A004248 -id:A004248 - cp's OEIS frontend

A292741 Number A(n,k) of partitions of n with k sorts of part 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 3, 2, 1, 4, 10, 11, 5, 2, 1, 5, 17, 31, 24, 7, 4, 1, 6, 26, 69, 95, 50, 11, 4, 1, 7, 37, 131, 278, 287, 104, 15, 7, 1, 8, 50, 223, 657, 1114, 865, 212, 22, 8, 1, 9, 65, 351, 1340, 3287, 4460, 2599, 431, 30, 12, 1, 10, 82, 521, 2459, 8042, 16439, 17844, 7804, 870, 42, 14

Offset: 0

Alois P. Heinz, Sep 22 2017

			A(1,3) = 3: 1a, 1b, 1c.
A(2,3) = 10: 2, 1a1a, 1a1b, 1a1c, 1b1a, 1b1b, 1b1c, 1c1a, 1c1b, 1c1c.
A(3,2) = 11: 3, 21a, 21b, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a, 1b1b1b.
Square array A(n,k) begins:
  1,  1,   1,    1,     1,     1,      1,      1, ...
  0,  1,   2,    3,     4,     5,      6,      7, ...
  1,  2,   5,   10,    17,    26,     37,     50, ...
  1,  3,  11,   31,    69,   131,    223,    351, ...
  2,  5,  24,   95,   278,   657,   1340,   2459, ...
  2,  7,  50,  287,  1114,  3287,   8042,  17215, ...
  4, 11, 104,  865,  4460, 16439,  48256, 120509, ...
  4, 15, 212, 2599, 17844, 82199, 289540, 843567, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-2 give: A002865, A000041, A090764.
Rows n=0-2 give: A000012, A001477, A002522, A071568.
Main diagonal gives A292462.
Cf. A003992, A004248, A009998, A051129, A292508, A292622, A292745.

b:= proc(n, i, k) option remember; `if`(n=0 or i<2, k^n,
      add(b(n-i*j, i-1, k), j=0..iquo(n, i)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[0, , ] = 1; b[n_, i_, k_] := b[n, i, k] = If[i < 2, k^n, Sum[b[n - i*j, i - 1, k], {j, 0, Quotient[n, i]}]];
A[n_, k_] := b[n, n, k];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 19 2018, translated from Maple *)

G.f. of column k: 1/(1-k*x) * 1/Product_{j>=2} (1-x^j).

A(n,k) = Sum_{j=0..n} A002865(j) * k^(n-j).

A344499 T(n, k) = F(n - k, k), where F(n, x) is the Fubini polynomial. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 13, 10, 3, 1, 0, 75, 74, 21, 4, 1, 0, 541, 730, 219, 36, 5, 1, 0, 4683, 9002, 3045, 484, 55, 6, 1, 0, 47293, 133210, 52923, 8676, 905, 78, 7, 1, 0, 545835, 2299754, 1103781, 194404, 19855, 1518, 105, 8, 1, 0, 7087261, 45375130, 26857659, 5227236, 544505, 39390, 2359, 136, 9, 1

Offset: 0

Peter Luschny, May 21 2021

The array rows are recursively generated by applying the Akiyama-Tanigawa algorithm to the powers (see the Python implementation below). In this way the array becomes the image of A004248 under the AT-transformation when applied to the columns of A004248. This makes the array closely linked to A371761, which is generated in the same way, but applied to the rows of A004248. - Peter Luschny, Apr 27 2024

			Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,      1;
[3] 0, 3,      2,       1;
[4] 0, 13,     10,      3,       1;
[5] 0, 75,     74,      21,      4,      1;
[6] 0, 541,    730,     219,     36,     5,     1;
[7] 0, 4683,   9002,    3045,    484,    55,    6,    1;
[8] 0, 47293,  133210,  52923,   8676,   905,   78,   7,   1;
[9] 0, 545835, 2299754, 1103781, 194404, 19855, 1518, 105, 8, 1;
.
Seen as an array A(n, k) = T(n + k, n):
[0] [1, 0,   0,    0,     0,       0,         0, ...  A000007
[1] [1, 1,   3,   13,    75,     541,      4683, ...  A000670
[2] [1, 2,  10,   74,   730,    9002,    133210, ...  A004123
[3] [1, 3,  21,  219,  3045,   52923,   1103781, ...  A032033
[4] [1, 4,  36,  484,  8676,  194404,   5227236, ...  A094417
[5] [1, 5,  55,  905, 19855,  544505,  17919055, ...  A094418
[6] [1, 6,  78, 1518, 39390, 1277646,  49729758, ...  A094419
[7] [1, 7, 105, 2359, 70665, 2646007, 118893705, ...  A238464

Variant of the array is A094416 (which has column 0 and row 0 missing).
The coefficients of the Fubini polynomials are A131689.
Cf. A094420 (main diagonal of array), A372346 (row sums), A004248, A371761.

