A000312 -id:A000312 - cp's OEIS frontend

A128433 Triangle, read by rows, T(n,k) = numerator of the maximum of the k-th Bernstein polynomial of degree n; denominator is A128434.

1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 27, 3, 27, 1, 1, 256, 216, 216, 256, 1, 1, 3125, 80, 5, 80, 3125, 1, 1, 46656, 37500, 34560, 34560, 37500, 46656, 1, 1, 823543, 5103, 590625, 35, 590625, 5103, 823543, 1, 1, 16777216, 13176688, 1792, 11200000, 11200000, 1792, 13176688, 16777216, 1

Offset: 0

Reinhard Zumkeller, Mar 03 2007

			Triangle begins as:
  1;
  1,        1;
  1,        1,        1;
  1,        4,        4,      1;
  1,       27,        3,     27,        1;
  1,      256,      216,    216,      256,        1;
  1,     3125,       80,      5,       80,     3125,     1;
  1,    46656,    37500,  34560,    34560,    37500, 46656,        1;
  1,   823543,     5103, 590625,       35,   590625,  5103,   823543,        1;
  1, 16777216, 13176688,   1792, 11200000, 11200000,  1792, 13176688, 16777216, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Bernstein Polynomial

Cf. A000169, A000312, A036505, A090878, A128434.

B[n_, k_]:= If[k==0 || k==n, 1, Binomial[n, k]*k^k*(n-k)^(n-k)/n^n];
T[n_, k_]= Numerator[B[n, k]];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 19 2021 *)

def B(n,k): return 1 if (k==0 or k==n) else binomial(n, k)*k^k*(n-k)^(n-k)/n^n
def T(n,k): return numerator(B(n,k))
flatten([[T(n,k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jul 19 2021

T(n,k)/A128434(n,k) = Binomial(n,k) * k^k * (n-k)^(n-k) / n^n.
For n>0: Sum_{k=0..n} T(n,k)/A128434(n,k) = A090878(n)/A036505(n-1).
T(n,n-k) = T(n,k).
T(n,0) = 1.
for n>0: T(n,1)/A128434(n,1) = A000312(n-1)/A000169(n).

A128434 Triangle, read by rows, T(n,k) = denominator of the maximum of the k-th Bernstein polynomial of degree n; numerator is A128433.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 9, 9, 1, 1, 64, 8, 64, 1, 1, 625, 625, 625, 625, 1, 1, 7776, 243, 16, 243, 7776, 1, 1, 117649, 117649, 117649, 117649, 117649, 117649, 1, 1, 2097152, 16384, 2097152, 128, 2097152, 16384, 2097152, 1, 1, 43046721, 43046721, 6561, 43046721, 43046721, 6561, 43046721, 43046721, 1

Offset: 0

Reinhard Zumkeller, Mar 03 2007

			Triangle begins as:
  1;
  1,       1;
  1,       2,      1;
  1,       9,      9,       1;
  1,      64,      8,      64,      1;
  1,     625,    625,     625,    625,       1;
  1,    7776     243,      16,    243,    7776,      1;
  1,  117649, 117649,  117649, 117649,  117649, 117649,       1;
  1, 2097152,  16384, 2097152,    128, 2097152,  16384, 2097152, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Bernstein Polynomial

Cf. A000169, A000312, A036505, A090878, A128433.

B[n_, k_]:= If[k==0 || k==n, 1, Binomial[n, k]*k^k*(n-k)^(n-k)/n^n];
T[n_, k_]= Denominator[B[n, k]];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 19 2021 *)

def B(n,k): return 1 if (k==0 or k==n) else binomial(n, k)*k^k*(n-k)^(n-k)/n^n
def T(n,k): return denominator(B(n,k))
flatten([[T(n,k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jul 19 2021

A128433(n,k)/T(n,k) = binomial(n,k) * k^k * (n-k)^(n-k) / n^n.
For n>0: Sum_{k=0..n} A128433(n,k)/T(n,k) = A090878(n)/A036505(n-1);
T(n, n-k) = T(n,k).
T(n, 0) = T(n, n) = 1.
for n>0: A128433(n,1)/T(n,1) = A000312(n-1)/A000169(n).

A246609 Number T(n,k) of endofunctions on [n] whose cycle lengths are multiples of k; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 4, 1, 0, 27, 6, 2, 0, 256, 57, 24, 6, 0, 3125, 680, 300, 120, 24, 0, 46656, 9945, 4480, 2160, 720, 120, 0, 823543, 172032, 78750, 41160, 17640, 5040, 720, 0, 16777216, 3438673, 1591296, 866460, 430080, 161280, 40320, 5040

Offset: 0

Alois P. Heinz, Aug 31 2014

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k <= n. T(0,k) = 1, T(n,k) = 0 for k > n and n > 0.

