A028872 -id:A028872 - cp's OEIS frontend

A361682 Array read by descending antidiagonals. A(n, k) is the number of multiset combinations of {0, 1} whose type is defined in the comments. Also A(n, k) = hypergeom([-k, -2], [1], n).

1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 10, 13, 7, 1, 1, 15, 25, 22, 9, 1, 1, 21, 41, 46, 33, 11, 1, 1, 28, 61, 79, 73, 46, 13, 1, 1, 36, 85, 121, 129, 106, 61, 15, 1, 1, 45, 113, 172, 201, 191, 145, 78, 17, 1, 1, 55, 145, 232, 289, 301, 265, 190, 97, 19, 1

Offset: 0

Peter Luschny, Mar 21 2023

A combination of a multiset M is an unordered selection of k objects of M, where every object can appear at most as many times as it appears in M.
A(n, k) = Cardinality(Union_{j=0..k} Combination(MultiSet(1^[j*n], 0^[(k-j)*n]))), where MultiSet(r^[s], u^[v]) denotes a set that contains the element r with multiplicity s and the element u with multiplicity v; thus the multisets under consideration have n*k elements. Since the base set is {1, 0} the elements can be represented as binary strings. Applying the combination operator to the multisets results in a set of binary strings where '0' resp. '1' can appear at most j*n resp. (k-j)*n times. 'At most' means that they do not have to appear; in other words, the resulting set always includes the empty string ''.
In contrast to the procedure in A361045 we consider here the cardinality of the set union and not the sum of the individual cardinalities. If you want to exclude the empty string, you will find the sequences listed in A361521. The same construction with multiset permutations instead of multiset combinations results in A361043.
A different view can be taken if one considers the hypergeometric representation, hypergeom([-k, -m], [1], n). This is a family of arrays that includes the 'rascal' triangle: the all 1's array A000012 (m = 0), the rascal array A077028 (m = 1), this array (m = 2), and A361731 (m = 3).

			Array A(n, k) starts:
   [0] 1,  1,   1,    1,   1,   1,   1,    1, ...  A000012
   [1] 1,  3,   6,   10,  15,  21,  28,   36, ...  A000217
   [2] 1,  5,  13,   25,  41,  61,  85,  113, ...  A001844
   [3] 1,  7,  22,   46,  79, 121, 172,  232, ...  A038764
   [4] 1,  9,  33,   73, 129, 201, 289,  393, ...  A081585
   [5] 1, 11,  46,  106, 191, 301, 436,  596, ...  A081587
   [6] 1, 13,  61,  145, 265, 421, 613,  841, ...  A081589
   [7] 1, 15,  78,  190, 351, 561, 820, 1128, ...  A081591
   000012  | A028872 | A239325 |
       A005408    A100536   A069133
.
Triangle T(n, k) starts:
   [0] 1;
   [1] 1,  1;
   [2] 1,  3,   1;
   [3] 1,  6,   5,   1;
   [4] 1, 10,  13,   7,   1;
   [5] 1, 15,  25,  22,   9,   1;
   [6] 1, 21,  41,  46,  33,  11,   1;
   [7] 1, 28,  61,  79,  73,  46,  13,  1;
   [8] 1, 36,  85, 121, 129, 106,  61, 15,  1;
   [9] 1, 45, 113, 172, 201, 191, 145, 78, 17, 1.
.
Row 4 of the triangle:
A(0, 4) =  1 = card('').
A(1, 3) = 10 = card('', 0, 00, 000, 1, 10, 100, 11, 110, 111).
A(2, 2) = 13 = card('', 0, 00, 000, 0000, 1, 10, 100, 11, 110, 1100, 111, 1111).
A(3, 1) =  7 = card('', 0, 00, 000, 1, 11, 111).
A(4, 0) =  1 = card('').

Rows: A000012, A000217, A001844, A038764, A081585, A081587, A081589, A081591.
Columns: A000012, A005408, A028872, A100536, A239325, A069133.
Cf. A239592 (main diagonal), A239331 (transposed array).
Cf. A361045, A361043, A361521, A361731, A077028.

