A079901 -id:A079901 - cp's OEIS frontend

A108396 Triangle read by rows: T(n,k) = n*(1+n^k)/2, 0<=k<=n.

0, 1, 1, 2, 3, 5, 3, 6, 15, 42, 4, 10, 34, 130, 514, 5, 15, 65, 315, 1565, 7815, 6, 21, 111, 651, 3891, 23331, 139971, 7, 28, 175, 1204, 8407, 58828, 411775, 2882404, 8, 36, 260, 2052, 16388, 131076, 1048580, 8388612, 67108868, 9, 45, 369, 3285, 29529, 265725

Offset: 0

Reinhard Zumkeller, Jun 02 2005

Row sums give A108397;
T(n,0) = A001477(n);
T(n,1) = A000217(n) for n>0;
T(n,2) = A006003(n) for n>1;
T(n,3) = A027441(n) for n>2;
T(n,4) = A021003(n) for n>3;
T(n,n) = A108398(n).

			.  0:  0
.  1:  1  1
.  2:  2  3   5
.  3:  3  6  15   42
.  4:  4 10  34  130   514
.  5:  5 15  65  315  1565   7815
.  6:  6 21 111  651  3891  23331  139971
.  7:  7 28 175 1204  8407  58828  411775  2882404
.  8:  8 36 260 2052 16388 131076 1048580  8388612  67108868
.  9:  9 45 369 3285 29529 265725 2391489 21523365 193710249 1743392205 .

Reinhard Zumkeller, Rows n = 0..124 of triangle, flattened

Cf. A079901, A000312, A033918, A001477, A000217, A006003, A027441, A021003, A108398, A108397 (row sums), A256512 (central terms).

a108396 n k = a108396_tabl !! n !! k
a108396_row n = a108396_tabl !! n
a108396_tabl = zipWith (\v ws -> map (flip div 2 . (* v) . (+ 1)) ws)
                       [0..] a079901_tabl
-- Reinhard Zumkeller, Mar 31 2015

Join[{0},Flatten[Table[n (1+n^k)/2,{n,10},{k,0,n}]]] (* Harvey P. Dale, Mar 19 2015 *)

Offset changed by Reinhard Zumkeller, Mar 31 2015

A158497 Triangle T(n,k) formed by the coordination sequences and the number of leaves for trees.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 12, 1, 4, 12, 36, 108, 1, 5, 20, 80, 320, 1280, 1, 6, 30, 150, 750, 3750, 18750, 1, 7, 42, 252, 1512, 9072, 54432, 326592, 1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 1, 9, 72, 576, 4608, 36864, 294912, 2359296, 18874368, 150994944, 1, 10, 90, 810, 7290, 65610, 590490, 5314410, 47829690, 430467210, 3874204890

Offset: 0

Thomas Wieder, Mar 20 2009

Consider the k-fold Cartesian products CP(n,k) of the vector A(n) = [1, 2, 3, ..., n].
An element of CP(n,k) is a n-tuple T_t of the form T_t = [i_1, i_2, i_3, ..., i_k] with t=1, .., n^k.
We count members T of CP(n,k) which satisfy some condition delta(T_t), so delta(.) is an indicator function which attains values of 1 or 0 depending on whether T_t is to be counted or not; the summation sum_{CP(n,k)} delta(T_t) over all elements T_t of CP produces the count.
For the triangle here we have delta(T_t) = 0 if for any two i_j, i_(j+1) in T_t one has i_j = i_(j+1): T(n,k) = Sum_{CP(n,k)} delta(T_t) = Sum_{CP(n,k)} delta(i_j = i_(j+1)).
The test on i_j > i_(j+1) generates A158498. One gets the Pascal triangle A007318 if the indicator function tests whether for any two i_j, i_(j+1) in T_t one has i_j >= i_(j+1).
Use of other indicator functions can also calculate the Bell numbers A000110, A000045 or A000108.

