A000169 -id:A000169 - cp's OEIS frontend

A369197 Number of labeled connected loop-graphs with n vertices, none isolated, and at most n edges.

1, 1, 3, 13, 95, 972, 12732, 202751, 3795864, 81609030, 1980107840, 53497226337, 1592294308992, 51758060711792, 1824081614046720, 69272000503031475, 2819906639193992192, 122488526636380368714, 5654657850859704139776, 276462849597009068108405, 14270030377126199463936000

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

			The a(0) = 0 through a(3) = 13 loop-graphs (loops shown as singletons):
  .  {{1}}  {{1,2}}      {{1,2},{1,3}}
            {{1},{1,2}}  {{1,2},{2,3}}
            {{2},{1,2}}  {{1,3},{2,3}}
                         {{1},{1,2},{1,3}}
                         {{1},{1,2},{2,3}}
                         {{1},{1,3},{2,3}}
                         {{2},{1,2},{1,3}}
                         {{2},{1,2},{2,3}}
                         {{2},{1,3},{2,3}}
                         {{3},{1,2},{1,3}}
                         {{3},{1,2},{2,3}}
                         {{3},{1,3},{2,3}}
                         {{1,2},{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

The minimal case is A000272.
Connected case of A066383 and A369196, loopless A369192 and A369193.
The loopless case is A129271, connected case of A369191.
The case of equality is A368951, connected case of A368597.
This is the connected case of A369194.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts (simple) graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A062740 counts connected loop-graphs.
A322661 counts covering loop-graphs, unlabeled A322700.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000169, A000666, A001429, A005703, A006649, A057500, A116508, A140638, A143543, A367863, A369145.

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(log(1/(1-t))/2 + 3*t/2 - 3*t^2/4 + 1 - x))} \\ Andrew Howroyd, Feb 02 2024

Logarithmic transform of A368927.
From Andrew Howroyd, Feb 02 2024: (Start)
a(n) = A000169(n) + A129271(n).
E.g.f.: log(1/(1-T(x)))/2 + 3*T(x)/2 - 3*T(x)^2/4 + 1 - x, where T(x) is the e.g.f. of A000169. (End)

a(0) changed to 1 and a(7) onwards from Andrew Howroyd, Feb 02 2024

A051129 Table T(n,k) = k^n read by upwards antidiagonals (n >= 1, k >= 1).

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 8, 9, 4, 1, 16, 27, 16, 5, 1, 32, 81, 64, 25, 6, 1, 64, 243, 256, 125, 36, 7, 1, 128, 729, 1024, 625, 216, 49, 8, 1, 256, 2187, 4096, 3125, 1296, 343, 64, 9, 1, 512, 6561, 16384, 15625, 7776, 2401, 512, 81, 10, 1, 1024, 19683, 65536, 78125, 46656, 16807, 4096, 729, 100, 11

Offset: 1

N. J. A. Sloane

(n-th term) = (n-th term of A002260)^(n-th term of A004736). Both A002260 and A004736 are related to A002024. - Robert A. Stump (bee_ess107(AT)yahoo.com), Aug 29 2002

			  1   2       3       4       5       6       7
  1   4       9      16      25      36      49
  1   8      27      64     125     216     343
  1  16      81     256     625    1296    2401
  1  32     243    1024    3125    7776   16807
  1  64     729    4096   15625   46656  117649
  1 128    2187   16384   78125  279936  823543

T. D. Noe, Rows n = 1..50 of triangle, flattened

Cf. A051128 (transposed), A003992 (transposed), A004248.
Cf. A002024, A002260, A004736.
Cf. A002260, A003101 (antidiagonal sums), A000169 (central terms), A003320 (row maxima), A247358 (sorted rows).

a051129 n k = k ^ (n - k)
a051129_row n = a051129_tabl !! (n-1)
a051129_tabl = zipWith (zipWith (^)) a002260_tabl $ map reverse a002260_tabl
-- Reinhard Zumkeller, Sep 14 2014

T:= (n, k)-> k^n:
seq(seq(T(1+d-k, k), k=1..d), d=1..11);  # Alois P. Heinz, Apr 18 2020