A := (n, k) -> 1 + n*k*(4 + n*(k - 1))/2:
for n from 0 to 7 do seq(A(n, k), k = 0..7) od;
# Alternative:
ogf := n -> (1 + (n - 1)*x)^2 / (1 - x)^3:
ser := n -> series(ogf(n), x, 12):
row := n -> seq(coeff(ser(n), x, k), k = 0..9):
seq(print(row(n)), n = 0..7);

def A(m: int, steps: int) -> int:
    if m == 0: return 1
    size = m * steps
    cset = set()
    for a in range(0, size + 1, m):
        S = [str(int(i < a)) for i in range(size)]
        C = Combinations(S)
        cset.update("".join(i for i in c) for c in C)
    return len(cset)
def ARow(n: int, size: int) -> list[int]:
    return [A(n, k) for k in range(size + 1)]
for n in range(8): print(ARow(n, 7))

A(n, k) = 1 + n*k*(4 + n*(k - 1))/2.
T(n, k) = 1 + k*(n - k)*(4 + k*(n - k - 1))/2.
A(n, k) = [x^k] (1 + (n - 1)*x)^2 / (1 - x)^3.
A(n, k) = hypergeom([-k, -2], [1], n).
A(n, k) = A361521(n, k) + 1.

A039824 Number of different coefficient values in expansion of Product (1+q^1+q^3...+q^(2i-1)), i=1 to n.

Original entry on oeis.org

1, 2, 4, 11, 20, 31, 46, 61, 78, 97, 118, 141, 166, 193, 222, 253, 286, 321, 358, 397, 438, 481, 526, 573, 622, 673, 726, 781, 838, 897, 958, 1021, 1086, 1153, 1222, 1293, 1366, 1441, 1518, 1597, 1678, 1761, 1846, 1933, 2022, 2113, 2206, 2301, 2398, 2497

Offset: 1

Olivier Gérard

nonn

J. Conrad, Table of n, a(n) for n = 1..180

Cf. A028872.

p[1] = 1 + q; p[n_] := p[n] = p[n - 1] (1 + Sum[q^k, {k, 1, 2 n - 1, 2}]) // Expand; a[1] = 1; a[n_] := p[n] // CoefficientList[#, q]& // Union // Length; Array[a, 180] (* Jean-François Alcover, May 04 2017 *)

def get(d, x): return d[x] if len(d) > x >= 0 else 0
def convolve(a, b):
    r = []
    for x in range(len(a) + len(b) - 1):
        n = 0
        for k in range(x + 1): n += get(a, k) * get(b, x - k)
        r.append(n)
    return r
def unique_in(d):
    out = list([])
    for elem in d:
        if elem not in out: out.append(elem)
    return len(out)
def A039824(x):
    seed = [0**k + k % 2 for k in range(2*(x+1))]
    product = seed[0:2]
    out = list([1])
    for k in range(2, x + 1):
        product = convolve(product, seed[0:2*k])
        out.append(unique_in(product))
    return out
# J. Conrad, May 02 2017

Conjecture: for n>6, a(n) = n^2 - 3. - Ralf Stephan, Mar 07 2004
Conjectures from Colin Barker, May 02 2017: (Start)
G.f.: x*(1 - x + x^2 + 4*x^3 - 3*x^4 + 2*x^6 - 4*x^7 + 2*x^8) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>9.
(End)

A110292 Riordan array (1-u, u) where u=(-1 + sqrt(1+8*x))/4.

Original entry on oeis.org

1, -1, 1, 2, -3, 1, -8, 12, -5, 1, 40, -60, 26, -7, 1, -224, 336, -148, 44, -9, 1, 1344, -2016, 896, -280, 66, -11, 1, -8448, 12672, -5664, 1824, -464, 92, -13, 1, 54912, -82368, 36960, -12144, 3240, -708, 122, -15, 1, -366080, 549120, -247104, 82368, -22704, 5280, -1020, 156, -17, 1

Offset: 0

Paul Barry, Jul 18 2005

Inverse of Riordan array (1/(1-x), x*(1+2*x)), A110291.

			Triangle begins as:
      1;
     -1,      1;
      2,     -3,     1;
     -8,     12,    -5,      1;
     40,    -60,    26,     -7,    1;
   -224,    336,  -148,     44,   -9,    1;
   1344,  -2016,   896,   -280,   66,  -11,   1;
  -8448,  12672, -5664,   1824, -464,   92, -13,   1;
  54912, -82368, 36960, -12144, 3240, -708, 122, -15,  1;

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

Cf. A000007, A028872, A110291, A181282.