			Array, A(n, k) = n*(n-1)^(k-1) for n > 1, A(n, k) = 1 otherwise, begins as:
  1,  1,   1,    1,     1,      1,       1,        1,        1, ... A000012;
  1,  1,   1,    1,     1,      1,       1,        1,        1, ... A000012;
  1,  2,   2,    2,     2,      2,       2,        2,        2, ... A040000;
  1,  3,   6,   12,    24,     48,      96,      192,      384, ... A003945;
  1,  4,  12,   36,   108,    324,     972,     2916,     8748, ... A003946;
  1,  5,  20,   80,   320,   1280,    5120,    20480,    81920, ... A003947;
  1,  6,  30,  150,   750,   3750,   18750,    93750,   468750, ... A003948;
  1,  7,  42,  252,  1512,   9072,   54432,   326592,  1959552, ... A003949;
  1,  8,  56,  392,  2744,  19208,  134456,   941192,  6588344, ... A003950;
  1,  9,  72,  576,  4608,  36864,  294912,  2359296, 18874368, ... A003951;
  1, 10,  90,  810,  7290,  65610,  590490,  5314410, 47829690, ... A003952;
  1, 11, 110, 1100, 11000, 110000, 1100000, 11000000, ............. A003953;
  1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, ............. A003954;
  1, 13, 156, 1872, 22464, 269568, 3234816, 38817792, ............. A170732;
  ... ;
The triangle begins as:
  1
  1, 1;
  1, 2,  2;
  1, 3,  6,  12;
  1, 4, 12,  36,  108;
  1, 5, 20,  80,  320,  1280;
  1, 6, 30, 150,  750,  3750,  18750;
  1, 7, 42, 252, 1512,  9072,  54432, 326592;
  1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344;
  ...;
T(3,3) = 12 counts the triples (1,2,1), (1,2,3), (1,3,1), (1,3,2), (2,1,2), (2,1,3), (2,3,1), (2,3,2), (3,1,2), (3,1,3), (3,2,1), (3,2,3) out of a total of 3^3 = 27 triples in the CP(3,3).

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

Array rows n: A170733 (n=14), ..., A170769 (n=50).
Columns k: A000012(n) (k=0), A000027(n) (k=1), A002378(n-1) (k=2), A011379(n-1) (k=3), A179824(n) (k=4), A101362(n-1) (k=5), 2*A168351(n-1) (k=6), 2*A168526(n-1) (k=7), 2*A168635(n-1) (k=8), 2*A168675(n-1) (k=9), 2*A170783(n-1) (k=10), 2*A170793(n-1) (k=11).
Diagonals k: A055897 (k=n), A055541 (k=n-1), A373395 (k=n-2), A379612 (k=n-3).
Sums: (-1)^n*A065440(n) (signed row).
Cf. A000045, A000108, A000110, A007318, A079901, A158498.

A158497:= func< n,k | k le 1 select n^k else n*(n-1)^(k-1) >;
[A158497(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 18 2025

A158497[n_, k_]:= If[n<2 || k==0, 1, n*(n-1)^(k-1)];
Table[A158497[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 18 2025 *)

def A158497(n,k): return n^k if k<2 else n*(n-1)^(k-1)
print(flatten([[A158497(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 18 2025

T(n, k) = (n-1)^(k-1) + (n-1)^k = n*A079901(n-1,k-1), k > 0.

Sum_{k=0..n} T(n,k) = (n*(n-1)^n - 2)/(n-2), n > 2.

Edited by R. J. Mathar, Mar 31 2009

More terms added by G. C. Greubel, Mar 18 2025

A384119 Array read by antidiagonals: T(n,m) is the number of minimum dominating sets in the n X m rook graph K_n X K_m.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 6, 3, 1, 1, 4, 9, 9, 4, 1, 1, 5, 16, 48, 16, 5, 1, 1, 6, 25, 64, 64, 25, 6, 1, 1, 7, 36, 125, 488, 125, 36, 7, 1, 1, 8, 49, 216, 625, 625, 216, 49, 8, 1, 1, 9, 64, 343, 1296, 6130, 1296, 343, 64, 9, 1, 1, 10, 81, 512, 2401, 7776, 7776, 2401, 512, 81, 10, 1

Offset: 0

Andrew Howroyd, May 20 2025

For m <= n, the minimum size of a dominating set is m. When m < n, solutions have exactly one vertex in each column. In the special case of n = m, solutions either have exactly one vertex in each column or have exactly one vertex in each row.

			Array begins:
=======================================================
n\m | 0 1  2   3    4     5      6       7        8 ...
----+--------------------------------------------------
  0 | 1 1  1   1    1     1      1       1        1 ...
  1 | 1 1  2   3    4     5      6       7        8 ...
  2 | 1 2  6   9   16    25     36      49       64 ...
  3 | 1 3  9  48   64   125    216     343      512 ...
  4 | 1 4 16  64  488   625   1296    2401     4096 ...
  5 | 1 5 25 125  625  6130   7776   16807    32768 ...
  6 | 1 6 36 216 1296  7776  92592  117649   262144 ...
  7 | 1 7 49 343 2401 16807 117649 1642046  2097152 ...
  8 | 1 8 64 512 4096 32768 262144 2097152 33514112 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Eric Weisstein's World of Mathematics, Minimum Dominating Set.
Eric Weisstein's World of Mathematics, Rook Graph.

Main diagonal is A248744.

Cf. A079901, A287274, A290632.

