A181567 -id:A181567 - cp's OEIS frontend

A181555 a(n) = A002110(n)^n.

1, 2, 36, 27000, 1944810000, 65774855015100000, 733384949590939374729000000, 9037114296609938214167920266348510000000, 78354300210436852307898467208663359164858967744100000000

Offset: 0

Matthew Vandermast, Oct 31 2010

For n>0, a(n)= first counting number whose prime signature consists of n repeated n times (cf. A002024). Subsequence of A025487.

			a(4) = 1944810000 = 210^4 = 2^4 * 3^4 * 5^4 * 7^4.

Amiram Eldar, Table of n, a(n) for n = 0..26

A061742(n) = A002110(n)^2. See also A006939, A066120, A166475, A167448.
A000005(a(n)) = A000169(n). The divisors of a(n) appear as the first A000169(n) terms of A178479, with A178479(A000169(n)) = a(n).
A071207(n, k) gives the number of divisors of n with (n-k) distinct prime factors, A181567(n, k) gives the number of divisors of n with k prime factors counted with multiplicity.
Cf. A001221, A001222, A079474, A146290, A146292.

a[0] = 1; a[n_] := Product[Prime[i], {i, 1, n}]^n; Array[a, 9, 0] (* Amiram Eldar, Aug 08 2019 *)

a(n) = A079474(2n,n). - Alois P. Heinz, Aug 22 2019

A071207 Triangular array T(n,k) read by rows, giving number of rooted trees on the vertex set {1..n+1} where k children of the root have a label smaller than the label of the root.

Original entry on oeis.org

1, 1, 1, 4, 4, 1, 27, 27, 9, 1, 256, 256, 96, 16, 1, 3125, 3125, 1250, 250, 25, 1, 46656, 46656, 19440, 4320, 540, 36, 1, 823543, 823543, 352947, 84035, 12005, 1029, 49, 1, 16777216, 16777216, 7340032, 1835008, 286720, 28672, 1792, 64, 1, 387420489

Offset: 0

Cedric Chauve (chauve(AT)lacim.uqam.ca), May 16 2002

The n-th term of the n-th binomial transform of a sequence {b} is given by {d} where d(n) = sum(k=0,n,T(n,k)*b(k)) and T(n,k)=binomial(n,k)*n^(n-k); such diagonals are related to the hyperbinomial transform (A088956). - Paul D. Hanna, Nov 04 2003
T(n,k) gives the number of divisors of A181555(n) with (n-k) distinct prime factors. See also A001221, A146289, A146290, A181567. - Matthew Vandermast, Oct 31 2010
T(n,k) is the number of partial functions on {1,2,...,n} leaving exactly k elements undefined.  Row sums = A000169. - Geoffrey Critzer, Jan 08 2012
As a triangular matrix, transforms rows into diagonals in the table of coefficients of successive iterations of x/(1-x). - Paul D. Hanna, Jan 19 2014
Also the number of rooted trees on n+1 labeled vertices in which some specified vertex (say, vertex 1) has k children. - Alan Sokal, Jul 22 2022

			1
1     1
4     4     1
27    27    9     1
256   256   96    16    1
3125  3125  1250  250   25    1
46656 46656 19440 4320  540   36    1

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
C. Chauve, S. Dulucq and O. Guibert, Enumeration of some labeled trees, Proceedings of FPSAC/SFCA 2000 (Moscow), Springer, pp. 146-157.
Alan D. Sokal, A remark on the enumeration of rooted labeled trees, arXiv:1910.14519 [math.CO], 2019 and Discrete Math. 343, 111865 (2020).

Cf. A000169, A000312, A089466, A088956, A166900.

T:= (n, k)-> binomial(n, k)*n^(n-k): seq(seq(T(n, k), k=0..n), n=0..10);

Prepend[Flatten[ Table[Table[Binomial[n, k] n^(n - k), {k, 0, n}], {n, 1, 8}]], 1]  (* Geoffrey Critzer, Jan 08 2012 *)

T(n,k)=if(k<0 || k>n,0,binomial(n,k)*n^(n-k))

/* Transforms rows into diagonals in the iterations of x/(1-x): */
{T(n, k)=local(F=x, M, N, P, m=n); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, x/(1-x+x*O(x^(m+2))))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, F=x; for(i=1, r, F=subst(F, x, x/(1-x+x*O(x^(m+2))))); polcoeff(F, c)); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Jan 19 2014

T(n,k) = binomial(n, k)*n^(n-k).