R:=PowerSeriesRing(Rationals(), 30);
F:= func< k | Coefficients(R!( (5-Sqrt(1+8*x))*(-1+Sqrt(1+8*x) )^k/4^(k+1) )) >;
A110292:= func< n,k | F(k)[n-k+1] >;
[A110292(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 04 2023

F[k_]:= CoefficientList[Series[(5-Sqrt[1+8*x])*(-1+Sqrt[1+8*x])^k/4^(k +1), {x,0,20}], x];
A110292[n_, k_]:= F[k][[n+1]];
Table[A110292[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 04 2023 *)

def p(k,x): return (5-sqrt(1+8*x))*(-1+sqrt(1+8*x))^k/4^(k+1)
def A110292(n,k): return ( p(k,x) ).series(x, 30).list()[n]
flatten([[A110292(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 04 2023

T(n, 0) = (-1)^n * 2^(n-1) * Catalan(n-1) + (3/2)*[n=0].
From G. C. Greubel, Jan 04 2023: (Start)
T(n, n) = 1.
T(n, n-1) = 1-2*n.
T(n, n-2) = 2*A028872(n).
T(n, 1) = (-1)^(n-1) * A181282(n-1), n >= 1.
Sum_{k=0..n} T(n, k) = A000007(n). (End)

A214859 Triangle read by rows, T(n,k) = n^2 - k(k+1)/2 if k(k+1)/2 <= n^2.

Original entry on oeis.org

0, 1, 0, 4, 3, 1, 9, 8, 6, 3, 16, 15, 13, 10, 6, 1, 25, 24, 22, 19, 15, 10, 4, 36, 35, 33, 30, 26, 21, 15, 8, 0, 49, 48, 46, 43, 39, 34, 28, 21, 13, 4, 64, 63, 61, 58, 54, 49, 43, 36, 28, 19, 9, 81, 80, 78, 75, 71, 66, 60, 53, 45, 36, 26, 15, 3, 100, 99, 97

Offset: 0

Philippe Deléham, Mar 09 2013

Row lengths are in A214857.

			Triangle begins:
    0;
    1,   0;
    4,   3,   1;
    9,   8,   6,   3;
   16,  15,  13,  10,   6,   1;
   25,  24,  22,  19,  15,  10,   4;
   36,  35,  33,  30,  26,  21,  15,  8,  0;
   49,  48,  46,  43,  39,  34,  28, 21, 13,  4;
   64,  63,  61,  58,  54,  49,  43, 36, 28, 19,  9;
   81,  80,  78,  75,  71,  66,  60, 53, 45, 36, 26, 15,  3;
  100,  99,  97,  94,  90,  85,  79, 72, 64, 55, 45, 34, 22,  9;
  121, 120, 118, 115, 111, 106, 100, 93, 85, 76, 66, 55, 43, 30, 16, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. Diagonals: A000217, A034856, A055999,

Columns: A000290, A005563, A028872, A028878.

Table[s = {}; k = 0; While[tri = k*(k + 1)/2; tri <= n^2, AppendTo[s, n^2 - tri]; k++]; s, {n, 0, 10}] (* T. D. Noe, Mar 11 2013 *)

T(2*n,n) = A022264(n).

T(n,n) = n*(n-1)/2 = A000217(n-1).

A239331 Square array, read by antidiagonals: column k has g.f. (1+(k-1)*x)^2/(1-x)^3.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 7, 13, 10, 1, 1, 9, 22, 25, 15, 1, 1, 11, 33, 46, 41, 21, 1, 1, 13, 46, 73, 79, 61, 28, 1, 1, 15, 61, 106, 129, 121, 85, 36, 1, 1, 17, 78, 145, 191, 201, 172, 113, 45, 1, 1, 19, 97, 190, 265, 301, 289, 232, 145, 55, 1, 1, 21

Offset: 0

Philippe Deléham, Mar 16 2014

			Square array begins:
n\k : 0......1......2......3......4......5......6......7......8......9
======================================================================
.0||  1......1......1......1......1......1......1......1......1......1
.1||  1......3......5......7......9.....11.....13.....15.....17.....19
.2||  1......6.....13.....22.....33.....46.....61.....78.....97....118
.3||  1.....10.....25.....46.....73....106....145....190....241....298
.4||  1.....15.....41.....79....129....191....265....351....449....559
.5||  1.....21.....61....121....201....301....421....561....721....901
.6||  1.....28.....85....172....289....436....613....820...1057...1324
.7||  1.....36....113....232....393....596....841...1128...1457...1828
.8||  1.....45....145....301....513....781...1105...1485...1921...2413
.9||  1.....55....181....379....649....991...1405...1891...2449...3079
10||  1.....66....221....466....801...1226...1741...2346...3041...3826
11||  1.....78....265....562....969...1486...2113...2850...3697...4654

Cf. Columns: A000012, A000217, A001844, A038764, A081585, A081587, A081589, A081591, A081593.