F := proc(n) option remember; if n = 0 then return 1 fi:
expand(add(binomial(n, k)*F(n - k)*x, k = 1..n)) end:
seq(seq(subs(x = k, F(n - k)), k = 0..n), n = 0..10);

F[n_] := F[n] = If[n == 0, 1,
   Expand[Sum[Binomial[n, k]*F[n - k]*x, {k, 1, n}]]];
Table[Table[F[n - k] /. x -> k, {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jun 06 2024, after Peter Luschny *)

# Computes the triangle.
@cached_function
def F(n):
    R. = PolynomialRing(ZZ)
    if n == 0: return R(1)
    return R(sum(binomial(n, k)*F(n - k)*x for k in (1..n)))
def Fval(n): return [F(n - k).substitute(x = k) for k in (0..n)]
for n in range(10): print(Fval(n))

# Computes the square array using the Akiyama-Tanigawa algorithm.
def ATFubini(n, len):
    A = [0] * len
    R = [0] * len
    for k in range(len):
        R[k] = (n + 1)**k  # Chancing this to R[k] = k**n generates A371761.
        for j in range(k, 0, -1):
            R[j - 1] = j * (R[j] - R[j - 1])
        A[k] = R[0]
    return A
for n in range(8): print([n], ATFubini(n, 7))  # Peter Luschny, Apr 27 2024

T(n, k) = (n - k)! * [x^(n - k)] (1 / (1 + k * (1 - exp(x)))).

T(2*n, n) = A094420(n).

A062275 Array A(n, k) = n^k * k^n, n, k >= 0, read by antidiagonals.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 16, 3, 0, 0, 4, 72, 72, 4, 0, 0, 5, 256, 729, 256, 5, 0, 0, 6, 800, 5184, 5184, 800, 6, 0, 0, 7, 2304, 30375, 65536, 30375, 2304, 7, 0, 0, 8, 6272, 157464, 640000, 640000, 157464, 6272, 8, 0, 0, 9, 16384, 750141, 5308416, 9765625

Offset: 0

Henry Bottomley, Jul 02 2001

Here 0^0 is defined to be 1. - Wolfdieter Lang, May 27 2018

			A(3, 2) = 3^2 * 2^3 = 9*8 = 72.
The array A(n, k) begins:
n\k 0 1   2   3    4     5      6      7       8        9       10 ...
0:  1 0   0   0    0     0      0      0       0        0        0 ...
1:  0 1   2   3    4     5      6      7       8        9       10 ...
2:  0 2  16  72  256   800   2304   6272   16384    41472   102400 ...
3:  0 3  72 729 5184 30375 157464 750141 3359232 14348907 59049000 ...
...
The triangle T(n, k) begins:
n\k  0  1    2      3      4      5      6    7  8  9 ...
0:   1
1:   0  0
2:   0  1    0
3:   0  2    2      0
4:   0  3   16      3      0
5:   0  4   72     72      4      0
6:   0  5  256    729    256      5      0
7:   0  6  800   5184   5184    800      6    0
8:   0  7 2304  30375  65536  30375   2304    7  0
9:   0  8 6272 157464 640000 640000 157464 6272  8  0
... - _Wolfdieter Lang_, May 22 2018

Eric Chen, Table of n, a(n) for n = 0..5049 (first 100 antidiagonals)

Columns and rows of A, or columns and diagonals of T, include A000007, A001477, A007758, A062074, A062075 etc. Diagonals of A include A062206, A051443, A051490. Sum of rows of T are A062817(n), for n >= 1

Cf. A055651, A055652, A303990.

{{1}}~Join~Table[(#^k k^#) &[n - k], {n, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, May 24 2018 *)

t1(n)=n-binomial(round(sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
a(n)=t1(n)^t2(n)*t2(n)^t1(n) \\ Eric Chen, Jun 09 2018

From Wolfdieter Lang, May 22 2018: (Start)
As a sequence: a(n) = A003992(n)*A004248(n).
As a triangle: T(n, k) = (n-k)^k * k^(n-k), for n >= 1 and k = 1..n. (End)

A110132 a(n) = floor(n/2)^ceiling(n/2).