E.g.f.: (-LambertW(-y)/y)^x/(1+LambertW(-y)). - Vladeta Jovovic

Name edited by Alan Sokal, Jul 22 2022

A273975 Three-dimensional array written by antidiagonals in k,n: T(k,n,h) with k >= 1, n >= 0, 0 <= h <= n*(k-1) is the coefficient of x^h in the polynomial (1 + x + ... + x^(k-1))^n = ((x^k-1)/(x-1))^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 3, 6, 7, 6, 3, 1, 1, 4, 6, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 4, 3, 2, 1, 1, 3, 6, 10, 12, 12, 10, 6, 3, 1, 1, 4, 10

Offset: 1

Andrey Zabolotskiy, Nov 10 2016

Equivalently, T(k,n,h) is the number of ordered sets of n nonnegative integers < k with the sum equal to h.
From Juan Pablo Herrera P., Nov 21 2016: (Start)
T(k,n,h) is the number of possible ways of randomly selecting h cards from k-1 sets, each with n different playing cards. It is also the number of lattice paths from (0,0) to (n,h) using steps (1,0), (1,1), (1,2), ..., (1,k-1).
Shallow diagonal sums of each triangle with fixed k give the k-bonacci numbers. (End)
T(k,n,h) is the number of n-dimensional grid points of a k X k X ... X k grid, which are lying in the (n-1)-dimensional hyperplane which is at an L1 distance of h from one of the grid's corners, and normal to the corresponding main diagonal of the grid. - Eitan Y. Levine, Apr 23 2023

			For first few k and for first few n, the rows with h = 0..n*(k-1) are given:
k=1:  1;  1;  1;  1;  1; ...
k=2:  1;  1, 1;  1, 2, 1;  1, 3, 3, 1;  1, 4, 6, 4, 1; ...
k=3:  1;  1, 1, 1;  1, 2, 3, 2, 1;  1, 3, 6, 7, 6, 3, 1; ...
k=4:  1;  1, 1, 1, 1;  1, 2, 3, 4, 3, 2, 1; ...
For example, (1 + x + x^2)^3 = 1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 3*x^5 + x^6, hence T(3,3,2) = T(3,3,4) = 6.
From _Eitan Y. Levine_, Apr 23 2023: (Start)
Example for the repeated cumulative sum formula, for (k,n)=(3,3) (each line is the cumulative sum of the previous line, and the first line is the padded, alternating 3rd row from Pascal's triangle):
  1  0  0 -3  0  0  3  0  0 -1
  1  1  1 -2 -2 -2  1  1  1
  1  2  3  1 -1 -3 -2 -1
  1  3  6  7  6  3  1
which is T(3,3,h). (End)

Andrey Zabolotskiy, slices with k+n = 1..20, flattened
Florentin Smarandache, K-Nomial Coefficients, arXiv:math/0612062 [math.GM], 2006 (originally published in French in: F. Smarandache, Généralisations et Généralités, Ed. Nouvelle, 1984, pp. 24-26).

k-nomial arrays for fixed k=1..10: A000012, A007318, A027907, A008287, A035343, A063260, A063265, A171890, A213652, A213651.
Arrays for fixed n=0..6: A000012, A000012, A004737, A109439, A277949, A277950, A277951.
Central n-nomial coefficients for n=1..9, i.e., sequences with h=floor(n*(k-1)/2) and fixed n: A000012, A000984 (A001405), A002426, A005721 (A005190), A005191, A063419 (A018901), A025012, (A025013), A025014, A174061 (A025015), A201549, (A225779), A201550. Arrays: A201552, A077042, see also cfs. therein.
Triangle n=k-1: A181567. Triangle n=k: A163181.

a = Table[CoefficientList[Sum[x^(h-1),{h,k}]^n,x],{k,10},{n,0,9}];
Flatten@Table[a[[s-n,n+1]],{s,10},{n,0,s-1}]
(* alternate program *)
row[k_, n_] := Nest[Accumulate,Upsample[Table[((-1)^j)*Binomial[n,j],{j,0,n}],k],n][[;;n*(k-1)+1]] (* Eitan Y. Levine, Apr 23 2023 *)