Table[ k^(n-k+1), {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 30 2012 *)

b(n) = floor(1/2 + sqrt(2 * n));
vector(100, n, (n - b(n) * (b(n) - 1) / 2)^(b(n) * (b(n) + 1) / 2 - n + 1)) \\ Altug Alkan, Dec 09 2015

a(n) = (n - b(n) * (b(n) - 1) / 2)^(b(n) * (b(n) + 1) / 2 - n + 1), where b(n) = [ 1/2 + sqrt(2 * n) ]. (b(n) is the n-th term of A002024.) - Robert A. Stump (bee_ess107(AT)yahoo.com), Aug 29 2002

More terms from James Sellers, Dec 11 1999

A055897 a(n) = n*(n-1)^(n-1).

Original entry on oeis.org

1, 2, 12, 108, 1280, 18750, 326592, 6588344, 150994944, 3874204890, 110000000000, 3423740047332, 115909305827328, 4240251492291542, 166680102383370240, 7006302246093750000, 313594649253062377472, 14890324713954061755186, 747581753430634213933056

Offset: 1

Christian G. Bower, Jun 12 2000

nonn

Total number of leaves in all labeled rooted trees with n nodes.
Number of endofunctions of [n] such that no element of [n-1] is fixed. E.g., a(3)=12: 123 -> 331, 332, 333, 311, 312, 313, 231, 232, 233, 211, 212, 213.
Number of functions f: {1, 2, ..., n} --> {1, 2, ..., n} such that f(1) != f(2), f(2) != f(3), ..., f(n-1) != f(n). - Warut Roonguthai, May 06 2006
Determinant of the n X n matrix ((2n, n^2, 0, ..., 0), (1, 2n, n^2, 0, ..., 0), (0, 1, 2n, n^2, 0, ..., 0), ..., (0, ..., 0, 1, 2n)). - Michel Lagneau, May 04 2010
For n > 1: a(n) = A240993(n-1) / A240993(n-2). - Reinhard Zumkeller, Aug 31 2014
Total number of points m such that f^(-1)(m) = {m}, (i.e., the preimage of m is the singleton set {m}) summed over all functions f:[n]->[n]. - Geoffrey Critzer, Jan 20 2022

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Frank Ellermann, Illustration of binomial transforms
Index entries for sequences related to rooted trees

Cf. A003227, A003228, A055314, A055540, A055541, A060226, A118537.

Cf. A240993. Diagonal of A265583.

List([1..20], n-> n*(n-1)^(n-1)); # G. C. Greubel, Aug 10 2019

a055897 n = n * (n - 1) ^ (n - 1) -- Reinhard Zumkeller, Aug 31 2014

[n*(n-1)^(n-1): n in [1..20]] // Wesley Ivan Hurt, Jun 26 2014

A055897:=n->`if`(n=1,1,n*(n-1)^(n-1)); seq(A055897(n), n=1..20); # Wesley Ivan Hurt, Jun 26 2014

Join[{1},Table[n(n-1)^(n-1), {n,2,20}]] (* Harvey P. Dale, Jul 18 2011 *)

{a(n)=polcoeff(1/(1-n*x+x*O(x^n))^2, n)} \\ Paul D. Hanna, Dec 27 2012

[n*(n-1)^(n-1) for n in (1..20)] # G. C. Greubel, Aug 10 2019

E.g.f.: x/(1-T), where T=T(x) is Euler's tree function (see A000169).
a(n) = Sum_{k=1..n} A055302(n, k)*k.
a(n) = the n-th term of the (n-1)-th binomial transform of {1, 1, 4, 18, 96, ..., (n-1)*(n-1)!, ...} (cf. A001563). - Paul D. Hanna, Nov 17 2003
a(n) = (n-1)^(n-1) + Sum_{i=2..n} (n-1)^(n-i)*binomial(n-1, i-1)*(i-1) *(i-1)!. - Paul D. Hanna, Nov 17 2003
a(n) = [x^(n-1)] 1/(1 - (n-1)*x)^2. - Paul D. Hanna, Dec 27 2012
a(n) ~ exp(-1) * n^n. - Vaclav Kotesovec, Nov 14 2014

Additional comments from Vladeta Jovovic, Mar 31 2001 and Len Smiley, Dec 11 2001

A081131 a(n) = n^(n-2) * binomial(n,2).