Cf. Rows: A000012, A005408, A028872, A100536, A239325, A069133.

T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k).
T(n,k) = 3*T(n,k-1) - 3*T(n,k-2) + T(n,k-3).
T(n,k) = (T(n,k-1) + T(n,k+1))/2 - A161680(n).
T(n,k) = (T(n-1,k) + T(n+1,k) - A000290(n))/2.

A193515 T(n,k) = number of ways to place any number of 3X1 tiles of k distinguishable colors into an nX1 grid.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 7, 7, 6, 1, 1, 6, 9, 10, 13, 9, 1, 1, 7, 11, 13, 22, 23, 13, 1, 1, 8, 13, 16, 33, 43, 37, 19, 1, 1, 9, 15, 19, 46, 69, 73, 63, 28, 1, 1, 10, 17, 22, 61, 101, 121, 139, 109, 41, 1, 1, 11, 19, 25, 78, 139, 181, 253, 268, 183, 60, 1, 1, 12

Offset: 1

R. H. Hardin, with proof and formula from Robert Israel in the Sequence Fans Mailing List, Jul 29 2011

Table starts:
..1...1...1...1...1....1....1....1....1....1....1....1.....1.....1.....1.....1
..1...1...1...1...1....1....1....1....1....1....1....1.....1.....1.....1.....1
..2...3...4...5...6....7....8....9...10...11...12...13....14....15....16....17
..3...5...7...9..11...13...15...17...19...21...23...25....27....29....31....33
..4...7..10..13..16...19...22...25...28...31...34...37....40....43....46....49
..6..13..22..33..46...61...78...97..118..141..166..193...222...253...286...321
..9..23..43..69.101..139..183..233..289..351..419..493...573...659...751...849
.13..37..73.121.181..253..337..433..541..661..793..937..1093..1261..1441..1633
.19..63.139.253.411..619..883.1209.1603.2071.2619.3253..3979..4803..5731..6769
.28.109.268.529.916.1453.2164.3073.4204.5581.7228.9169.11428.14029.16996.20353

			Some solutions for n=7 k=3; colors=1,2,3 and empty=0
..3....0....0....2....0....1....3....0....0....0....1....0....3....1....0....0
..3....0....0....2....2....1....3....2....1....0....1....3....3....1....0....0
..3....1....0....2....2....1....3....2....1....2....1....3....3....1....0....3
..1....1....3....0....2....0....0....2....1....2....3....3....0....2....0....3
..1....1....3....0....0....2....2....2....1....2....3....2....1....2....1....3
..1....0....3....0....0....2....2....2....1....0....3....2....1....2....1....0
..0....0....0....0....0....2....2....2....1....0....0....2....1....0....1....0

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A000930,
Column 2 is A003229(n-1),
Column 3 is A084386,
Column 4 is A089977,
Column 10 is A178205,
Row 6 is A028872(n+2),
Row 7 is A144390(n+1),
Row 8 is A003154(n+1).

T:= proc(n, k) option remember;
      `if`(n<0, 0,
      `if`(n<3 or k=0, 1, k*T(n-3, k) +T(n-1, k)))
    end:
seq(seq(T(n, d+1-n), n=1..d), d=1..13); # Alois P. Heinz, Jul 29 2011

nmax = 13; t[?Negative, ] = 0; t[n_, k_] /; (n < 3 || k == 0) = 1; t[n_, k_] := t[n, k] = k*t[n-3, k] + t[n-1, k]; Flatten[ Table[ t[n-k+1, k], {n , 1, nmax}, {k, n, 1, -1}]](* Jean-François Alcover, Nov 28 2011, after Maple *)