T(n,m) = {if(n<=m, m^n) + if(m<=n, n^m) - if(m==n, n!)}

T(n,m) = T(m,n).

T(n,m) = n^m for m < n.

A376878 Triangle read by rows: T(n, k) = n^k * n! * [x^k][y^n]((sec(y) + tan(y)) * exp(x*y)).

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 2, 9, 27, 27, 5, 32, 96, 256, 256, 16, 125, 500, 1250, 3125, 3125, 61, 576, 2700, 8640, 19440, 46656, 46656, 272, 2989, 16464, 60025, 168070, 352947, 823543, 823543, 1385, 17408, 109312, 458752, 1433600, 3670016, 7340032, 16777216, 16777216

Offset: 0

Peter Luschny, Oct 13 2024

			Triangle starts:
  [0]    1;
  [1]    1,     1;
  [2]    1,     4,      4;
  [3]    2,     9,     27,     27;
  [4]    5,    32,     96,    256,     256;
  [5]   16,   125,    500,   1250,    3125,    3125;
  [6]   61,   576,   2700,   8640,   19440,   46656,   46656;
  [7]  272,  2989,  16464,  60025,  168070,  352947,  823543,   823543;
  [8] 1385, 17408, 109312, 458752, 1433600, 3670016, 7340032, 16777216, 16777216;

Cf. A000111, A000312, A079901, A109449, A292976 (row sums).

P := n -> coeff(series((sec(y) + tan(y)) * exp(x*y), y, 12), y, n):
seq(seq(coeff(P(n), x,  k) * n^k * n!, k = 0..n), n = 0..8);
T := (n, k) -> ifelse(n = k, n^n, (-1)^binomial(n - k, 2)*n^k*binomial(n, k)*(euler(n - k) - euler(n - k, 0)*2^(n - k))):
seq(print([n], seq(T(n, k), k = 0..n)), n = 0..8);

from math import comb, isqrt
from sympy import bernoulli, euler
def A000111(n): return abs(((1<A376878(n): return comb(a:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),b:=n-comb(a+1,2))*a**b*A000111(a-b) # Chai Wah Wu, Nov 13 2024

T(n, k) = (-1)^binomial(n-k, 2)*n^k*binomial(n, k)*(Euler(n-k) - Euler(n-k, 0)*2^(n - k)) for 0 <= k < n and n^n for n = k.

T(n, k) = n^k*A109449(n, k) = n^k*binomial(n, k)*A000111(n - k).

A364870 Array read by ascending antidiagonals: A(n, k) = (n + k)^n, with k >= 0.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 27, 9, 3, 1, 256, 64, 16, 4, 1, 3125, 625, 125, 25, 5, 1, 46656, 7776, 1296, 216, 36, 6, 1, 823543, 117649, 16807, 2401, 343, 49, 7, 1, 16777216, 2097152, 262144, 32768, 4096, 512, 64, 8, 1, 387420489, 43046721, 4782969, 531441, 59049, 6561, 729, 81, 9, 1

Offset: 0

Stefano Spezia, Aug 11 2023

			The array begins:
     1,    1,     1,     1,     1,      1, ...
     1,    2,     3,     4,     5,      6, ...
     4,    9,    16,    25,    36,     49, ...
    27,   64,   125,   216,   343,    512, ...
   256,  625,  1296,  2401,  4096,   6561, ...
  3125, 7776, 16807, 32768, 59049, 100000, ...
  ...

Cf. A000012 (n=0), A000169, A000272, A000312 (k=0), A007830 (k=3), A008785 (k=4), A008786 (k=5), A008787 (k=6), A031973 (antidiagonal sums), A052746 (2nd superdiagonal), A052750, A062971 (main diagonal), A079901 (read by descending antidiagonals), A085527 (1st superdiagonal), A085528 (1st subdiagonal), A085532, A099753.

A[n_,k_]:=(n+k)^n; Join[{1},Table[A[n-k,k],{n,9},{k,0,n}]]//Flatten

E.g.f. of k-th column: LambertW(-x)^k/(x^k*(1 + LambertW(-x))).

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A108396 Triangle read by rows: T(n,k) = n*(1+n^k)/2, 0<=k<=n.

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A158497 Triangle T(n,k) formed by the coordination sequences and the number of leaves for trees.

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A384119 Array read by antidiagonals: T(n,m) is the number of minimum dominating sets in the n X m rook graph K_n X K_m.

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A376878 Triangle read by rows: T(n, k) = n^k * n! * [x^k][y^n]((sec(y) + tan(y)) * exp(x*y)).

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A364870 Array read by ascending antidiagonals: A(n, k) = (n + k)^n, with k >= 0.

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