Original entry on oeis.org

0, 0, 1, 9, 96, 1250, 19440, 352947, 7340032, 172186884, 4500000000, 129687123005, 4086546038784, 139788510734886, 5159146026151936, 204350482177734375, 8646911284551352320, 389289535005334947848, 18580248257778920521728, 937146152681201173795569

Offset: 0

Paul Barry, Mar 08 2003

Main diagonal of A081130.
a(n) is the number of partial functions f: {1,2,...,n} -> {1,2,...,n} that have exactly 2 undefined elements. - Geoffrey Critzer, Feb 08 2012
a(n+1) is the determinant of the circulant matrix having (n-1, n-2, ..., 0) as first row, for n >= 1. See A070896 for a variant, and A303260 for a related sequence. - M. F. Hasler, Apr 23 2018
a(n) is the number of birooted labeled trees on n nodes. - Brendan McKay, May 01 2018

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

Cf. A070896, A081129, A081130, A303260.

Sequences of the form (n+m)^n*binomial(n+2,2): A081133 (m=0), A081132 (m=1), this sequence (m=2), A053507 (m=3), A081196 (m=4).

[n lt 2 select 0 else n^(n-2)*Binomial(n,2): n in [0..20]]; // G. C. Greubel, May 18 2021

Join[{0},Table[n^(n-2) Binomial[n, 2], {n, 1, 20}]] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)

[0 if (n<2) else n^(n-2)*binomial(n,2) for n in (0..20)] # G. C. Greubel, May 18 2021

a(0) = a(1) = 0, a(n) = n^(n-2)*binomial(n,2).

E.g.f.: T(x)^2/(2*(1-T(x))) where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, Feb 08 2012

A275549 Number of classes of endofunctions of [n] under reversal.

Original entry on oeis.org

1, 1, 3, 18, 136, 1625, 23436, 412972, 8390656, 193739769, 5000050000, 142656721086, 4458051717120, 151437584670385, 5556003465485760, 218946946471875000, 9223372039002259456, 413620131002462320337, 19673204037747448432896, 989209827833222327690890

Offset: 0

Olivier Gérard, Aug 01 2016

f and g are in the same class if function g(i) = f(n+1-i) for all i.
Decomposition by class size
.
n    1      2
---------------
1    1      0
2    2      1
3    9      9
4    16     120
5    125    1500
6    216    23220
7    2401   410571
.
Demonstration for the formula: the classes are either of size 1 or 2.
The classes of size 1 is for functions invariant by reversal. They are specified by half their values, including one more if n is odd. Their number is n^(ceiling(n/2)).
So the number of classes under this symmetry is half (the number of functions + the number of classes of size 1).
a(n) is the number of unoriented length n strings with a maximum of n colors. - Andrew Howroyd, Sep 13 2019

Andrew Howroyd, Table of n, a(n) for n = 0..200

Main diagonal of A277504.
Cf. A000312 All endofunctions
Cf. A000169 Classes under translation mod n
Cf. A001700 Classes under sort
Cf. A056665 Classes under rotation
Cf. A168658 Classes under complement to n+1
Cf. A130293 Classes under translation and rotation
Cf. A081721 Classes under rotation and reversal
Cf. A275550 Classes under reversal and complement
Cf. A275551 Classes under translation and reversal
Cf. A275552 Classes under translation and complement
Cf. A275553 Classes under translation, complement and reversal
Cf. A275554 Classes under translation, rotation and complement
Cf. A275555 Classes under translation, rotation and reversal
Cf. A275556 Classes under translation, rotation, complement and reversal
Cf. A275557 Classes under rotation and complement
Cf. A275558 Classes under rotation, complement and reversal
Cf. A078707 Endofunctions symmetric around their middle (stable by reversal).

a(n) = {(n^n + n^((n+1)\2))/2} \\ Andrew Howroyd, Sep 13 2019

a(n) = (n^n+n^ceiling(n/2))/2.