Column k > 1 is asymptotic to n^(n - 1/2 + 1/(2*k)) * sqrt(2*Pi) / (2^(1/(2*k)) * k^(1/k) * Gamma(1/(2*k))) * (1 - (3*k-1)*(k-1) * sqrt(2/n) * Gamma(1/(2*k)) / (12 * k^2 * Gamma(1/2+1/(2*k)))). - Vaclav Kotesovec, Sep 01 2014

			Triangle T(n,k) begins:
  1;
  0,      1;
  0,      4,      1;
  0,     27,      6,     2;
  0,    256,     57,    24,     6;
  0,   3125,    680,   300,   120,    24;
  0,  46656,   9945,  4480,  2160,   720,  120;
  0, 823543, 172032, 78750, 41160, 17640, 5040, 720;
  ...

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

Columns k=0-10 give: A000007, A000312, A060435, A246610, A246611, A246612, A246613, A246614, A246615, A246616, A246617.
Main diagonal gives A000142(n-1) for n > 0.
T(2n,n) gives A246618.

with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i=0 or i>n, 0, add(b(n-i*j, i+k, k)*(i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
T:= (n, k)->add(b(j, k$2)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(seq(T(n,k), k=0..n), n=0..10);

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i == 0 || i > n, 0, Sum[b[n-i*j, i+k, k]*(i-1)!^j*multinomial[n, {n-i*j, Sequence @@ Table[i, {j}]}]/j!, {j, 0, n/i}]]]; T[0, 0] = 1; T[n_, k_] := Sum[b[j, k, k]*n^(n-j)*Binomial[n-1, j-1], {j, 0, n}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 06 2015, after Alois P. Heinz *)

E.g.f. for column k > 0: 1 / (1 - (-1)^k * LambertW(-x)^k)^(1/k). - Vaclav Kotesovec, Sep 01 2014

A008786 a(n) = (n+5)^n.

Original entry on oeis.org

1, 6, 49, 512, 6561, 100000, 1771561, 35831808, 815730721, 20661046784, 576650390625, 17592186044416, 582622237229761, 20822964865671168, 799006685782884121, 32768000000000000000, 1430568690241985328321, 66249952919459433152512, 3244150909895248285300369

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000169, A000272, A000312, A007778, A007830, A008785, this sequence, A008787, A008788, A008789, A008790, A008791.

List([0..20], n-> (n+5)^n); # G. C. Greubel, Sep 11 2019

[(n+5)^n: n in [0..20]]; // Vincenzo Librandi, Jun 11 2013

Table[(n+5)^n,{n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)

vector(20, n, (n+4)^(n-1)) \\ G. C. Greubel, Sep 11 2019

[(n+5)^n for n in (0..20)] # G. C. Greubel, Sep 11 2019

E.g.f.(x) for b(n) = n^(n-5) = a(n-5): T - (15/16)*T^2 + (85/216)T^3 - (25/288)*T^4 + (1/120)*T^5, where T=T(x) is Euler's tree function. - Len Smiley, Nov 17 2001
E.g.f.: LambertW(-x)^5/((-x)^5*(1+LambertW(-x))). - Vladeta Jovovic, Nov 07 2003
E.g.f.: (1/4)*d/dx((LambertW(-x)/(-x))^4). - Wolfdieter Lang, Oct 25 2022

A008787 a(n) = (n + 6)^n.

Original entry on oeis.org

1, 7, 64, 729, 10000, 161051, 2985984, 62748517, 1475789056, 38443359375, 1099511627776, 34271896307633, 1156831381426176, 42052983462257059, 1638400000000000000, 68122318582951682301, 3011361496339065143296

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000169, A000272, A000312, A007778, A007830, A008785, A008786, this sequence, A008788, A008789, A008790, A008791.

