A089258 Transposed version of A080955: T(n,k) = A080955(k,n), n>=0, k>=-1.

1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 5, 6, 9, 1, 4, 10, 16, 24, 44, 1, 5, 17, 38, 65, 120, 265, 1, 6, 26, 78, 168, 326, 720, 1854, 1, 7, 37, 142, 393, 872, 1957, 5040, 14833, 1, 8, 50, 236, 824, 2208, 5296, 13700, 40320, 133496, 1, 9, 65, 366, 1569, 5144, 13977, 37200, 109601, 362880, 1334961

Offset

0,8

Comments

Can be extended to columns with negative indices k<0 via T(n,k) = A292977(n,-k). - Max Alekseyev, Mar 06 2018

Links

Table of n, a(n) for n=0..65.

Formula

For n > 0, k >= -1, T(n,k) is the permanent of the n X n matrix with k+1 on the diagonal and 1 elsewhere.

T(0,k) = 1.

T(n,k) = Sum_{j>=0} A008290(n,j) * (k+1)^j.

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

T(n,k) = n! * Sum_{j=0..n} k^j/j!.

E.g.f. for k-th column: exp(k*x)/(1-x).

Assuming n >= 0, k >= 0: T(n, k) = exp(k-1)*Gamma(n+1, k-1). - Peter Luschny, Dec 24 2021

Example

n\k -1   0   1    2    3    4     5     6  ...
----------------------------------------------
0  | 1,  1,  1,   1,   1,   1,    1,    1, ...
1  | 0,  1,  2,   3,   4,   5,    6,    7, ...
2  | 1,  2,  5,  10,  17,  26,   37,   50, ...
3  | 2,  6, 16,  38,  78, 152,  236,  366, ...
4  | 9, 24, 65, 168, 393, 824, 1569, 2760, ...
...

Mathematica

(* Assuming offset (0, 0): *)
T[n_, k_] := Exp[k - 1] Gamma[n + 1, k - 1];
Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten  (* Peter Luschny, Dec 24 2021 *)

Crossrefs

Columns: A000166, A000142, A000522, A010842, A053486, A053487, A080954.

Main diagonal gives A217701.

Cf. A080955, A008290, A292977.

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Dec 12 2003

Extensions

Edited and changed offset for k to -1 by Max Alekseyev, Mar 08 2018

Status

approved

A352682 Array read by ascending antidiagonals. A(n, k) = (n-1)*Gould(k-1) + Bell(k) for n >= 0 and k >= 1, A(n, 0) = 1.

1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 3, 5, 6, 1, 4, 4, 8, 15, 21, 1, 5, 5, 11, 24, 52, 82, 1, 6, 6, 14, 33, 83, 203, 354, 1, 7, 7, 17, 42, 114, 324, 877, 1671, 1, 8, 8, 20, 51, 145, 445, 1400, 4140, 8536, 1, 9, 9, 23, 60, 176, 566, 1923, 6609, 21147, 46814

Offset

0,8

Comments

The array defines a family of Bell-like sequences. The case n = 1 are the Bell numbers A000110, case n = 0 is A032347 and case n = 2 is A038561. The n-th sequence r(k) = T(n, k) is defined for k >= 0 by the recurrence r(k) = Sum_{j=0..k-1} binomial(k-1, j)*r(j) with r(0) = 1 and r(1) = n.

Links

Table of n, a(n) for n=0..65.

Formula

Given a list T let PS(T) denote the list of partial sums of T. Given two list S and T let [S, T] denote the concatenation of the lists. Further let P[end] denote the last element of the list P. Row n of the array with length k can be computed by the following procedure:

A = [n], P = [1], R = [1];

Repeat k-1 times: R = [R, A], P = PS([A, P]), A = [P[end]];

Return R.