Original entry on oeis.org

1, 0, 1, 1, 4, 8, 27, 81, 256, 1024, 3125, 15625, 46656, 279936, 823543, 5764801, 16777216, 134217728, 387420489, 3486784401, 10000000000, 100000000000, 285311670611, 3138428376721, 8916100448256, 106993205379072, 302875106592253, 3937376385699289

Offset: 0

Paul Barry, Jul 13 2005

Central coefficients T(n,floor(n/2)) of number triangle A004248. The central coefficients T(2n,n) of this triangle are n^n or A000312.

a(n) is the number of partitions of [n] such that each block has exactly one even element: a(5) = 8: 1235|4, 123|45, 125|34, 12|345, 1345|2, 134|25, 145|23, 14|235. - Alois P. Heinz, Jun 01 2023

Alois P. Heinz, Table of n, a(n) for n = 0..772

Cf. A000312, A004248, A110138.

Join[{1},Table[Floor[n/2]^Ceiling[n/2],{n,30}]] (* Harvey P. Dale, Feb 16 2025 *)

A329940 Square array read by antidiagonals upwards: T(n,k) is the number of right unique relations between set A with n elements and set B with k elements.

Original entry on oeis.org

1, 3, 2, 7, 8, 3, 15, 26, 15, 4, 31, 80, 63, 24, 5, 63, 242, 255, 124, 35, 6, 127, 728, 1023, 624, 215, 48, 7, 255, 2186, 4095, 3124, 1295, 342, 63, 8, 511, 6560, 16383, 15624, 7775, 2400, 511, 80, 9, 1023, 19682, 65535, 78124, 46655, 16806, 4095, 728, 99, 10

Offset: 1

Roy S. Freedman, Nov 24 2019

A relation R between set A with n elements and set B with k elements is a subset of the Cartesian product A x B. A relation R is right unique if (a, b1) in R and (a,b2) in R implies b1=b2. T(n,k) is the number of right unique relations and T(k,n) is the number of left unique relations: relation R is left unique if (a1,b) in R and (a2,b) in R implies a1=a2.

			T(n,k) begins:
    1,    2,     3,      4,       5,       6,        7,        8, ...
    3,    8,    15,     24,      35,      48,       63,       80, ...
    7,   26,    63,    124,     215,     342,      511,      728, ...
   15,   80,   255,    624,    1295,    2400,     4095,     6560, ...
   31,  242,  1023,   3124,    7775,   16806,    32767,    59048, ...
   63,  728,  4095,  15624,   46655,  117648,   262143,   531440, ...
  127, 2186, 16383,  78124,  279935,  823542,  2097151,  4782968, ...
  255, 6560, 65535, 390624, 1679615, 5764800, 16777215, 43046720, ...

Roy S. Freedman, Some New Results on Binary Relations, arXiv:1501.01914 [cs.DM], 2015.

Cf. A037205 (main diagonal).

Cf. A003992, A004248, A009998, A009999, A051128, A051129, A095884.

T:= (n, k)-> (k+1)^n-1:
seq(seq(T(1+d-k, k), k=1..d), d=1..12);

T[n_, k_] := (k + 1)^n - 1; Table[T[n - k + 1, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 25 2019 *)

MuPAD
```
T:=(n,k)->(k+1)^n-1:
```

T(n,k) = (k+1)^n - 1.

A361390 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is carryless n^k base 10.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 8, 9, 4, 1, 0, 1, 6, 7, 6, 5, 1, 0, 1, 2, 1, 4, 5, 6, 1, 0, 1, 4, 3, 6, 5, 6, 7, 1, 0, 1, 8, 9, 4, 5, 6, 9, 8, 1, 0, 1, 6, 7, 6, 5, 6, 3, 4, 9, 1, 0, 1, 2, 1, 4, 5, 6, 1, 2, 1, 10, 1, 0, 1, 4, 3, 6, 5, 6, 7, 6, 9, 100, 11, 1, 0, 1, 8, 9, 4, 5, 6, 9, 8, 1, 1000, 121, 12, 1

Offset: 0

Seiichi Manyama, Mar 10 2023

			4 * 4 = 16, so T(4,2) = 6. 6 * 4 = 24, so T(4,3) = 4.
Square array begins:
  1, 0, 0, 0, 0, 0, 0, 0, ...
  1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 2, 4, 8, 6, 2, 4, 8, ...
  1, 3, 9, 7, 1, 3, 9, 7, ...
  1, 4, 6, 4, 6, 4, 6, 4, ...
  1, 5, 5, 5, 5, 5, 5, 5, ...
  1, 6, 6, 6, 6, 6, 6, 6, ...
  1, 7, 9, 3, 1, 7, 9, 3, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
Index entries for sequences related to carryless arithmetic

Columns k=0..4 give A000012, A001477, A059729, A169885, A169886.
Rows n=0..4 give A000007, A000012, A000689, A001148, A168428.
T(11,k) gives A059734.
Main diagonal gives A361351.
Cf. A004248, A359697.

