k=4), A001593 (k=5), this sequence (k=6), A001596 (k=7), A198401 (k=8), A185277 (k=9), A177068 (k=10), A177069 (k=11).

[6^n+n^6: n in [0..30]]; // Vincenzo Librandi, Oct 27 2011

seq(seq(k^n+n^k, k=6..6), n=0..19); # Zerinvary Lajos, Jun 29 2007

Table[6^n + n^6, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - 6 x + 72 x^2 - 75 x^3 - 1475 x^4 - 1776 x^5 - 334 x^6 - 7 x^7)/((1-x)^7 (1-6 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 28 2014 *)
LinearRecurrence[{13,-63,161,-245,231,-133,43,-6},{1,7,100,945,5392,23401,93312,397585},20] (* Harvey P. Dale, Jan 07 2023 *)

a(n)=6^n+n^6 \\ Charles R Greathouse IV, Feb 14 2011

[6^n+n^6 for n in (0..30)] # Bruno Berselli, Aug 28 2014

G.f.: (1 - 6*x + 72*x^2 - 75*x^3 - 1475*x^4 - 1776*x^5 - 334*x^6 - 7*x^7)/((1-x)^7*(1-6*x)). - Vincenzo Librandi, Aug 28 2014

A001596 a(n) = 7^n + n^7.

Original entry on oeis.org

1, 8, 177, 2530, 18785, 94932, 397585, 1647086, 7861953, 45136576, 292475249, 1996813914, 13877119009, 96951758924, 678328486353, 4747732369318, 33233199005057, 232630924325880, 1628414210130481

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (15,-84,252,-462,546,-420,204,-57,7).

Cf. A001580, A001585, A001589, A001593, A001594.

[7^n+n^7: n in [0..30]]; // Vincenzo Librandi, Oct 27 2011

seq(seq(k^n+n^k, k=7..7), n=0..18); # Zerinvary Lajos, Jun 29 2007

f[n_]:=7^n+n^7;f[Range[0,40]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)
LinearRecurrence[{15,-84,252,-462,546,-420,204,-57,7},{1,8,177,2530,18785,94932,397585,1647086,7861953},20] (* Harvey P. Dale, Sep 16 2018 *)

PARI
```
a(n)=7^n+n^7
```

A055652 Table T(m,k)=m^k+k^m (with 0^0 taken to be 1) as square array read by antidiagonals.

Original entry on oeis.org

2, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 8, 4, 1, 1, 5, 17, 17, 5, 1, 1, 6, 32, 54, 32, 6, 1, 1, 7, 57, 145, 145, 57, 7, 1, 1, 8, 100, 368, 512, 368, 100, 8, 1, 1, 9, 177, 945, 1649, 1649, 945, 177, 9, 1, 1, 10, 320, 2530, 5392, 6250, 5392, 2530, 320, 10, 1, 1, 11, 593, 7073

Offset: 0

Henry Bottomley, Jun 08 2000

Columns and rows are A000012 (apart from first term), A000027, A001580, A001585, A001589, A001593 etc. Diagonals include A013499 (apart from first two terms), A051442, A051489.
Cf. A055651.
Contribution from Franklin T. Adams-Watters, Oct 26 2009: (Start)
Rows/columns: A000027, A001580, A001585, A001589, A001593, A001594, A001596.
Main diagonal is 2 * A000312. More diagonals: A051442, A051489, A155539.
Cf. A076980, A156353, A156354. (End)

E.g.f. Sum(n,m, T(n,m)/(n! m!)) = e^(x e^y) + e^(y e^x). [From Franklin T. Adams-Watters, Oct 26 2009]

A185277 a(n) = n^9 + 9^n.

Original entry on oeis.org

1, 10, 593, 20412, 268705, 2012174, 10609137, 45136576, 177264449, 774840978, 4486784401, 33739007300, 287589316833, 2552470327702, 22897453501745, 205929575454024, 1853088908328577, 16677300287543066, 150094833656289489

Offset: 0

Vladimir Joseph Stephan Orlovsky, Feb 19 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (19,-135,525,-1290,2142,-2478,2010,-1125,415,-91,9).

Cf. sequences of the form k^n+n^k: A001580 (k=2), A001585 (k=3), A001589 (k=4), A001593 (k=5), A001594 (k=6), A001596 (k=7), A198401 (k=8), this sequence (k=9), A177068 (k=10), A177069 (k=11).

[9^n+n^9: n in [0..30]]; // Vincenzo Librandi, Oct 27 2011

Table[9^n + n^9, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - 9 x + 538 x^2 + 9970 x^3 - 43028 x^4 - 638168 x^5 - 1317266 x^6 - 779618 x^7 - 130925 x^8 - 4527 x^9 - 8 x^10)/((1 - x)^10 (1 - 9 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 28 2014 *)
LinearRecurrence[{19,-135,525,-1290,2142,-2478,2010,-1125,415,-91,9},{1,10,593,20412,268705,2012174,10609137,45136576,177264449,774840978,4486784401},20] (* Harvey P. Dale, Jun 08 2023 *)

for(n=0,25, print1(n^9 + 9^n, ", ")) \\ G. C. Greubel, Jun 25 2017

[9^n+n^9 for n in (0..30)] # Bruno Berselli, Aug 28 2014

G.f.: (1 - 9*x + 538*x^2 + 9970*x^3 - 43028*x^4 - 638168*x^5-1317266*x^6 - 779618*x^7 - 130925*x^8 - 4527*x^9 - 8*x^10)/((1-x)^10*(1-9*x)). - Vincenzo Librandi, Aug 28 2014

A177069 11^n + n^11.