Example

Array starts:
n\k 0, 1,  2,  3,  4,   5,    6,    7,     8,      9, ...
---------------------------------------------------------
[0] 1, 0,  1,  2,  6,  21,   82,  354,  1671,   8536, ... A032347
[1] 1, 1,  2,  5, 15,  52,  203,  877,  4140,  21147, ... A000110
[2] 1, 2,  3,  8, 24,  83,  324, 1400,  6609,  33758, ... A038561
[3] 1, 3,  4, 11, 33, 114,  445, 1923,  9078,  46369, ... A038559
[4] 1, 4,  5, 14, 42, 145,  566, 2446, 11547,  58980, ... A352683
[5] 1, 5,  6, 17, 51, 176,  687, 2969, 14016,  71591, ...
[6] 1, 6,  7, 20, 60, 207,  808, 3492, 16485,  84202, ...
[7] 1, 7,  8, 23, 69, 238,  929, 4015, 18954,  96813, ...
[8] 1, 8,  9, 26, 78, 269, 1050, 4538, 21423, 109424, ...
[9] 1, 9, 10, 29, 87, 300, 1171, 5061, 23892, 122035, ...

Maple

alias(PS = ListTools:-PartialSums):
BellRow := proc(n, len) local a, k, P, T;
a := n; P := [1]; T := [1];
for k from 1 to len-1 do
   T := [op(T), a]; P := PS([a, op(P)]); a := P[-1] od;
T end: seq(lprint(BellRow(n, 10)), n = 0..9);

Mathematica

nmax = 10;
BellRow[n_, len_] := Module[{a, k, P, T}, a = n; P = {1}; T = {1};
   For[k = 1, k <= len - 1, k++,
      T = Append[T, a]; P = Accumulate[Join[{a}, P]]; a = P[[-1]]];
   T];
rows = Table[BellRow[n, nmax + 1], {n, 0, nmax}];
A[n_, k_] := rows[[n + 1, k + 1]];
Table[A[n - k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 15 2024, after Peter Luschny *)

Prog

(Julia)
function BellRow(m, len)
    a = m; P = BigInt[1]; T = BigInt[1]
    for n in 1:len
        T = vcat(T, a)
        P = cumsum(vcat(a, P))
        a = P[end]
    end
T end
for n in 0:9 BellRow(n, 9) |> println end

Crossrefs

Rows: A032347, A000110 (Bell), A038561, A038559, A352683.

Diagonals: A352684 (main).

Cf. A040027 (Gould), A352686 (subtriangle).

Compare A352680 for a similar array based on the Catalan numbers.

Keyword

nonn,tabl

Author

Peter Luschny, Mar 28 2022

Status

approved

A372014 T(n,k) is the total number of levels in all Motzkin paths of length n containing exactly k path nodes; triangle T(n,k), n>=0, 1<=k<=n+1, read by rows.

1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 4, 6, 4, 3, 1, 8, 14, 12, 7, 4, 1, 18, 32, 33, 21, 11, 5, 1, 44, 74, 84, 64, 34, 16, 6, 1, 113, 180, 208, 181, 111, 52, 22, 7, 1, 296, 457, 520, 485, 344, 179, 76, 29, 8, 1, 782, 1195, 1334, 1273, 1000, 599, 274, 107, 37, 9, 1

Offset

0,7

Comments

A Motzkin path of length n has n+1 nodes.

Links

Alois P. Heinz, Rows n = 0..28, flattened

Wikipedia, Motzkin number

Formula

Sum_{k=1..n+1} k * T(n,k) = A005717(n+1) = (n+1) * A001006(n).

Example

In the A001006(3) = 4 Motzkin paths of length 3 there are 2 levels with 1 node, 2 levels with 2 nodes, 2 levels with 3 nodes, and 1 level with 4 nodes.
  2  _     1        1
  2 / \    3 /\_    3 _/\    4 ___    .
  So row 3 is [2, 2, 2, 1].
Triangle T(n,k) begins:
    1;
    0,    1;
    1,    1,    1;
    2,    2,    2,    1;
    4,    6,    4,    3,    1;
    8,   14,   12,    7,    4,   1;
   18,   32,   33,   21,   11,   5,   1;
   44,   74,   84,   64,   34,  16,   6,   1;
  113,  180,  208,  181,  111,  52,  22,   7,  1;
  296,  457,  520,  485,  344, 179,  76,  29,  8, 1;
  782, 1195, 1334, 1273, 1000, 599, 274, 107, 37, 9, 1;
  ...