T(n, k) = fromdigits(Vec(Pol(digits(n))^k)%10);

A362885 Array read by ascending antidiagonals: A(n, k) = (1 + 2n)k^n.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 5, 6, 1, 0, 7, 20, 9, 1, 0, 9, 56, 45, 12, 1, 0, 11, 144, 189, 80, 15, 1, 0, 13, 352, 729, 448, 125, 18, 1, 0, 15, 832, 2673, 2304, 875, 180, 21, 1, 0, 17, 1920, 9477, 11264, 5625, 1512, 245, 24, 1, 0, 19, 4352, 32805, 53248, 34375, 11664, 2401, 320, 27, 1

Offset: 0

Stefano Spezia, May 08 2023

			The array begins:
    1,  1,   1,    1,     1,     1, ...
    0,  3,   6,    9,    12,    15, ...
    0,  5,  20,   45,    80,   125, ...
    0,  7,  56,  189,   448,   875, ...
    0,  9, 144,  729,  2304,  5625, ...
    0, 11, 352, 2673, 11264, 34375, ...
    ...

Cf. A000007 (k=0), A000012 (n=0), A004248, A005408 (k=1), A008585 (n=1), A014480 (k=2), A033429 (n=2), A058962 (k=4), A124647 (k=3), A155988 (k=9), A171220 (k=5), A176043, A199299 (k=6), A199300 (k=7), A199301 (k=8), A244727 (n=3), A362886 (antidiagonal sums).

A[n_,k_]:=(1+2n)k^n; Join[{1}, Table[A[n-k,k],{n,10},{k,0,n}]]//Flatten (* or *)
A[n_,k_]:=SeriesCoefficient[(1+k*x)/(1-k*x)^2,{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
A[n_,k_]:=n!SeriesCoefficient[Exp[k*x](1+2k*x),{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

A(n, k) = A005408(n)*A004248(n, k).
O.g.f. of column k: (1 + k*x)/(1 - k*x)^2.
E.g.f. of column k: exp(k*x)*(1 + 2*k*x).
A(n, n) = A176043(n+1).

A372118 Square array A(n, k) = ((k+2)^(n+2) - 2 * (k+1)^(n+2) + k^(n+2))/2 for k, n >= 0 read by ascending antidiagonals.

Original entry on oeis.org

1, 3, 1, 7, 6, 1, 15, 25, 9, 1, 31, 90, 55, 12, 1, 63, 301, 285, 97, 15, 1, 127, 966, 1351, 660, 151, 18, 1, 255, 3025, 6069, 4081, 1275, 217, 21, 1, 511, 9330, 26335, 23772, 9751, 2190, 295, 24, 1, 1023, 28501, 111645, 133057, 70035, 19981, 3465, 385, 27, 1

Offset: 0

Werner Schulte, Apr 19 2024

Depending on some fixed integer m >= 0 we define a family of square arrays A(m; n, k) = (Sum_{i=0..m} (-1)^i * binomial(m, i) * (k + m - i)^(n+m)) / m! for k, n >= 0. Special cases are: A004248 (m=0), A343237 (m=1) and this array (m=2). The A(m; n, k) satisfy: A(m; n, k) = (k+m) * A(m; n-1, k) + A(m-1; n, k) with initial values A(0; n, k) = k^n and A(m; 0, k) = 1.
Further properties are conjectures:
  (1) O.g.f. of column k is Prod_{i=k..k+m} 1 / (1 - i * t);
  (2) E.g.f. of row n is exp(x) * (Sum_{k=0..n} binomial(k+m, m) * A048993(n+m, k+m) * x^k);
  (3) The LU decompositions of these arrays are given by the upper triangular matrix U which is the transpose of A007318 and the lower triangular matrix L, where L is defined L(m; n, k) = A048993(n+m, k+m) * (k+m)! / m!, i.e., A(m; n, k) = Sum_{i=0..k} L(m; n, i) * binomial(k, i).
The three conjectures are true, see links. - Sela Fried, Jul 07 2024

