A036216 -id:A036216 - cp's OEIS frontend

A050982 5-idempotent numbers.

1, 30, 525, 7000, 78750, 787500, 7218750, 61875000, 502734375, 3910156250, 29326171875, 213281250000, 1510742187500, 10458984375000, 70971679687500, 473144531250000, 3105010986328125, 20091247558593750, 128360748291015625, 810699462890625000

Offset: 5

Eric W. Weisstein

Number of n-permutations of 6 objects: t,u,v,z,x, y with repetition allowed, containing exactly five u's. Example: a(6)=30 because we have uuuuut, uuuutu, uuutuu, uutuuu, utuuuu, tuuuuu, uuuuuv, uuuuvu, uuuvuu, uuvuuu, uvuuuu, vuuuuu, uuuuuz, uuuuzu, uuuzuu, uuzuuu, uzuuuu, zuuuuu, uuuuux, uuuuxu, uuuxuu, uuxuuu, uxuuuu, xuuuuu, uuuuuy, uuuuyu, uuuyuu, uuyuuu, uyuuuu, yuuuuu. - Zerinvary Lajos, Jun 16 2008

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43.

Vincenzo Librandi, Table of n, a(n) for n = 5..1000
Eric Weisstein's World of Mathematics, Idempotent Number.
Index entries for linear recurrences with constant coefficients, signature (30,-375,2500,-9375,18750,-15625).

Cf. A001788, A036216, A040075, A050988, A050989, A000389, A054849, A036219, A045543, A036084, A140404, A000389, A054849, A036219, A045543, A036084, A140404.

[Binomial(n, 5)*5^(n-5): n in [5..25]]; // Vincenzo Librandi, Aug 12 2017

seq(binomial(n, 5)*5^(n-5), n=5..32); # Zerinvary Lajos, Jun 16 2008

CoefficientList[Series[1 / (1 - 5 x)^6, {x, 0, 33}], x] (* Vincenzo Librandi, Aug 12 2017 *)

a(n)=binomial(n, 5)*5^(n-5) \\ Charles R Greathouse IV, Sep 03 2011

a(n) = C(n, 5)*5^(n-5).
G.f.: x^5/(1-5*x)^6. - Zerinvary Lajos, Aug 06 2008
From Amiram Eldar, Apr 17 2022: (Start)
Sum_{n>=5} 1/a(n) = 6400*log(5/4) - 17125/12.
Sum_{n>=5} (-1)^(n+1)/a(n) = 32400*log(6/5) - 23625/4. (End)

A172362 a(n) = binomial(n+10, 10)*3^n.

Original entry on oeis.org

1, 33, 594, 7722, 81081, 729729, 5837832, 42532776, 287096238, 1818276174, 10909657044, 62482581252, 343654196886, 1824010737318, 9380626649064, 46903133245320, 228652774570935, 1089463220014455, 5084161693400790

Offset: 0

Zerinvary Lajos, Feb 01 2010

With a different offset, number of n-permutations (n>=10) of 4 objects: u, v, z, x with repetition allowed, containing exactly ten, (10) u's.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (33,-495,4455,-26730,112266,-336798,721710,-1082565,1082565,-649539,177147).

Cf. A027472, A036216, A036217, A036219, A036220, A036221, A036222, A036223.

[3^n*Binomial(n+10, 10): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

Maple
```
seq(binomial(n+10, 10)*3^n, n=0..30);
```

Table[Binomial[n + 10, 10]*3^n, {n, 0, 20}]

G.f.: 1/(1-3*x)^11. - Vincenzo Librandi, Oct 15 2011
From Amiram Eldar, Aug 28 2022: (Start)
Sum_{n>=0} 1/a(n) = 261617/42 - 15360*log(3/2).
Sum_{n>=0} (-1)^n/a(n) = 7864320*log(4/3) - 47510881/21. (End)

A050988 6-idempotent numbers.

Original entry on oeis.org

1, 42, 1008, 18144, 272160, 3592512, 43110144, 480370176, 5043886848, 50438868480, 484213137408, 4489976365056, 40409787285504, 354362750042112, 3037395000360960, 25514118003032064, 210491473525014528, 1708695490967764992, 13669563927742119936, 107917609955858841600

Offset: 6

Eric W. Weisstein

Number of n-permutations of 7 objects: s, t, u, v, z, x, y with repetition allowed, containing exactly six u's. - Zerinvary Lajos, Jun 16 2008

Vincenzo Librandi, Table of n, a(n) for n = 6..400
Eric Weisstein's World of Mathematics, Idempotent Number.