Maple

g:= proc(x, y, p) (h-> `if`(x=0, add(z^coeff(h, z, i)
          , i=0..degree(h)), b(x, y, h)))(p+z^y) end:
b:= proc(x, y, p) option remember; `if`(y+2<=x, g(x-1, y+1, p), 0)
      +`if`(y+1<=x, g(x-1, y, p), 0)+`if`(y>0, g(x-1, y-1, p), 0)
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=1..n+1))(g(n, 0$2)):
seq(T(n), n=0..10);

Crossrefs

Columns k=1-2 give: A088457, A051485.

Row sums give A372033 = A001006 + A333498.

Cf. A005717, A371928.

Keyword

nonn,tabl

Author

Alois P. Heinz, Apr 15 2024

Status

approved

A194821 a(n) = 1+floor(sum{<((-1)^k)*k*sqrt(2)> : 1<=k<=n}), where < > = fractional part.

0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 1, 2, 1, 2, 2, 2, 2, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 1, 2, 1, 2, 2, 2, 2, 3, 3

Offset

1,6

Comments

Does 0 occur infinitely many times? Is the sequence unbounded?

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Mathematica

r = Sqrt[2]; p[x_] := FractionalPart[x];
f[n_] := 1 + Floor[Sum[p[k*r] (-1)^k, {k, 1, n}]]
Table[f[n], {n, 1, 100}]   (* A194821 *)

Prog

(PARI) for(n=1, 50, print1(1 + floor(sum(k=1, n, (-1)^k*frac(k*sqrt(2))), ", ")) \\ G. C. Greubel, Apr 02 2018
(Magma) [1 + Floor((&+[(-1)^k*(k*Sqrt(2) - Floor(k*Sqrt(2))) :k in [1..n]])) : n in [1..50]]; // G. C. Greubel, Apr 02 2018

Crossrefs

Cf. A194822, A194823, A194824.

Keyword

nonn

Author

Clark Kimberling, Sep 03 2011

Status

approved

A264051 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A264078(n)) is the number of integer partitions of n having k standard Young tableaux such that no entries i and i+1 appear in the same row.

0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 2, 3, 0, 2, 4, 2, 1, 1, 1, 1, 1, 4, 3, 1, 0, 0, 2, 2, 0, 1, 0, 1, 0, 0, 0, 1, 7, 2, 0, 0, 1, 0, 3, 0, 1, 0, 2, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 7, 3, 1, 2, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1

Offset

0,8

Comments

Row sums give A000041.

Column k=0 gives A025065(n-2) for n>=2.

Links

Alois P. Heinz, Rows n = 0..14, flattened

S. Dulucq and O. Guibert, Stack words, standard tableaux and Baxter permutations, Disc. Math. 157 (1996), 91-106.

FindStat - Combinatorial Statistic Finder, Number of standard Young tableaux of an integer partition such that no k and k+1 appear in the same row.

Formula

Sum_{k=1..A264078(n)} k*T(n,k) = A237770(n). - Alois P. Heinz, Nov 02 2015

Example

Triangle begins:
0,1,
0,1,
1,1,
1,2,
2,2,1,
2,3,0,2,
4,2,1,1,1,1,1,
4,3,1,0,0,2,2,0,1,0,1,0,0,0,1,
7,2,0,0,1,0,3,0,1,0,2,1,0,0,0,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0,1,
...