List([0..20], n-> (n+6)^n); # G. C. Greubel, Sep 11 2019

[(n+6)^n: n in [0..20]]; // Vincenzo Librandi, Jun 11 2013

Maple
```
a:= n-> (n+6)^n: seq(a(n), n=0..20);
```

Table[(n+6)^n,{n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)

vector(20, n, (n+5)^(n-1)) \\ G. C. Greubel, Sep 11 2019

[(n+6)^n for n in (0..20)] # G. C. Greubel, Sep 11 2019

E.g.f.(x) for b(n) = n^(n-6) = a(n-6): T - (31/32)*T^2 + (575/1296)*T^3 - (415/3456)*T^4 + (137/7200)*T^5 - (1/720)*T^6; where T=T(x) is Euler's tree function (see A000169). - Len Smiley, Nov 17 2001
E.g.f.: LambertW(-x)^6/(x^6*(1+LambertW(-x))). - Vladeta Jovovic, Nov 07 2003
E.g.f.: (1/5)*d/dx(LambertW(-x)/(-x))^5. - Wolfdieter Lang, Oct 25 2022

A008790 a(n) = n^(n+4).

Original entry on oeis.org

0, 1, 64, 2187, 65536, 1953125, 60466176, 1977326743, 68719476736, 2541865828329, 100000000000000, 4177248169415651, 184884258895036416, 8650415919381337933, 426878854210636742656, 22168378200531005859375

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Cf. A000169, A000272, A000312, A007778, A007830, A008785, A008786, A008787, A008788, A008789, A008791.

List([0..20], n-> n^(n+4)); # G. C. Greubel, Sep 11 2019

[n^(n+4): n in [0..20]]; // Vincenzo Librandi, Jun 11 2013

a:=n->mul(n,k=-3..n):seq(a(n),n=0..20); # Zerinvary Lajos, Jan 26 2008

Table[n^(n+4),{n,0,20}](* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)

vector(20, n, (n-1)^(n+3)) \\ G. C. Greubel, Sep 11 2019

[n^(n+4) for n in (0..20)] # G. C. Greubel, Sep 11 2019

E.g.f.: T*(1 +22*T +58*T^2 +24*T^3)*(1-T)^(-9); where T is Euler's tree function (see A000169). - Len Smiley, Nov 17 2001
See A008517 and A134991 for similar e.g.f.s and diagonals of A048993. - Tom Copeland, Oct 03 2011
E.g.f.: d^4/dx^4 {x^4/(T(x)^4*(1-T(x)))}, where T(x) = Sum_{n>=1} n^(n-1)*x^n/n! is the tree function of A000169. - Peter Bala, Aug 05 2012

A056788 a(n) = n^n + (n-1)^(n-1).

Original entry on oeis.org

2, 5, 31, 283, 3381, 49781, 870199, 17600759, 404197705, 10387420489, 295311670611, 9201412118867, 311791207040509, 11414881932150269, 449005897206417391, 18884637964090410991, 845687005960046315793, 40173648337182874339601, 2017766063735610126699403

Offset: 1

Walter Nissen, Aug 20 2000

nonn

For even n > 1, the absolute value of the discriminant of the polynomial x^n+x-1. [Corrected by Artur Jasinski, May 07 2010]

The largest known prime in this sequence is a(4) = 283.

			a(3) = 2^2 + 3^3 = 4 + 27 = 31.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see equation (6.7).

Pierre Letouzey, Generalized Hofstadter functions G, H and beyond: numeration systems and discrepancy, arXiv:2502.12615 [cs.DM], 2025. See p. 18.
Walter Nissen, post on np ( n ) = n^n + (n+1)^(n+1), on home page "Up for the count!". (Updated Oct 02 2012)

Cf. A000312 (n^n), A086797 (discriminant of the polynomial x^n-x-1).

Cf. A056187, A056790, A192397 (smallest & largest prime factor of a(n), records of the latter), A217435 = bigomega(a(n)).

Join[{2}, Table[n^n+(n-1)^(n-1), {n, 2, 20}]] (* T. D. Noe, Aug 13 2004 *)
Join[{2},Total/@Partition[Table[n^n,{n,20}],2,1]] (* Harvey P. Dale, Jun 26 2017 *)

A056788(n)=n^n+(n-1)^(n-1)  \\ M. F. Hasler, Oct 02 2012

Minor corrections by M. F. Hasler, Oct 02 2012

A057075 Table read by antidiagonals of T(n,k)=floor(n^n/k) with n,k >= 1.