			Square array A(n, k) starts:
n\k :    0     1       2       3        4         5         6         7
=======================================================================
  0 :    1     1       1       1        1         1         1         1
  1 :    3     6       9      12       15        18        21        24
  2 :    7    25      55      97      151       217       295       385
  3 :   15    90     285     660     1275      2190      3465      5160
  4 :   31   301    1351    4081     9751     19981     36751     62401
  5 :   63   966    6069   23772    70035    170898    365001    706104
  6 :  127  3025   26335  133057   481951   1398097   3463615   7628545
  7 :  255  9330  111645  724260  3216795  11075670  31794105  79669320
  etc.

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).
Sela Fried, On an integer sequence related to Euler's formula for the Stirling numbers of the second kind, 2024.

Rows: A000012 (n=0), A008585 (n=1), A227776 (n=2).
Columns: A000225 (k=0), A000392 (k=1), A016269 (k=2), A016753 (k=3), A016103 (k=4), A019757 (k=5), A020570 (k=6), A020782 (k=7).
Main diagonal: A281596(n+2).
Cf. A007318, A048993, A004248, A343237.

A372118[n_, k_] := ((k+2)^(n+2) - 2*(k+1)^(n+2) + k^(n+2))/2;
Table[A372118[n-k, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jul 10 2024 *)

A(n, k) = ((k+2)^(n+2) - 2 * (k+1)^(n+2) + k^(n+2))/2

A(n, k) = (k+2) * A(n-1, k) + (k+1)^(n+1) - k^(n+1) for n > 0.
Conjectures:
  (1) O.g.f. of column k is Prod_{i=k..k+2} 1 / (1 - i * t);
  (2) E.g.f. of row n is exp(x) * (Sum_{k=0..n} binomial(k+2, 2) * A048993(n+2, k+2) * x^k);
  (3) The LU decomposition of this array is given by the upper triangular matrix U which is the transpose of A007318 and the lower triangular matrix L, where L is defined L(n, k) = A048993(n+2, k+2) * (k+2)! / 2!, i.e., A(n, k) = Sum_{i=0..k} L(n, i) * binomial(k, i).
The three conjectures are true. See comments. - Sela Fried, Jul 09 2024

A375577 Array read by ascending antidiagonals: A(n,k) = k^n + k*n + 1.

Original entry on oeis.org

2, 1, 2, 1, 3, 2, 1, 4, 5, 2, 1, 5, 9, 7, 2, 1, 6, 15, 16, 9, 2, 1, 7, 25, 37, 25, 11, 2, 1, 8, 43, 94, 77, 36, 13, 2, 1, 9, 77, 259, 273, 141, 49, 15, 2, 1, 10, 143, 748, 1045, 646, 235, 64, 17, 2, 1, 11, 273, 2209, 4121, 3151, 1321, 365, 81, 19, 2

Offset: 0

Stefano Spezia, Aug 19 2024

			Array begins:
  2, 2,  2,   2,    2,     2, ...
  1, 3,  5,   7,    9,    11, ...
  1, 4,  9,  16,   25,    36, ...
  1, 5, 15,  37,   77,   141, ...
  1, 6, 25,  94,  273,   646, ...
  1, 7, 43, 259, 1045,  3151, ...
  1, 8, 77, 748, 4121, 15656, ...
  ...

Cf. A000290, A004247, A004248, A005408 (n=1), A005491 (n=3), A007395 (n=0), A054977 (k=0), A176691 (k=2), A176805 (k=3), A176916 (k=5), A176972 (k=7), A214647.

Cf. A375578 (antidiagonal sums).

A[0,0]=2; A[n_,k_]:=k^n+k*n+1;Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

G.f. for the k-th column: (2*x^2 - 3*x - k^2 + k + 1)/((x - 1)^2*(x - k)).
E.g.f. for the k-th column: exp(x)*(1 + exp((k-1)*x) + k*x).
A(n,1) = n + 2.
A(2,n) = A000290(n+1).
A(n,n) = 2*A214647(n) + 1.