Maple

h:= proc(l, j) option remember; `if`(l=[], 1,
      `if`(l[1]=0, h(subsop(1=[][], l), j-1), add(
      `if`(i<>j and l[i]>0 and (i=1 or l[i]>l[i-1]),
       h(subsop(i=l[i]-1, l), i), 0), i=1..nops(l))))
    end:
g:= proc(n, i, l) `if`(n=0 or i=1, x^h([1$n, l[]], 0),
      `if`(i<1, 0, g(n, i-1, l)+ `if`(i>n, 0,
       g(n-i, i, [i, l[]]))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n$2, [])):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 02 2015

Mathematica

h[l_, j_] := h[l, j] = If[l == {}, 1, If[l[[1]] == 0, h[ReplacePart[l, 1 -> Sequence[]], j - 1], Sum[If[i != j && l[[i]] > 0 && (i == 1 || l[[i]] > l[[i - 1]]), h[ReplacePart[l, i -> l[[i]] - 1], i], 0], {i, 1, Length[l]} ]]]; g[n_, i_, l_] := If[n == 0 || i == 1, x^h[Join[Array[1 &, n], l], 0], If[i < 1, 0, g[n, i - 1, l] + If[i > n, 0, g[n - i, i, Join[{i}, l]]] ]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][g[n, n, {}]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 22 2016, after Alois P. Heinz *)

Crossrefs

Cf. A000041, A001181, A237770, A264078.

Keyword

nonn,tabf

Author

Christian Stump, Nov 01 2015

Status

approved

A356997 a(n) = b(n) - b(n - b(n - b(n))) for n >= 2, where b(n) = A356988(n).

0, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 3, 3, 3, 3, 3, 2, 2, 2, 3, 4, 5, 5, 5, 5, 5, 5, 5, 4, 3, 3, 3, 3, 4, 5, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 6, 5, 5, 5, 5, 5, 5, 6, 7, 8, 9, 10, 11, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 12, 11, 10, 9, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 11

Offset

2,9

Comments

The line graph of the sequence consists of a series of local plateaus and local troughs joined at each end by lines of slope 1 and slope -1. More precisely, for k >= 3 the graph of the sequence consists of

local plateaus: on the integer interval [2*F(k), 2*F(k) + 2*F(k-3)] the sequence has the constant value F(k-2)

descent to a trough: on the integer interval [2*F(k) + 2*F(k-3), F(k+2)] the line graph of the sequence has slope -1

local troughs: on the integer interval [F(k+2), F(k+2) + F(k-3)] the sequence has the constant value F(k-3)

ascent to a plateau: on the integer interval [F(k+2) + F(k-3), 2*F(k+1)] the line graph of the sequence has slope 1.

Links

Table of n, a(n) for n=2..100.

Formula

a(n+1) - a(n) = 1, 0 or -1.

Let F(n) = A000045(n) with F(-1) = 1 and let L(n) = A000032(n).

For k >= 5, a(F(k) + j) = F(k-5) for 0 <= j <= F(k-5) (troughs).

For k >= 4, a(2*F(k) + j) = F(k-2) for 0 <= j <= 2*F(k-3) (plateaus).

Example

The sequence is arranged to show the local plateaus (P) and the local troughs (T):
    0,
    1,
    1,
T   0,
P   1, 1, 1
    1,
P   2, 2, 2,
T   1,1,
    2,
P   3, 3, 3, 3, 3,
T   2, 2, 2,
    3,
    4,
P   5, 5, 5, 5, 5, 5, 5,
    4,
T   3, 3, 3, 3,
    4,
    5,
    6,
    7,
P   8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8,
    7,
    6,
T   5, 5, 5, 5, 5, 5,
    6,
    7,
    8,
    9,
    10,
    11,
    12,
P   13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13,
    12,
    11,
    10,
    9,
T   8, 8, 8, 8, 8, 8, 8, 8, 8,
    9,
    10,
    11,
    ...