A369194 Number of labeled loop-graphs covering n vertices with at most n edges.

Original entry on oeis.org

1, 1, 4, 23, 199, 2313, 34015, 606407, 12712643, 306407645, 8346154699, 253476928293, 8490863621050, 310937199521774, 12356288017546937, 529516578044589407, 24339848939829286381, 1194495870124420574751, 62332449791125883072149, 3446265450868329833016605

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

Row-sums of left portion of A369199.

			The a(0) = 1 through a(3) = 23 loop-graphs (loops shown as singletons):
  {}  {{1}}  {{1,2}}      {{1},{2,3}}
             {{1},{2}}    {{2},{1,3}}
             {{1},{1,2}}  {{3},{1,2}}
             {{2},{1,2}}  {{1,2},{1,3}}
                          {{1,2},{2,3}}
                          {{1},{2},{3}}
                          {{1,3},{2,3}}
                          {{1},{2},{1,3}}
                          {{1},{2},{2,3}}
                          {{1},{3},{1,2}}
                          {{1},{3},{2,3}}
                          {{2},{3},{1,2}}
                          {{2},{3},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1},{1,3},{2,3}}
                          {{2},{1,2},{1,3}}
                          {{2},{1,2},{2,3}}
                          {{2},{1,3},{2,3}}
                          {{3},{1,2},{1,3}}
                          {{3},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1,2},{1,3},{2,3}}

The minimal case is A001862, without loops A053530.
This is the covering case of A066383 and A369196, cf. A369192 and A369193.
The case of equality is A368597, without loops A367863.
The version without loops is A369191.
The connected case is A369197, without loops A129271.
The unlabeled version is A370169, equality A368599, non-covering A368598.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts simple graphs; also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A133686 counts choosable graphs, covering A367869.
A322661 counts covering loop-graphs, unlabeled A322700.
A367867 counts non-choosable graphs, covering A367868.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000169, A000272, A000666, A003465, A005703, A006649, A057500, A062740, A116508, A367862, A367916.

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]], Length[Union@@#]==n&&Length[#]<=n&]],{n,0,5}]

Inverse binomial transform of A369196.

A003025 Number of n-node labeled acyclic digraphs with 1 out-point.

Original entry on oeis.org

1, 2, 15, 316, 16885, 2174586, 654313415, 450179768312, 696979588034313, 2398044825254021110, 18151895792052235541515, 299782788128536523836784628, 10727139906233315197412684689421

Offset: 1

N. J. A. Sloane

nonn

From Gus Wiseman, Jan 02 2024: (Start)
Also the number of n-element sets of finite nonempty subsets of {1..n}, including a unique singleton, such that there is exactly one way to choose a different element from each. For example, the a(0) = 0 through a(3) = 15 set-systems are:
  .  {{1}}  {{1},{1,2}}  {{1},{1,2},{1,3}}
            {{2},{1,2}}  {{1},{1,2},{2,3}}
                         {{1},{1,3},{2,3}}
                         {{2},{1,2},{1,3}}
                         {{2},{1,2},{2,3}}
                         {{2},{1,3},{2,3}}
                         {{3},{1,2},{1,3}}
                         {{3},{1,2},{2,3}}
                         {{3},{1,3},{2,3}}
                         {{1},{1,2},{1,2,3}}
                         {{1},{1,3},{1,2,3}}
                         {{2},{1,2},{1,2,3}}
                         {{2},{2,3},{1,2,3}}
                         {{3},{1,3},{1,2,3}}
                         {{3},{2,3},{1,2,3}}
These set-systems are all connected.
The case of labeled graphs is A000169.
(End)

			a(2) = 2: o-->--o (2 ways)
a(3) = 15: o-->--o-->--o (6 ways) and
o ... o o-->--o
.\ . / . \ . /
. v v ... v v
.. o ..... o
(3 ways) (6 ways)

R. W. Robinson, Counting labeled acyclic digraphs, pp. 239-273 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..50
Marcel et al., Is there a formula for the number of st-dags (DAG with 1 source and 1 sink) with n vertices?, MathOverflow, 2021.