A338131 Triangle read by rows, T(n, k) = k^(n - k) + Sum_{i = 1..n-k} k^(n - k - i)*2^(i - 1), for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 8, 8, 8, 4, 1, 16, 16, 20, 14, 5, 1, 32, 32, 48, 46, 22, 6, 1, 64, 64, 112, 146, 92, 32, 7, 1, 128, 128, 256, 454, 376, 164, 44, 8, 1, 256, 256, 576, 1394, 1520, 828, 268, 58, 9, 1, 512, 512, 1280, 4246, 6112, 4156, 1616, 410, 74, 10, 1

Offset: 0

Werner Schulte, Oct 11 2020

This number triangle is case s = 2 of the triangles T(s; n,k) depending on some fixed integer s. Here are several (generalized) formulas and properties (attention: negative values are possible if s < 0):
  (1) T(s; n,k) = k^(n-k) + Sum_{i=1..n-k} k^(n-k-i)*s^(i-1) for 0<=k<=n;
  (2) T(s; n,n) = 1 for n >= 0, and T(s; n,n-1) = n for n > 0;
  (3) T(s; n+1,k) = k * T(s; n,k) + s^(n-k) for 0<=k<=n;
  (4) T(s; n,k) = (k+s) * T(s; n-1,k) - s*k * T(s; n-2,k) for 0<=k<=n-2;
  (5) G.f. of column k: Sum_{n>=k} T(s; n,k)*t^n = ((1-(s-1)*t)/(1-s*t))
      *(t^k/(1-k*t)) when t^k/(1-k*t) is g.f. of column k>=0 of A004248.

			The number triangle T(n, k) for 0 <= k <= n starts:
n\ k :    0     1      2      3      4      5      6     7    8    9   10
=========================================================================
   0 :    1
   1 :    1     1
   2 :    2     2      1
   3 :    4     4      3      1
   4 :    8     8      8      4      1
   5 :   16    16     20     14      5      1
   6 :   32    32     48     46     22      6      1
   7 :   64    64    112    146     92     32      7     1
   8 :  128   128    256    454    376    164     44     8    1
   9 :  256   256    576   1394   1520    828    268    58    9    1
  10 :  512   512   1280   4246   6112   4156   1616   410   74   10   1

Cf. A004248.

For columns k = 0, 1, 2, 3, 4 see A011782, A000079, A001792, A027649, A010036 respectively.

T := proc(n, k) if k = 0 then `if`(n = 0, 1, 2^(n-1)) elif k = 2 then n*2^(n-3)
else (k^(n-k)*(1-k) + 2^(n-k))/(2-k) fi end:
seq(seq(T(n, k), k=0..n), n=0..10); # Peter Luschny, Oct 29 2020

T(n,k) = k^(n-k) + sum(i=1, n-k, k^(n-k-i) * 2^(i-1));
matrix(7,7, n, k, if(n>=k, T(n-1,k-1), 0)) \\ to see the triangle \\ Michel Marcus, Oct 12 2020

T(n,k) = ((k-1) * k^(n-k) - 2^(n-k)) / (k-2) if k <> 2, and T(n,2) = n * 2^(n-3) for n >= k.
T(n,n) = 1 for n >= 0, and T(n,n-1) = n for n > 0.
T(n+1,k) = k * T(n,k) + 2^(n-k) for 0 <= k <= n.
T(n,k) = (k+2) * T(n-1,k) - 2*k * T(n-2,k) for 0 <= k <= n-2.
T(n,k) = k * T(n-1,k) + T(n-1,k-1) - (k-1) * T(n-2,k-1) for 0 < k < n.
G.f. of column k >= 0: Sum_{n>=k} T(n,k) * t^n = ((1-t) / (1-2*t)) * (t^k / (1-k*t)) when t^k / (1-k*t) is g. f. of column k of A004248.
G.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n = ((1-t) / (1-2*t)) * (Sum_{k>=0} (x*t)^k / (1-k*t)).

cp's OEIS Frontend

A292741 Number A(n,k) of partitions of n with k sorts of part 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A344499 T(n, k) = F(n - k, k), where F(n, x) is the Fubini polynomial. Triangle read by rows, T(n, k) for 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

SageMath

SageMath

Formula

A062275 Array A(n, k) = n^k * k^n, n, k >= 0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A110132 a(n) = floor(n/2)^ceiling(n/2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A329940 Square array read by antidiagonals upwards: T(n,k) is the number of right unique relations between set A with n elements and set B with k elements.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

MuPAD

Formula

A361390 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is carryless n^k base 10.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A362885 Array read by ascending antidiagonals: A(n, k) = (1 + 2*n)*k^n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A372118 Square array A(n, k) = ((k+2)^(n+2) - 2 * (k+1)^(n+2) + k^(n+2))/2 for k, n >= 0 read by ascending antidiagonals.

Views

Author

A362885 Array read by ascending antidiagonals: A(n, k) = (1 + 2n)k^n.