With z X 1 tiles of k colors on an n X 1 grid (with n >= z), either there is a tile (of any of the k colors) on the first spot, followed by any configuration on the remaining (n-z) X 1 grid, or the first spot is vacant, followed by any configuration on the remaining (n-1) X 1. So T(n,k) = T(n-1,k) + k*T(n-z,k), with T(n,k) = 1 for n=0,1,...,z-1. The solution is T(n,k) = sum_r r^(-n-1)/(1 + z k r^(z-1)) where the sum is over the roots of the polynomial k x^z + x - 1.
T(n,k) = sum {s=0..[n/3]} (binomial(n-2*s,s)*k^s).
For z X 1 tiles, T(n,k,z) = sum{s=0..[n/z]} (binomial(n-(z-1)*s,s)*k^s). - R. H. Hardin, Jul 31 2011

A214870 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 10, 9, 8, 13, 17, 16, 6, 14, 21, 26, 25, 11, 12, 22, 31, 37, 36, 18, 15, 20, 32, 43, 50, 49, 27, 24, 23, 30, 44, 57, 65, 64, 38, 35, 19, 33, 42, 58, 73, 82, 81, 51, 48, 28, 29, 45, 56, 74, 91, 101, 100, 66, 63, 39, 34, 41, 59, 72, 92, 111

Offset: 1

Boris Putievskiy, Mar 11 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1).
Enumeration table T(n,k) layer by layer. The order of the list:
  T(1,1)=1;
  T(1,2), T(2,1), T(2,2);
  . . .
  T(1,n), T(3,n), ... T(n,3), T(n,1); T(n,2), T(n,4), ... T(4,n), T(2,n);
  . . .

			The start of the sequence as table:
   1   2   5  10  17  26 ...
   3   4   9  16  25  36 ...
   7   8   6  11  18  27 ...
  13  14  12  15  24  35 ...
  21  22  20  23  19  28 ...
  31  32  30  33  29  34 ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   2,  3;
   5,  4,  7;
  10,  9,  8, 13;
  17, 16,  6, 14, 21;
  26, 25, 11, 12, 22, 31;
  ...

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A060736, A185725, A213921, A213922; table T(n,k) contains: in rows A002522, A000290, A059100, A005563, A117950, A008865, A087475, A028872, A117951, A028347, A114949, A028875, A117619, A028878, A189833, A028881, A189834, A028884, A114948, A028560, A189836; in columns A002061, A014206, A002378, A027688, A028387, A027689, A028552, A027690, A014209, A027691, A027692, A082111, A027693, A028557, A027694, A108195, A187710, A048058, A048840.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i > j:
   result=i*i-i+(j%2)*(2-(j+1)/2)+((j+1)%2)*(j/2+1)
else:
   result=j*j-2*(i%2)*j + (i%2)*((i+1)/2+1) + ((i+1)%2)*(-i/2+1)

As table
T(n,k) = k*k-2*(n mod 2)*k+(n mod 2)*((n+1)/2+1)+((n+1) mod 2)*(-n/2+1), if n<=k;
T(n,k) = n*n-n+(k mod 2)*(2-(k+1)/2)+((k+1) mod 2)*(k/2+1), if n>k.
As linear sequence
a(n) = j*j-2*(i mod 2)*j+(i mod 2)*((i+1)/2+1)+((i+1) mod 2)*(-i/2+1), if i<=j;
a(n) = i*i-i+(j mod 2)*(2-(j+1)/2)+((j+1) mod 2)*(j/2+1), if i>j; where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A248093 Triangle read by rows: TR(n,k) is the number of unordered vertex pairs at distance k of the hexagonal triangle T_n, defined in the He et al. reference (1<=k<=2n+1).

Original entry on oeis.org

1, 0, 6, 6, 6, 3, 13, 15, 21, 21, 15, 6, 22, 27, 42, 48, 45, 36, 24, 9, 33, 42, 69, 84, 87, 81, 69, 51, 33, 12, 46, 60, 102, 129, 141, 141, 132, 114, 93, 66, 42, 15, 61, 81, 141, 183, 207, 216, 213, 198, 177, 147, 117, 81, 51, 18, 78, 105, 186, 246, 285

Offset: 0

Emeric Deutsch, Nov 14 2014

Number of entries in row n is 2*n+2.
The entries in row n are the coefficients of the Hosoya polynomial of T_n.
TR(n,0) = A028872(n+2) = number of vertices of T_n.
TR(n,1) = A140091(n) = number of edges of T_n.
sum(j*TR(n,j), j=0..2n+1) = A033544(n) = the Wiener index of T_n.
(1/2)*sum(j*(j+1)TR(n,j), j=0..2n+1) = A248094(n) = the hyper-Wiener index of T_n.
sum((-1)^j*TR(n,j), j=0..2n+1) = A002061(n). - Peter Luschny, Nov 15 2014