Cf. A000027, A001788, A036216, A040075, A050982, A050989.

seq(binomial(n, 6)*6^(n-6), n=6..32); # Zerinvary Lajos, Jun 16 2008
A050988:=x^6/(1-6*x)^7; series(A050988,x,32); # Zerinvary Lajos, Aug 09 2008

a[n_] := Binomial[n, 6]*6^(n - 6); Array[a, 20, 6] (* Amiram Eldar, Apr 17 2022 *)

a(n) = binomial(n,6)*6^(n-6).
G.f.: x^6/(1-6*x)^7. - Zerinvary Lajos, Aug 09 2008
From Amiram Eldar, Apr 17 2022: (Start)
Sum_{n>=6} 1/a(n) = 102561/5 - 112500*log(6/5).
Sum_{n>=6} (-1)^n/a(n) = 605052*log(7/6) - 466341/5. (End)

A081143 5th binomial transform of (0,0,0,1,0,0,0,0,......).

Original entry on oeis.org

0, 0, 0, 1, 20, 250, 2500, 21875, 175000, 1312500, 9375000, 64453125, 429687500, 2792968750, 17773437500, 111083984375, 683593750000, 4150390625000, 24902343750000, 147857666015625, 869750976562500, 5073547363281250

Offset: 0

Paul Barry, Mar 08 2003

Starting at 1, four-fold convolution of A000351 (powers of 5).

With a different offset, number of n-permutations (n=4)of 6 objects u, v, w, z, x, y with repetition allowed, containing exactly three u's. Example: a(4)=20 because we have uuuv, uuvu, uvuu, vuuu, uuuw, uuwu, uwuu, wuuu, uuuz, uuzu, uzuu, zuuu, uuux, uuxu, uxuu, xuuu, uuuy, uuyu, uyuu and yuuu. - Zerinvary Lajos, Jun 03 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (20,-150,500,-625).

Cf. A038846, A036216, A001789, A081144, A081135.

[5^(n-3) * Binomial(n, 3): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013

seq(binomial(n,3)*5^(n-3), n=0..25); # Zerinvary Lajos, Jun 03 2008

CoefficientList[Series[x^3/(1-5x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{20,-150,500,-625}, {0,0,0,1}, 30] (* Harvey P. Dale, Dec 24 2015 *)

vector(31, n, my(m=n-1); 5^(m-3)*binomial(m,3)) \\ G. C. Greubel, Mar 05 2020

[lucas_number2(n, 5, 0)*binomial(n,3)/5^3 for n in range(0, 22)] # Zerinvary Lajos, Mar 12 2009

a(n) = 20*a(n-1) - 150*a(n-2) + 500*a(n-3) - 625*a(n-4), with a(0)=a(1)=a(2)=0, a(3)=1.
a(n) = 5^(n-3)*binomial(n,3).
G.f.: x^3/(1-5*x)^4.
E.g.f.: x^3*exp(x)/6. - G. C. Greubel, Mar 05 2020
From Amiram Eldar, Jan 04 2022: (Start)
Sum_{n>=3} 1/a(n) = 240*log(5/4) - 105/2.
Sum_{n>=3} (-1)^(n+1)/a(n) = 540*log(6/4) - 195/2. (End)

A059300 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 4.

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 12, 24, 4, 1, 20, 90, 80, 5, 1, 30, 240, 540, 240, 6, 1, 42, 525, 2240, 2835, 672, 7, 1, 56, 1008, 7000, 17920, 13608, 1792, 8, 1, 72, 1764, 18144, 78750, 129024, 61236, 4608, 9, 1, 90, 2880, 41160, 272160, 787500, 860160, 262440, 11520, 10

Offset: 0

N. J. A. Sloane, Jan 25 2001

			Triangle begins:
1;
1,  2;
1,  6,   3;
1, 12,  24,    4;
1, 20,  90,   80,    5;
1, 30, 240,  540,  240,   6;
1, 42, 525, 2240, 2835, 672, 7;
...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
John Riordan and N. J. A. Sloane, Correspondence, 1974

There are 4 versions: A059297-A059300. Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc. Row sums are A000248.

/* As triangle: */ [[Binomial(n+1,n-k+1)*(n-k+1)^k: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 22 2015

t[n_, k_] := Binomial[n + 1, k]*(n - k + 1)^k; Flatten@Table[t[n, k], {n, 0, 9}, {k, 0, n}] (* Arkadiusz Wesolowski, Mar 23 2013 *)

for(n=0, 25, for(k=0, n, print1(binomial(n+1,k)*(n-k+1)^k, ", "))) \\ G. C. Greubel, Jan 05 2017

T(n,k) = binomial(n+1,n-k+1)*(n-k+1)^k. - R. J. Mathar, Mar 14 2013

A141054 8-idempotent numbers: a(n) = binomial(n+8,8)*8^n.