Original entry on oeis.org

1, 0, 4, 0, 2, 27, 0, 1, 13, 256, 0, 1, 9, 128, 3125, 0, 0, 6, 85, 1562, 46656, 0, 0, 5, 64, 1041, 23328, 823543, 0, 0, 4, 51, 781, 15552, 411771, 16777216, 0, 0, 3, 42, 625, 11664, 274514, 8388608, 387420489, 0, 0, 3, 36, 520, 9331, 205885, 5592405, 193710244, 10000000000

Offset: 1

Henry Bottomley, Jul 31 2000

			From _Seiichi Manyama_, Aug 12 2023: (Start)
Square array begins:
      1,     0,     0,     0,    0,    0, ...
      4,     2,     1,     1,    0,    0, ...
     27,    13,     9,     6,    5,    4, ...
    256,   128,    85,    64,   51,   42, ...
   3125,  1562,  1041,   781,  625,  520, ...
  46656, 23328, 15552, 11664, 9331, 7776, ... (End)

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Rows are: A000007 (essentially), A033324, A033347, A057066-A057074.
Columns include A000312 and A057065.
Leading diagonal is A000169.
Cf. A060155.

T(n, k) = n^n\k; \\ Seiichi Manyama, Aug 12 2023

A070896 Determinant of the Cayley addition table of Z_{n}.

Original entry on oeis.org

0, -1, -9, 96, 1250, -19440, -352947, 7340032, 172186884, -4500000000, -129687123005, 4086546038784, 139788510734886, -5159146026151936, -204350482177734375, 8646911284551352320, 389289535005334947848, -18580248257778920521728

Offset: 1

Santi Spadaro, May 23 2002

sign

a(n) is the determinant of the n X n matrix M_(i,j) = ((i+j) mod n) where i and j range from 0 to n-1. - Benoit Cloitre, Nov 29 2002

|a(n)| = number of labeled mappings from n points to themselves (endofunctions) with an even number of cycles. E.g.f.: (1/2)*LambertW(-x)^2/(1+LambertW(-x)). - Vladeta Jovovic, Mar 30 2006

			a(3) = -9 because the determinant of {{0,1,2}, {1,2,0}, {2,0,1}} is -9.

G. C. Greubel, Table of n, a(n) for n = 1..385

Cf. A000312, A052182, A060281, A060435.

[(-1)^Floor(n/2)*(1/2)*(n-1)*n^(n-1): n in [1..50]]; // G. C. Greubel, Nov 14 2017

Table[(-1)^Floor[n/2]*(1/2)*(n - 1)*n^(n - 1), {n, 1, 50}] (* G. C. Greubel, Nov 14 2017 *)

a(n)=(-1)^floor(n/2)*(1/2)*(n-1)*n^(n-1)

a(n) = (-1)^floor(n/2)*(1/2)*(n-1)*n^(n-1). - Benoit Cloitre, Nov 29 2002

A133018 Partition number of n, raised to power n.

Original entry on oeis.org

1, 1, 4, 27, 625, 16807, 1771561, 170859375, 54875873536, 19683000000000, 17080198121677824, 16985107389382393856, 43439888521963583647921, 113809328043328941786781301, 667840509835890864312744140625, 4816039244598889571670527496421376

Offset: 0

Omar E. Pol, Oct 31 2007

nonn

			a(6)=1771561 because the partition number of 6 is 11 and 11^6=1771561.

Cf. A000312, A058694, A062457, A133032, A259373, A265094. Partition numbers: A000041.

A000041 := proc(n) combinat[numbpart](n) ; end: A133018 := proc(n) A000041(n)^n ; end: seq(A133018(n),n=0..18) ; # R. J. Mathar, Jan 13 2008

Table[PartitionsP[n]^n,{n,0,15}] (* James C. McMahon, Mar 10 2025 *)

a(n) = A000041(n)^n.

a(n) ~ exp(1/24 - 3/(4*Pi^2) - (72+Pi^2)*sqrt(n)/(24*sqrt(6)*Pi) + sqrt(2/3)*Pi*n^(3/2)) / (3^(n/2) * 4^n * n^n). - Vaclav Kotesovec, Jun 23 2015

More terms from R. J. Mathar, Jan 13 2008

a(15) from James C. McMahon, Mar 10 2025

cp's OEIS Frontend

A128433 Triangle, read by rows, T(n,k) = numerator of the maximum of the k-th Bernstein polynomial of degree n; denominator is A128434.

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A128434 Triangle, read by rows, T(n,k) = denominator of the maximum of the k-th Bernstein polynomial of degree n; numerator is A128433.

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A246609 Number T(n,k) of endofunctions on [n] whose cycle lengths are multiples of k; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

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A008786 a(n) = (n+5)^n.

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A008787 a(n) = (n + 6)^n.

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A008790 a(n) = n^(n+4).

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A056788 a(n) = n^n + (n-1)^(n-1).

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