A diagonal of A058876.
Row sums of A350487.
The unlabeled version is A350415.
Column k=1 of A361718.
Cf. A003026, A003028, A345258.
For any number of sinks we have A003024, unlabeled A003087.
For n-1 sinks we have A058877.
For a fixed sink we have A134531 (up to sign), column k=1 of A368602.
Cf. A000169, A058891, A059201, A082402, A088957, A323818, A334282, A367904, A368600, A368601.

Table[Length[Select[Subsets[Subsets[Range[n]],{n}],Count[#,{}]==1&&Length[Select[Tuples[#],UnsameQ@@#&]]==1&]],{n,0,4}] (* _Gus Wiseman, Jan 02 2024 *)

\\ requires A058876.
my(T=A058876(20)); vector(#T, n, T[n][1]) \\ Andrew Howroyd, Dec 27 2021

\\ see Marcel et al. link (formula for a').
seq(n)={my(a=vector(n)); a[1]=1; for(n=2, #a, a[n]=sum(k=1, n-1, (-1)^(k-1) * binomial(n,k) * (2^(n-k)-1)^k * a[n-k])); a} \\ Andrew Howroyd, Jan 01 2022

a(n) = (-1)^(n-1) * n * A134531(n). - Gus Wiseman, Jan 02 2024

More terms from Vladeta Jovovic, Apr 10 2001

A052752 a(n) = (3*n+1)^(n-1).

Original entry on oeis.org

1, 1, 7, 100, 2197, 65536, 2476099, 113379904, 6103515625, 377801998336, 26439622160671, 2064377754059776, 177917621779460413, 16777216000000000000, 1718264124282290785243, 189937030341242876870656, 22539340290692258087863249, 2857942574656970690381479936

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..333
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 708
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.

Cf. A000169, A052750, A052774, A052782, A000272, A001711.

[(3*n+1)^(n-1): n in [0..50]]; // G. C. Greubel, Nov 16 2017

spec := [S,{B=Prod(S,S,S,Z),S=Set(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Table[(3n+1)^(n-1),{n,0,20}] (* Harvey P. Dale, Aug 14 2015 *)
With[{nmax = 50}, CoefficientList[Series[Exp[-LambertW[-3*x]/3], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 16 2017 *)

for(n=0,50, print1((3*n+1)^(n-1), ", ")) \\ G. C. Greubel, Nov 16 2017

x='x+O('x^50); Vec(serlaplace(exp(-lambertw(-3*x)/3))) \\ G. C. Greubel, Nov 16 2017

E.g.f.: exp(-(1/3)*LambertW(-3*x)).
From Peter Bala, Dec 19 2013: (Start)
The e.g.f. A(x) = 1 + x + 7*x^2/2! + 100*x^3/3! + 2197*x^4/4! + ... satisfies:
1) A(x*exp(-3*x)) = exp(x) = 1/A(-x*exp(3*x));
2) A^3(x) = 1/x*series reversion(x*exp(-3*x));
3) A(x^3) = 1/x*series reversion(x*exp(-x^3));
4) A(x) = exp(x*A(x)^3);
5) A(x) = 1/A(-x*A(x)^6). (End)
E.g.f.: (-LambertW(-3*x)/(3*x))^(1/3). - Vaclav Kotesovec, Dec 07 2014
Related to A001711 by Sum_{n >= 1} a(n)*x^n/n! = series reversion( 1/(1 + x)^3*log(1 + x) ) = series reversion(x - 7*x^2/2! + 47*x^3/3! - 342*x^4/4! + ...). Cf. A000272, A052750. - Peter Bala, Jun 15 2016
a(n) = Sum_{k=1..n} (-1)^(n-k)*(2n+k)^(n-1)*binomial(n,k-1), a(0)=1. - Vladimir Kruchinin, Aug 14 2025

Better description from Vladeta Jovovic, Sep 02 2003

A060072 a(n) = (n^(n-1) - 1)/(n-1) for n>1, a(1) = 0.