Original entry on oeis.org

1, 72, 2880, 84480, 2027520, 42172416, 787218432, 13495173120, 215922769920, 3262832967680, 46984794734592, 649244436332544, 8656592484433920, 111869810568069120, 1406363332855726080, 17251390216363573248, 207016682596362878976, 2435490383486622105600

Offset: 0

Zerinvary Lajos, Aug 01 2008

With a different offset, number of n-permutations of 9 objects:
p, r, s, t, u, v, z, x, y with repetition allowed, containing exactly eight (8) u's. Example: a(1)=72 because we have
uuuuuuuup, uuuuuuupu, uuuuuupuu, uuuuupuuu, uuuupuuuu, uuupuuuuu, uupuuuuuu, upuuuuuuu, puuuuuuuu,
uuuuuuuur, uuuuuuuru, uuuuuuruu, uuuuuruuu, uuuuruuuu, uuuruuuuu, uuruuuuuu, uruuuuuuu, ruuuuuuuu,
uuuuuuuus, uuuuuuusu, uuuuuusuu, uuuuusuuu, uuuusuuuu, uuusuuuuu, uusuuuuuu, usuuuuuuu, suuuuuuuu,
uuuuuuuut, uuuuuuutu, uuuuuutuu, uuuuutuuu, uuuutuuuu, uuutuuuuu, uutuuuuuu, utuuuuuuu, tuuuuuuuu,
uuuuuuuuv, uuuuuuuvu, uuuuuuvuu, uuuuuvuuu, uuuuvuuuu, uuuvuuuuu, uuvuuuuuu, uvuuuuuuu, vuuuuuuuu,
uuuuuuuuz, uuuuuuuzu, uuuuuuzuu, uuuuuzuuu, uuuuzuuuu, uuuzuuuuu, uuzuuuuuu, uzuuuuuuu, zuuuuuuuu,
uuuuuuuux, uuuuuuuxu, uuuuuuxuu, uuuuuxuuu, uuuuxuuuu, uuuxuuuuu, uuxuuuuuu, uxuuuuuuu, xuuuuuuuu,
uuuuuuuuy, uuuuuuuyu, uuuuuuyuu, uuuuuyuuu, uuuuyuuuu, uuuyuuuuu, uuyuuuuuu, uyuuuuuuu, yuuuuuuuu.

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A000027, A001788, A036216, A040075, A050982, A050988, A050989, A059297, A059300.

[8^n* Binomial(n+8, 8): n in [0..20]]; // Vincenzo Librandi, Oct 16 2011

Maple
```
seq(binomial(n+8,8)*8^n, n=0..17);
```

Table[Binomial[n + 8, 8] 8^n, {n, 0, 15}] (* Michael De Vlieger, Jul 24 2017 *)

vector(15,n,binomial(n+7,8)*8^(n-1)) \\ Derek Orr, Jul 24 2017

a(n) = binomial(n+8,8)*8^n.
G.f.: 1/(1-8*x)^9. - Vincenzo Librandi, Oct 16 2011
From Amiram Eldar, Apr 17 2022: (Start)
Sum_{n>=0} 1/a(n) = 738990736/105 - 52706752*log(8/7).
Sum_{n>=0} (-1)^n/a(n) = 306110016*log(9/8) - 1261909808/35. (End)

A050989 7-idempotent numbers.

Original entry on oeis.org

1, 56, 1764, 41160, 792330, 13311144, 201885684, 2826399576, 37096494435, 461645264080, 5493578642552, 62926446269232, 697434779483988, 7510836086750640, 78863778910881720, 809668130151718992, 8147285559651672357, 80514351413028291528, 782778416515552834300

Offset: 7

Eric W. Weisstein

Vincenzo Librandi, Table of n, a(n) for n = 7..400
Eric Weisstein's World of Mathematics, Idempotent Number.
Index entries for linear recurrences with constant coefficients, signature (56,-1372,19208,-168070,941192,-3294172,6588344,-5764801).

Cf. A001788, A036216, A040075, A050982, A050988, A059297, A059300.