			Row n=1 is 6, 6, 6, 3; indeed, T_1 is a hexagon ABCDEF; it has 6 distances equal to 0 (= number of vertices), 6 distances equal to 1 (= number of edges), 6 distances equal to 2 (AC, BD, CE, DA, EA, FB), and 3 distances equal to 3 (AD, BE, CF).
Triangle starts:
1, 0;
6, 6, 6, 3;
13, 15, 21, 21, 15, 6;
22, 27, 42, 48, 45, 36, 24, 9;
33, 42, 69, 84, 87, 81, 69, 51, 33, 12;

Q. H. He, J. Z. Gu, S. J. Xu, and W. H. Chan, Hosoya polynomials of hexagonal triangles and trapeziums, MATCH, Commun. Math. Comput. Chem. 72, 2014, 835-843.

Cf. A028872, A140091, A033544, A248094.

G := (1+(3+6*t+4*t^2+3*t^3)*z-(1+t+2*t^2)*(2+t-2*t^2)*z^2+t^2*(1-3*t^2)*z^3+t^4*z^4)/((1-z)^3*(1-t^2*z)^2): Gser := simplify(series(G, z = 0, 25)): for n from 0 to 22 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, j), j = 0 .. 2*n+1) end do; # yields sequence in triangular form

G.f.: (1 + (3 + 6*t + 4*t^2 + 3*t^3)*z - (1 + t + 2*t^2)*(2 + t - 2*t^2)*z^2 +t^2*(1 - 3*t^2)*z^3 + t^4*z^4)/((1-z)^3*(1 - t^2*z^2)^2); follows from Theorem 3.6 of the He et al. reference.

A332700 A(n, k) = Sum_{j=0..n} j!Stirling2(n, j)(k-1)^(n-j), for n >= 0 and k >= 0, read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 3, 1, 1, 1, 24, 13, 4, 1, 1, 1, 120, 75, 22, 5, 1, 1, 1, 720, 541, 160, 33, 6, 1, 1, 1, 5040, 4683, 1456, 285, 46, 7, 1, 1, 1, 40320, 47293, 15904, 3081, 456, 61, 8, 1, 1, 1, 362880, 545835, 202672, 40005, 5656, 679, 78, 9, 1, 1

Offset: 0

Peter Luschny, Feb 28 2020

			Array begins:
[0] 1, 1,       1,       1,        1,         1,         1, ...    A000012
[1] 1, 1,       1,       1,        1,         1,         1, ...    A000012
[2] 1, 2,       3,       4,        5,         6,         7, ...    A000027
[3] 1, 6,       13,      22,       33,        46,        61, ...   A028872
[4] 1, 24,      75,      160,      285,       456,       679, ...
[5] 1, 120,     541,     1456,     3081,      5656,      9445, ...
[6] 1, 720,     4683,    15904,    40005,     84336,     158095, ...
[7] 1, 5040,    47293,   202672,   606033,    1467376,   3088765, ...
[8] 1, 40320,   545835,  2951680,  10491885,  29175936,  68958295, ...
[9] 1, 362880,  7087261, 48361216, 204343641, 652606336, 1731875605, ...
       A000142, A000670, A122704,  A255927,   A326324, ...
Seen as a triangle:
[0] [1]
[1] [1, 1]
[2] [1, 1,     1]
[3] [1, 2,     1,     1]
[4] [1, 6,     3,     1,     1]
[5] [1, 24,    13,    4,     1,    1]
[6] [1, 120,   75,    22,    5,    1,   1]
[7] [1, 720,   541,   160,   33,   6,   1,  1]
[8] [1, 5040,  4683,  1456,  285,  46,  7,  1, 1]
[9] [1, 40320, 47293, 15904, 3081, 456, 61, 8, 1, 1]

Paolo Xausa, Table of n, a(n) for n = 0..11475 (antidiagonals 0..150 of the square, flattened).

The matrix transpose of A326323.

Cf. A173018, A000012, A000142, A000670, A122704, A255927, A326324, A000027, A028872.