T(k,n,h) = Sum_{i = 0..floor(h/k)} (-1)^i*binomial(n,i)*binomial(n+h-1-k*i,n-1). [Corrected by Eitan Y. Levine, Apr 23 2023]
From Eitan Y. Levine, Apr 23 2023: (Start)
(T(k,n,h))_{h=0..n*(k-1)} = f(f(...f(g(P))...)), where:
(x_i)_{i=0..m} denotes a tuple (in particular, the LHS contains the values for 0 <= h <= n*(k-1)),
f repeats n times,
f((x_i){i=0..m}) = (Sum{j=0..i} x_j)_{i=0..m} is the cumulative sum function,
g((x_i){i=0..m}) = (x(i/k) if k|i, otherwise 0)_{i=0..m*k} is adding k-1 zeros between adjacent elements,
and P=((-1)^i*binomial(n,i))_{i=0..n} is the n-th row of Pascal's triangle, with alternating signs. (End)
From Eitan Y. Levine, Jul 27 2023: (Start)
Recurrence relations, the first follows from the sequence's defining polynomial as mentioned in the Smarandache link:
T(k,n+1,h) = Sum_{i = 0..s-1} T(k,n,h-i)
T(k+1,n,h) = Sum_{i = 0..n} binomial(n,i)*T(k,n-i,h-i*k) (End)

A163181 T(n,k) is the number of weak compositions of k into n parts no greater than (n-1) for n>=1, 0<=k<=n(n-1).

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 6, 7, 6, 3, 1, 1, 4, 10, 20, 31, 40, 44, 40, 31, 20, 10, 4, 1, 1, 5, 15, 35, 70, 121, 185, 255, 320, 365, 381, 365, 320, 255, 185, 121, 70, 35, 15, 5, 1, 1, 6, 21, 56, 126, 252, 456, 756, 1161, 1666, 2247, 2856, 3431, 3906, 4221, 4332, 4221, 3906, 3431

Offset: 1

Geoffrey Critzer, Jul 22 2009

T(n,k) is the number of length n sequences on an alphabet of {0,1,2,...,n-1} that have a sum of k. Equivalently T(n,k) is the number of functions f:{1,2,...,n}->{0,1,2,...,n-1} such that Sum(f(i)=k, i=1...n).

Row n is also row n of the array of q-nomial coefficients. - Matthew Vandermast, Oct 31 2010

			T(3,4) = 6 because there are 6 ternary sequences of length three that sum to 4: [0, 2, 2], [1, 1, 2], [1, 2, 1], [2, 0, 2], [2, 1, 1], [2, 2, 0].

Alois P. Heinz, Rows n = 1..32, flattened

The maximum of row n is in column k=n(n-1)/2 = A000217(n-1).

For q-nomial arrays, see A000012, A007318, A027907, A008287, A035343, A063260, A063265, A171890. See also A181567. - Matthew Vandermast, Oct 31 2010

b:= proc(n, k, l) option remember; `if`(k=0, 1,
      `if`(l=0, 0, add(b(n, k-j, l-1), j=0..min(n-1, k))))
    end:
T:= (n, k)-> b(n, k, n):
seq(seq(T(n, k), k=0..n*(n-1)), n=1..8);  # Alois P. Heinz, Feb 21 2013

(*warning very inefficient*) Table[Distribution[Map[Total, Strings[Range[n], n]]], {n, 1, 6}]//Grid
nn=100;Table[CoefficientList[Series[Sum[x^i,{i,0,n-1}]^n,{x,0,nn}],x],{n,1,10}]//Grid (* Geoffrey Critzer, Feb 21 2013*)

O.g.f. for row n is ((1-x^n)/(1-x))^n. For k<=(n-1), T(n,k) = C(n+k-1,k).

A234574 T(n,k) is the number of size k ordered submultisets of the regular multiset {1_1,1_2,...,1_(n-1),1_n, ... ,i_1,i_2,...,i_(n-1),i_n, ... ,n_1,n_2,...,n_(n-1),n_n} (which contains n copies of i for 1 <= i <= n).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 6, 6, 1, 3, 9, 27, 78, 210, 510, 1050, 1680, 1680, 1, 4, 16, 64, 256, 1020, 4020, 15540, 58380, 210840, 722400, 2310000, 6745200, 17417400, 37837800, 63063000, 63063000, 1, 5, 25, 125, 625, 3125, 15620, 77980, 388220, 1923180, 9454620

Offset: 0

Thomas Wieder, Dec 29 2013

A181567 gives the case for unordered submultisets.