[7^(n-7)* Binomial(n, 7): n in [7..30]]; // Vincenzo Librandi, Oct 16 2011

seq(binomial(n, 7)*7^(n-7), n=7..33); # Zerinvary Lajos, Aug 01 2008

LinearRecurrence[{56,-1372,19208,-168070,941192,-3294172,6588344,-5764801}, {1,56,1764,41160,792330,13311144,201885684,2826399576},20] (* Harvey P. Dale, May 31 2014 *)

a(n)=binomial(n, 7)*7^(n-7) \\ Charles R Greathouse IV, Sep 03 2011

a(n) = C(n, 7)*7^(n-7).
G.f.: x^7/(1-7*x)^8.
From Amiram Eldar, Apr 17 2022: (Start)
Sum_{n>=7} 1/a(n) = 2286144*log(7/6) - 10572289/30.
Sum_{n>=7} (-1)^(n+1)/a(n) = 12845056*log(8/7) - 51456517/30. (End)

A059298 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 2.

Original entry on oeis.org

1, 2, 1, 3, 6, 1, 4, 24, 12, 1, 5, 80, 90, 20, 1, 6, 240, 540, 240, 30, 1, 7, 672, 2835, 2240, 525, 42, 1, 8, 1792, 13608, 17920, 7000, 1008, 56, 1, 9, 4608, 61236, 129024, 78750, 18144, 1764, 72, 1, 10, 11520, 262440, 860160, 787500, 272160, 41160

Offset: 0

N. J. A. Sloane, Jan 25 2001

The inverse triangle is the signed version 1,-2,1,9,-6,1,.. of triangle A061356. - Peter Luschny, Mar 13 2009
T(n,k) is the sum of the products of the cardinality of the blocks (cells) in the set partitions of {1,2,..,n} into exactly k blocks.
From Peter Bala, Jul 22 2014: (Start)
Exponential Riordan array [(1+x)*exp(x), x*exp(x)].
Let M = A093375, the exponential Riordan array [(1+x)*exp(x), x], and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
  /I_k 0\
  \ 0  M/
having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. The present triangle equals the infinite matrix product M(0)*M(1)*M(2)*... - see the Example section. (End)
The Bell transform of n+1. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			Triangle begins
1;
2, 1;
3, 6, 1;
4, 24, 12, 1; ...
From _Peter Bala_, Jul 22 2014: (Start)
With the arrays M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins
/1          \/1        \/1        \      /1           \
|2  1       ||0 1      ||0 1      |      |2  1        |
|3  4  1    ||0 2 1    ||0 0 1    |... = |3  6  1     |
|4  9  6 1  ||0 3 4 1  ||0 0 2 1  |      |4 24 12  1  |
|5 16 18 8 1||0 4 9 6 1||0 0 3 4 1|      |5 80 90 20 1|
|...        ||...      ||...      |      |...         | (End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Eric Weisstein's World of Mathematics, Idempotent Number

There are 4 versions: A059297, A059298, A059299, A059300.
Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc.
Row sums are A000248. A093375.

/* As triangle */ [[Binomial(n,k)*k^(n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Aug 22 2015

T:= (n, k)-> binomial(n+1,k+1)*(k+1)^(n-k): seq(seq(T(n, k), k=0..n), n=0..10); # Georg Fischer, Oct 27 2021

t = Transpose[ Table[ Range[0, 11]! CoefficientList[ Series[(x Exp[x])^n/n!, {x, 0, 11}], x], {n, 11}]]; Table[ t[[n, k]], {n, 2, 11}, {k, n - 1}] // Flatten (* or simply *)
t[n_, k_] := Binomial[n, k]*k^(n - k); Table[t[n, k], {n, 10}, {k, n}] // Flatten

for(n=1, 25, for(k=1, n, print1(binomial(n,k)*k^(n-k), ", "))) \\ G. C. Greubel, Jan 05 2017

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: n+1, 10) # Peter Luschny, Jan 18 2016

A099097 Riordan array (1, 3+x).

Original entry on oeis.org

1, 0, 3, 0, 1, 9, 0, 0, 6, 27, 0, 0, 1, 27, 81, 0, 0, 0, 9, 108, 243, 0, 0, 0, 1, 54, 405, 729, 0, 0, 0, 0, 12, 270, 1458, 2187, 0, 0, 0, 0, 1, 90, 1215, 5103, 6561, 0, 0, 0, 0, 0, 15, 540, 5103, 17496, 19683, 0, 0, 0, 0, 0, 1, 135, 2835, 20412, 59049, 59049, 0, 0, 0, 0, 0, 0, 18, 945, 13608, 78732, 196830, 177147

Offset: 0

Paul Barry, Sep 25 2004

Row sums are A006190(n+1). Diagonal sums are A052931. The Riordan array (1, s+tx) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).

Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1/3, -1/3, 0, 0, 0, 0, 0, ...] DELTA [3, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 10 2008

			Triangle begins:
  1;
  0, 3;
  0, 1, 9;
  0, 0, 6, 27;
  0, 0, 1, 27,  81;
  0, 0, 0,  9, 108, 243;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A027465.

Diagonals are of the form 3^n*binomial(n+m, m): A000244 (m=0), A027471 (m=1), A027472 (m=2), A036216 (m=3), A036217 (m=4), A036219 (m=5), A036220 (m=6), A036221 (m=7), A036222 (m=8), A036223 (m=9), A172362 (m=10).

Table[3^(2*k-n)*Binomial[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 19 2021 *)

flatten([[3^(2*k-n)*binomial(k, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 19 2021

Triangle: T(n, k) = binomial(k, n-k)*3^k*(1/3)^(n-k).
G.f. of column k: (3*x + x^2)^k.
G.f.: 1/(1 - 3*y*x - y*x^2). - Philippe Deléham, Nov 21 2011
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A006190(n+1), A135030(n+1), A181353(n+1) for x = 0,1,2,3 respectively. - Philippe Deléham, Nov 21 2011

A120981 Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k vertices of outdegree 1 (n >= 0, k >= 0).

Original entry on oeis.org

1, 0, 3, 3, 0, 9, 1, 27, 0, 27, 18, 12, 162, 0, 81, 15, 270, 90, 810, 0, 243, 138, 270, 2430, 540, 3645, 0, 729, 189, 2898, 2835, 17010, 2835, 15309, 0, 2187, 1218, 4536, 34776, 22680, 102060, 13608, 61236, 0, 6561, 2280, 32886, 61236, 312984, 153090, 551124

Offset: 0

Emeric Deutsch, Jul 21 2006

A ternary tree is a rooted tree in which each vertex has at most three children and each child of a vertex is designated as its left or middle or right child.

			T(2,0)=3 because we have (Q,L,M), (Q,L,R) and (Q,M,R), where Q denotes the root and L (M,R) denotes a left (middle, right) child of Q.
Triangle starts:
   1;
   0,   3;
   3,   0,   9;
   1,  27,   0,  27;
  18,  12, 162,   0, 81;
  15, 270,  90, 810,  0, 243;

Andrew Howroyd, Table of n, a(n) for n = 0..1274
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.

Diagonals include A129530, A036216.

Cf. A001764 (row sums), A004319, A120429, A120982, A120983, A120984.

T:=proc(n,k) if k<=n then (1/(n+1))*binomial(n+1,k)*sum(3^(3*j-n+2*k)*binomial(n+1-k,j)*binomial(j,n-k-2*j),j=0..n+1-k) else 0 fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

T[n_, k_] := (1/(n+1))*Binomial[n+1, k]*Sum[3^(2k - n + 3j)*Binomial[n + 1 - k, j]*Binomial[j, n - k - 2j], {j, 0, n - k + 1}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 02 2018 *)

T(n,k) = binomial(n+1, k)*sum(j=0, n+1-k, 3^(2*k-n+3*j)*binomial(n+1-k, j)*binomial(j, n-k-2*j))/(n+1); \\ Andrew Howroyd, Nov 06 2017

from sympy import binomial
def T(n, k): return binomial(n + 1, k)*sum([3**(2*k - n + 3*j)*binomial(n + 1 - k, j)*binomial(j, n - k - 2*j) for j in range(n + 2 - k)])//(n + 1)
for n in range(21): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Nov 07 2017

T(n,0) = A120984(n).
Sum_{k>=1} k*T(n,k) = 3*binomial(3n,n-1) = 3*A004319(n).
T(n,k) = (1/(n+1))*binomial(n+1,k)*Sum_{j=0..n+1-k} 3^(2k-n+3j)*binomial(n+1-k,j)*binomial(j,n-k-2j).
G.f.: G=G(t,z) satisfies G = 1 + 3tzG + 3z^2*G^2 + z^3*G^3.

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A050982 5-idempotent numbers.

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A172362 a(n) = binomial(n+10, 10)*3^n.

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A050988 6-idempotent numbers.

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A081143 5th binomial transform of (0,0,0,1,0,0,0,0,......).

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PARI

Sage

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A059300 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 4.

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A141054 8-idempotent numbers: a(n) = binomial(n+8,8)*8^n.

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Maple

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A050989 7-idempotent numbers.

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