# Prints array by row.
A := (n, k) -> add(combinat:-eulerian1(n, j)*k^j, j=0..n):
seq(print(seq(A(n,k), k=0..10)), n=0..8);
# Alternative:
egf := n -> `if`(n=1, 1/(1-x), (n-1)/(n - exp((n-1)*x))):
ser := n -> series(egf(n), x, 21):
for n from 0 to 6 do seq(n!*coeff(ser(k), x, n), k=0..9) od;
# Or:
A := (n, k) -> if k = 0 or n = 0 then 1 elif k = 1 then n! else
polylog(-n, 1/k)*(k-1)^(n+1)/k fi:
for n from 0 to 6 do seq(A(n, k), k=0..9) od;

A332700[n_, k_] := n! + Sum[j! StirlingS2[n, j] (k-1)^(n-j), {j, n-1}];
Table[A332700[n-k, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Feb 01 2024 *)

def T(n, k):
    return sum(factorial(j)*stirling_number2(n, j)*(k-1)^(n-j) for j in range(n+1))
for n in range(8): print([T(n, k) for k in range(8)])

A(n, k) = Sum_{j=0..n} E(n, j)*k^j, where E(n, k) = A173018(n, k).
A(n, 1) = n!*[x^n] 1/(1-x).
A(n, k) = n!*[x^n] (k-1)/(k - exp((k-1)*x)) for k != 1.
A(n, k) = PolyLog(-n, 1/k)*(k-1)^(n+1)/k for k >= 2.

A386206 Triangle read by rows: T(n,k) = n^2 - k, with 0 <= k <= n.

Original entry on oeis.org

0, 1, 0, 4, 3, 2, 9, 8, 7, 6, 16, 15, 14, 13, 12, 25, 24, 23, 22, 21, 20, 36, 35, 34, 33, 32, 31, 30, 49, 48, 47, 46, 45, 44, 43, 42, 64, 63, 62, 61, 60, 59, 58, 57, 56, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 100, 99, 98, 97, 96, 95, 94, 93, 92, 91, 90

Offset: 0

Stefano Spezia, Jul 15 2025

			The triangle begins as:
   0;
   1,  0;
   4,  3,  2;
   9,  8,  7,  6;
  16, 15, 14, 13, 12;
  25, 24, 23, 22, 21, 20;
  36, 35, 34, 33, 32, 31, 30;
  49, 48, 47, 46, 45, 44, 43, 42;
  64, 63, 62, 61, 60, 59, 58, 57, 56;
  ...

Vincenzo Librandi, Table of n, a(n) for n = 0..3320 (rows 0..80)

Cf. A000290 (k=0), A002414 (row sums), A005563, A008865, A028347 (k=4), A028872 (k=3), A028875 (k=5), A279019 (diagonal).

[[n^2-k: k in [0..n]]: n in [0..9]]; // Vincenzo Librandi, Jul 17 2025

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

G.f.: x*(1 + x + 2*x*y^2 + 5*x^3*y^2 - x^2*y*(4 + 5*y))/((1 - x)^3*(1 - x*y)^3).
T(n,1) = A005563(n-1) for n > 0.
T(n,2) = A008865(n) for n > 1.

cp's OEIS Frontend

A361682 Array read by descending antidiagonals. A(n, k) is the number of multiset combinations of {0, 1} whose type is defined in the comments. Also A(n, k) = hypergeom([-k, -2], [1], n).

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A039824 Number of different coefficient values in expansion of Product (1+q^1+q^3...+q^(2i-1)), i=1 to n.

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A110292 Riordan array (1-u, u) where u=(-1 + sqrt(1+8*x))/4.

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A214859 Triangle read by rows, T(n,k) = n^2 - k*(k+1)/2 if k*(k+1)/2 <= n^2.

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A239331 Square array, read by antidiagonals: column k has g.f. (1+(k-1)*x)^2/(1-x)^3.

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A193515 T(n,k) = number of ways to place any number of 3X1 tiles of k distinguishable colors into an nX1 grid.

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A214870 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

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A248093 Triangle read by rows: TR(n,k) is the number of unordered vertex pairs at distance k of the hexagonal triangle T_n, defined in the He et al. reference (1<=k<=2n+1).

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A214859 Triangle read by rows, T(n,k) = n^2 - k(k+1)/2 if k(k+1)/2 <= n^2.

A332700 A(n, k) = Sum_{j=0..n} j!Stirling2(n, j)(k-1)^(n-j), for n >= 0 and k >= 0, read by ascending antidiagonals.