			For n=2 we have the regular multiset L = [1,1,2,2].
We get the following ordered submultisets from L:
For k=0 1 multiset: []
For k=1 2 multisets: [1], [2]
For k=2 4 multisets: [1,1], [1,2], [2,1], [2,2]
For k=3 6 multisets: [1,1,2], [1,2,1], [2,1,1], [1,2,2], [2,1,2], [2,2,1]
For k=4 6 multisets: [1,1,2,2], [1,2,1,2], [1,2,2,1], [2,1,1,2], [2,1,2,1], [2,2,1,1].
Triangle begins with:
  1;
  1, 1;
  1, 2, 4, 6, 6;
  1, 3, 9, 27, 78, 210, 510, 1050, 1680, 1680;
  1, 4, 16, 64, 256, 1020, 4020, 15540, 58380, 210840, 722400, 2310000, 6745200, 17417400, 37837800, 63063000, 63063000;
  ...

Alois P. Heinz, Rows n = 0..26, flattened
Thomas Wieder, Maple program for A234574

Cf. A181567.

Row sums give A274762.

# first Maple program: see link above
# second Maple program:
b:= proc(n, k, i) option remember; `if`(k=0, 1,
     `if`(i<1, 0, add(b(n, k-j, i-1)/j!, j=0..n)))
    end:
T:= (n, k)-> b(n, k, n)*k!:
seq(seq(T(n, k), k=0..n^2), n=0..5); # Alois P. Heinz, Jul 04 2016

More terms from Alois P. Heinz, Jul 04 2016

A329709 Triangle, read by rows, where the n-th row lists the (n^2+1) coefficients of (1+2x+...+(n+1)x^n)^n.

Original entry on oeis.org

1, 1, 2, 1, 4, 10, 12, 9, 1, 6, 21, 56, 111, 174, 219, 204, 144, 64, 1, 8, 36, 120, 330, 768, 1544, 2728, 4275, 5920, 7256, 7848, 7386, 5880, 3900, 2000, 625, 1, 10, 55, 220, 715, 2002, 4970, 11120, 22685, 42570, 73953, 119340, 179305, 251230, 328450, 400304, 453695, 476870, 462815, 411740, 332045, 239070, 150840, 79920, 32400, 7776

Offset: 0

Seiichi Manyama, Feb 29 2020

			Triangle begins:
  1;
  1, 2;
  1, 4, 10, 12,   9;
  1, 6, 21, 56, 111, 174, 219, 204, 144, 64;
  ...

Seiichi Manyama, Rows n = 0..30, flattened

Cf. A181567, A329708.

for(n=0, 5, print(Vecrev(sum(k=0, n, (k+1)*x^k)^n), ", "))

cp's OEIS Frontend

A181555 a(n) = A002110(n)^n.

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A071207 Triangular array T(n,k) read by rows, giving number of rooted trees on the vertex set {1..n+1} where k children of the root have a label smaller than the label of the root.

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A273975 Three-dimensional array written by antidiagonals in k,n: T(k,n,h) with k >= 1, n >= 0, 0 <= h <= n*(k-1) is the coefficient of x^h in the polynomial (1 + x + ... + x^(k-1))^n = ((x^k-1)/(x-1))^n.

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A163181 T(n,k) is the number of weak compositions of k into n parts no greater than (n-1) for n>=1, 0<=k<=n(n-1).

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A234574 T(n,k) is the number of size k ordered submultisets of the regular multiset {1_1,1_2,...,1_(n-1),1_n, ... ,i_1,i_2,...,i_(n-1),i_n, ... ,n_1,n_2,...,n_(n-1),n_n} (which contains n copies of i for 1 <= i <= n).

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A329709 Triangle, read by rows, where the n-th row lists the (n^2+1) coefficients of (1+2*x+...+(n+1)*x^n)^n.

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A329709 Triangle, read by rows, where the n-th row lists the (n^2+1) coefficients of (1+2x+...+(n+1)x^n)^n.