A040075 -id:A040075 - cp's OEIS frontend

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

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

A059299 Triangle of idempotent numbers (version 3), T(n, k) = binomial(n, k) * (n - k)^k.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 12, 24, 4, 0, 1, 20, 90, 80, 5, 0, 1, 30, 240, 540, 240, 6, 0, 1, 42, 525, 2240, 2835, 672, 7, 0, 1, 56, 1008, 7000, 17920, 13608, 1792, 8, 0, 1, 72, 1764, 18144, 78750, 129024, 61236, 4608, 9, 0, 1, 90, 2880, 41160

Offset: 0

N. J. A. Sloane, Jan 25 2001

			Triangle begins:
1,
1,  0,
1,  2,   0,
1,  6,   3,    0,
1, 12,  24,    4,    0,
1, 20,  90,   80,    5,   0,
1, 30, 240,  540,  240,   6, 0,
1, 42, 525, 2240, 2835, 672, 7, 0,
...

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

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

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

T := (n, k) -> binomial(n, k) * (n - k)^k:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

t[n_, k_] := Binomial[n, k]*(n - k)^k; Prepend[Flatten@Table[t[n, k], {n, 10}, {k, 0, n}], 1] (* Arkadiusz Wesolowski, Mar 23 2013 *)

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

Name corrected by Peter Luschny, Nov 12 2023

A172978 a(n) = binomial(n+10, 10)*4^n.

Original entry on oeis.org

1, 44, 1056, 18304, 256256, 3075072, 32800768, 318636032, 2867724288, 24216338432, 193730707456, 1479398129664, 10848919617536, 76776969601024, 526470648692736, 3509804324618240, 22813728110018560, 144934272698941440, 901813252348968960, 5505807224867389440

Offset: 0

Zerinvary Lajos, Feb 06 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..157
Index entries for linear recurrences with constant coefficients, signature (44,-880,10560,-84480,473088,-1892352,5406720,-10813440,14417920,-11534336,4194304).

Cf. A002697, A038845, A038846, A040075, A045543, A054337, A054338, A054339, A054340

[Binomial(n+10, 10)*4^n: n in [0..30]]; // Vincenzo Librandi, Jun 06 2011

Table[Binomial[n + 10, 10]*4^n, {n, 0, 20}]

From Amiram Eldar, Mar 27 2022: (Start)
G.f.: 1/(1 - 4*x)^11.
Sum_{n>=0} 1/a(n) = 14269429/63 - 787320*log(4/3).
Sum_{n>=0} (-1)^n/a(n) = 78125000*log(5/4) - 1098284605/63. (End)

A305833 Triangle read by rows: T(0,0)=1; T(n,k) = 4*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 4, 16, 1, 64, 8, 256, 48, 1, 1024, 256, 12, 4096, 1280, 96, 1, 16384, 6144, 640, 16, 65536, 28672, 3840, 160, 1, 262144, 131072, 21504, 1280, 20, 1048576, 589824, 114688, 8960, 240, 1, 4194304, 2621440, 589824, 57344, 2240, 24, 16777216, 11534336, 2949120, 344064, 17920, 336, 1

Offset: 0

Shara Lalo, Jun 11 2018

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013611 ((1+4*x)^n).
The coefficients in the expansion of 1/(1-4x-x^2) are given by the sequence generated by the row sums.
The row sums are A001076 (Denominators of continued fraction convergent to sqrt(5)).
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 4.236067977...; a metallic mean (see A098317), when n approaches infinity.

			Triangle begins:
         1;
         4;
        16,        1;
        64,        8;
       256,       48,        1;
      1024,      256,       12;
      4096,     1280,       96,       1;
     16384,     6144,      640,      16;
     65536,    28672,     3840,     160,      1;
    262144,   131072,    21504,    1280,     20;
   1048576,   589824,   114688,    8960,    240,    1;
   4194304,  2621440,   589824,   57344,   2240,   24;
  16777216, 11534336,  2949120,  344064,  17920,  336,  1;
  67108864, 50331648, 14417920, 1966080, 129024, 3584, 28;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72, 90, 373.

Shara Lalo, Left justified triangle
Shara Lalo, Skew diagonals in triangle A013611

Row sums give A001076.
Cf. A000302 (column 0), A002697 (column 1), A038845 (column 2), A038846 (column 3), A040075 (column 4).
Cf. A013611.
Cf. A098317.

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 4 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten

G.f.: 1 / (1 - 4*t*x - t^2).

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

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A059299 Triangle of idempotent numbers (version 3), T(n, k) = binomial(n, k) * (n - k)^k.

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A172978 a(n) = binomial(n+10, 10)*4^n.

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A305833 Triangle read by rows: T(0,0)=1; T(n,k) = 4*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

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