Maple

# b(n) = A356988
b := proc(n) option remember; if n = 1 then 1 else n - b(b(n - b(b(b(n-1))))) end if; end proc:
seq( b(n) - b(n - b(n - b(n))), n = 2
..100);

Crossrefs

Cf. A000045, A356988, A356991 - A356999.

Keyword

nonn,easy

Author

Peter Bala, Sep 11 2022

Status

approved

A124369 Riordan array (1/((1-x-x^2)(1+x+x^2)),x(1+x)/((1-x-x^2)(1+x+x^2))).

1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 2, 6, 4, 3, 1, 4, 9, 12, 7, 4, 1, 7, 17, 24, 21, 11, 5, 1, 10, 34, 48, 50, 34, 16, 6, 1, 17, 58, 103, 110, 91, 52, 22, 7, 1, 28, 104, 200, 250, 220, 152, 76, 29, 8, 1, 44, 188, 385, 534, 530, 400, 239, 107, 37, 9, 1

Offset

0,7

Comments

Row sums are A123392. Diagonal sums are A124370. First column is A094686. Product of A026729 and abs(A049310).

Links

Table of n, a(n) for n=0..65.

Formula

Number triangle T(n,k)=sum{j=0..n, C(j,n-j)*C((j+k)/2,(j-k)/2)*(1+(-1)^(j-k))/2};

T(n,k) = T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + 2*T(n-3,k) + T(n-4,k), T(0,0) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 22 2014

Example

Triangle begins
1,
0, 1,
1, 1, 1,
2, 2, 2, 1,
2, 6, 4, 3, 1,
4, 9, 12, 7, 4, 1,
7, 17, 24, 21, 11, 5, 1,
10, 34, 48, 50, 34, 16, 6, 1,
17, 58, 103, 110, 91, 52, 22, 7, 1

Keyword

easy,nonn,tabl

Author

Paul Barry, Oct 27 2006

Status

approved

A105241 Vector triangular array of Fibonacci tensor Markov.

0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 1, 1, 2, 2, 2, 1, 3, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 2, 2, 2, 4, 3, 3, 1, 4, 0

Offset

1,16

Comments

This is the triangle form from {6,2,2}. T[n,k,j] levels j: {0, 1, 1, 1} {0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2} {0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3} {0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 1, 1, 2, 2, 2, 1, 3, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 2, 2, 2, 4, 3, 3, 1, 4}

Links

Table of n, a(n) for n=1..105.

Formula

v[n]=M.v[n-1] M={M1, M2} M1={{0, 1}, {1, 0}} M2={{0, 1}, {1, 1}} Selective flattening and expression to get a vector triangle representation =a[n]

Mathematica

v[1] = {{0, 1}, {1, 1}} M = {{{0, 1}, {1, 1}}, {{0, 1}, {1, 1}}} v[n_] := v[n] = M.v[n - 1] a = Table[v[n], {n, 1, 6}] Dimensions[a aa = Table[Flatten[Table[Table[a[[n, j]], {j, 1, 2}], {n, 1, m}]], {m, 1, 6}] aout= Flatten[aa]

Keyword

nonn,uned,obsc

Author

Roger L. Bagula, Apr 12 2005

Status

approved

A105242 A nonsense sequence.

0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 1, 1, 2, 2, 2, 1, 3, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 2, 2, 2, 4, 3, 3, 1, 4, 1, 1, 1, 2, 2, 2, 1, 3, 2, 2, 2, 4, 3, 3, 1, 4, 1, 1, 1, 2, 2, 2, 1, 3, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 2, 2, 2, 4, 3

Offset

1,12

Links

Table of n, a(n) for n=1..105.

Mathematica

v[1] = {{0, 1}, {1, 1}}
M = {{{0, 1}, {1, 0}}, {{0, 1}, {1, 1}}}
v[n_] := v[n] = M.v[n - 1]
a = Table[v[n], {n, 1, 6}]
aa = Flatten[a]
Length[aa]

Keyword

nonn,uned,obsc,less

Author

Roger L. Bagula, Apr 12 2005

Status

approved