Original entry on oeis.org

1, 12, 2169, 178478, 4208945, 48989176, 364568617, 1996813914, 8804293473, 33739007300, 125937424601, 570623341222, 3881436747409, 36314872537968, 383799398752905, 4185897925275026, 45967322049616577, 505481300395601404

Offset: 0

Vincenzo Librandi, May 31 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..950
Index entries for linear recurrences with constant coefficients, signature (23,-198,946,-2915,6237,-9636,10956,-9207,5665,-2486,738,-133,11).

Cf. sequences of the form k^n+n^k: A001580 (k=2), A001585 (k=3), A001589 (k=4), A001593 (k=5), A001594 (k=6), A001596 (k=7), A198401 (k=8), A185277 (k=9), A177068 (k=10), this sequence (k=11).

Magma
```
[11^n+n^11: n in [0..20]]
```

Table[11^n + n^11, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - 11 x + 2091 x^2 + 130021 x^3 + 524976 x^4 -14501046 x^5 - 91394082 x^6 - 163229406 x^7 - 104915271 x^8 - 24085951 x^9 - 1676905 x^10 - 22407 x^11 - 10 x^12)/((1 - x)^12 (1 - 11 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 28 2014 *)

a(n)= 11^n+n^11 \\ Charles R Greathouse IV, Jan 11 2012

[11^n+n^11 for n in (0..30)] # Bruno Berselli, Aug 28 2014

G.f.: (1 - 11*x + 2091*x^2 + 130021*x^3 + 524976*x^4 - 14501046*x^5 - 91394082*x^6 - 163229406*x^7 - 104915271*x^8 - 24085951*x^9 - 1676905*x^10 - 22407*x^11 - 10*x^12) / ((1-x)^12*(1-11*x)). - Vincenzo Librandi, Aug 28 2014

A198401 a(n) = 8^n + n^8.

Original entry on oeis.org

1, 9, 320, 7073, 69632, 423393, 1941760, 7861953, 33554432, 177264449, 1173741824, 8804293473, 69149458432, 550571544609, 4399522300160, 35186934979457, 281479271677952, 2251806789442689, 18014409529442560, 144115205059418913

Offset: 0

Vincenzo Librandi, Oct 27 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (17,-108,372,-798,1134,-1092,708,-297,73,-8).

Cf. A001580, A001585, A001589, A001593, A001594, A001596, A177068, A177069, A185277.

Magma
```
[8^n+n^8: n in [0..20]]
```

f[n_] := 8^n + n^8; f[Range[0, 30]]
LinearRecurrence[{17,-108,372,-798,1134,-1092,708,-297,73,-8},{1,9,320,7073,69632,423393,1941760,7861953,33554432,177264449},30] (* Harvey P. Dale, Aug 26 2023 *)

a(n)=8^n+n^8 \\ Charles R Greathouse IV, Oct 16 2015

G.f.: -(9*x^9 +1966*x^8 +34133*x^7 +120575*x^6 +109459*x^5 +18599*x^4 -2233*x^3 -275*x^2 +8*x -1) / ((x -1)^9*(8*x -1)). - Colin Barker, Sep 13 2013

A193094 Augmentation of the triangular array P=A130296 whose n-th row is (n+1,1,1,1,1...,1) for 0<=k<=n. See Comments.

Original entry on oeis.org

1, 2, 1, 6, 4, 3, 24, 18, 16, 13, 120, 96, 90, 84, 71, 720, 600, 576, 558, 532, 461, 5040, 4320, 4200, 4128, 4050, 3908, 3447, 40320, 35280, 34560, 34200, 33888, 33462, 32540, 29093, 362880, 322560, 317520, 315360, 313800, 312096, 309330, 302436

Offset: 0

Clark Kimberling, Jul 30 2011

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
Regarding W=A193093:
col 1:  A000142, n!
col 2:  A001593, n*n!
col 3:  A130744, n*(n+2)*n!
diag (1,1,3,13,71,...):  A003319, indecomposable permutations.
It appears that T(n,k) is the number of indecomposable permutations p of [n+2] for which p(k+2) = 1. For example, T(2,1) = 4 counts 2413, 3412, 4213, 4312. - David Callan, Aug 27 2014

			First 5 rows:
1
2.....1
6.....4....3
24....18...16...13
120...96...90...84...71

Cf. A193091, A130296, A193093.

p[n_, k_] := If[k == 0, n + 1, 1]
Table[p[n, k], {n, 0, 5}, {k, 0, n}] (* A130296 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]] (* A193094 *)
Flatten[Table[v[n], {n, 0, 9}]]

A309421 Numbers k such that 5^k + k^5 is prime.

Original entry on oeis.org

24, 1036, 104824

Offset: 1

Hugo Pfoertner, Jul 30 2019

a(4) > 125000.

Cf. A001593, A114973, A309422.

cp's OEIS Frontend

A001594 a(n) = 6^n + n^6.

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A001596 a(n) = 7^n + n^7.

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Maple

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A055652 Table T(m,k)=m^k+k^m (with 0^0 taken to be 1) as square array read by antidiagonals.

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A185277 a(n) = n^9 + 9^n.

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Magma

Mathematica

PARI

Sage

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A177069 11^n + n^11.

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Magma

Mathematica

PARI

Sage

Formula

A198401 a(n) = 8^n + n^8.

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Magma

Mathematica

PARI

Formula

A193094 Augmentation of the triangular array P=A130296 whose n-th row is (n+1,1,1,1,1...,1) for 0<=k<=n. See Comments.

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Mathematica

A309421 Numbers k such that 5^k + k^5 is prime.

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