Original entry on oeis.org

0, 1, 4, 21, 156, 1555, 19608, 299593, 5380840, 111111111, 2593742460, 67546215517, 1941507093540, 61054982558011, 2085209001813616, 76861433640456465, 3041324492229179280, 128583032925805678351, 5784852794328402307380, 275941052631578947368421

Offset: 1

Henry Bottomley, Feb 21 2001

nonn

(n-1)-digit repunits in base n written in decimal.

			a(10)=111111111; i.e., just nine 1's (converted from base 10 to decimal).

Harry J. Smith, Table of n, a(n) for n=1..200

Cf. A000169, A037205, A055869, A060073, A125118, A228275.

Cf. other sequences of generalized repunits, such as A053696, A055129, A031973, A125598, A173468, A023037, A119598, A085104, and A162861.

[0] cat [ (n^(n-1) -1)/(n-1) : n in [2..25]]; // G. C. Greubel, Aug 15 2022

Join[{0},Array[(#^(#-1)-1)/(#-1)&,20,2]] (* Harvey P. Dale, Jun 04 2013 *)

a(n) = if (n==1, 0, (n^(n - 1) - 1)/(n - 1)); \\ Harry J. Smith, Jul 01 2009

[0]+[(n^(n-1) -1)/(n-1) for n in (2..25)] # G. C. Greubel, Aug 15 2022

a(n+1) = Sum_{k=1..n} n^(k-1)*C(n, k). - Olivier Gérard, Jun 26 2001 [Corrected by Mathew Englander, Dec 15 2020]
a(n) = Sum_{j=2..n} n^(n-j). - Zerinvary Lajos, Sep 11 2006
a(n+1) = A125118(n,n). - Reinhard Zumkeller, Nov 21 2006
a(n) = Integral_{x=1/n..1} 1/x^n dx. - Francesco Daddi, Aug 01 2011
a(n) = A037205(n-1)/(n-1) = A060073(n)*(n-1) = A023037(n) - A000169(n).
a(n) = [x^n] x^2/((1 - x)*(1 - n*x)). - Ilya Gutkovskiy, Oct 04 2017
a(n) = 1 + A228275(n, n-2) for n >= 2. - Mathew Englander, Dec 14 2020

Name edited by Michel Marcus, Dec 14 2020

A079901 Triangle of powers, T(n,k) = n^k, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 3, 9, 27, 1, 4, 16, 64, 256, 1, 5, 25, 125, 625, 3125, 1, 6, 36, 216, 1296, 7776, 46656, 1, 7, 49, 343, 2401, 16807, 117649, 823543, 1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 1, 9, 81, 729, 6561, 59049, 531441, 4782969, 43046721

Offset: 0

Reinhard Zumkeller, Feb 21 2003

Matrix inverse equals the triangle R where R(n,k) = A107045(n,k)/A107046(n,k) are coefficients with exponential-like properties. - Paul D. Hanna, May 22 2005

			Triangle begins:
  1;
  1,1;
  1,2,4;
  1,3,9,27;
  1,4,16,64,256;
  1,5,25,125,625,3125;

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A107045/A107046 (matrix inverse).

Cf. A002262, A000290, A000578, A000583, A000272, A000169, A000312.

a079901 n k = a079901_tabl !! n !! k
a079901_row n = a079901_tabl !! n
a079901_tabl = zipWith (map . (^)) [0..] a002262_tabl
-- Reinhard Zumkeller, Mar 31 2015

Join[{1},Flatten[Table[n^k,{n,9},{k,0,n}]]] (* Harvey P. Dale, Feb 08 2013 *)

row(n) = vector(n+1, k, n^(k-1)); \\ Amiram Eldar, May 09 2025

T(n,k) = if k=0 then 1 else T(n,k-1)*n.
T(n,0) = 1; T(n,1) = n for n>0; T(n,2) = A000290(n) for n > 1; T(n,3) = A000578(n) for n > 2; T(n,4) = A000583(n) for n>3.
T(n,n-2) = A000272(n) for n>2; T(n,n-1) = A000169(n) for n>1; T(n,n) = A000312(n).

cp's OEIS Frontend

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