A000027 -id:A000027

A354127

Triangle read by rows: T(n, k) is the number of graphs obtained by adding k pierced circles to a path graph P_n.

+10
0

1, 1, 0, 2, 2, 0, 12, 10, 3, 0, 82, 82, 28, 4, 0, 646, 738, 315, 60, 5, 0, 5574, 7198, 3636, 900, 110, 6, 0, 51386, 74086, 43225, 13020, 2135, 182, 7, 0, 498026, 793490, 524784, 185920, 37940, 4452, 280, 8, 0, 5019720, 8761906, 6475959, 2634912, 642180, 95508, 8442, 408, 9, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..54.

Nicholas Owad and Anastasiia Tsvietkova, Random meander model for links, arXiv:2205.03451 [math.GT], 2022.

FORMULA

T(n, k) = Sum_{m=k..n} (-1)^(m+k)*binomial(m, k)*O(m, n), with O(k, s) = binomial(2*s-k-1, k)*C(s-k)^2 (see Lemma 3.3 at page 7 in Owad and Tsvietkova).

T(n, n-2) = A006331(n-1).

EXAMPLE

The triangle begins

1;

1, 0;

2, 2, 0;

12, 10, 3, 0;

82, 82, 28, 4, 0;

646, 738, 315, 60, 5, 0;

...

MATHEMATICA

bigO[k_, s_]:=Binomial[2s-k-1, k]CatalanNumber[s-k]^2; T[n_, k_]:=Sum[(-1)^(m+k)Binomial[m, k]bigO[m, n], {m, k, n}]; Flatten[Table[T[n, k], {n, 0, 9}, {k, 0, n}]]

CROSSREFS

Cf. A000007 (k = n), A000027 (k = n - 1), A000108, A001246 (row sums), A006331, A007318, A052553.

KEYWORD

nonn,tabl

AUTHOR

Stefano Spezia, May 18 2022

STATUS

approved

A355668

Array read by upwards antidiagonals T(n,k) = J(k) + n*J(k+1) where J(n) = A001045(n) is the Jacobsthal numbers.

+10
0

0, 1, 1, 2, 2, 1, 3, 3, 4, 3, 4, 4, 7, 8, 5, 5, 5, 10, 13, 16, 11, 6, 6, 13, 18, 27, 32, 21, 7, 7, 16, 23, 38, 53, 64, 43, 8, 8, 19, 28, 49, 74, 107, 128, 85, 9, 9, 22, 33, 60, 95, 150, 213, 256, 171, 10, 10, 25, 38, 71, 116, 193, 298, 427, 512, 341

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

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

FORMULA

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

G.f.: (x*(y-1) - y)/((x - 1)^2*(y + 1)*(2*y - 1)). - Stefano Spezia, Jul 13 2022

EXAMPLE

Row n=0 is A001045(k), then for further rows we successively add A001045(k+1).

k=0 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k=10

n=0: 0 1 1 3 5 11 21 43 85 171 ... = A001045

n=1: 1 2 4 8 16 32 64 128 256 512 ... = A000079

n=2: 2 3 7 13 27 53 107 213 427 853 ... = A048573

n=3: 3 4 10 18 38 74 150 298 598 1194 ... = A171160

n=4: 4 5 13 23 49 95 193 383 769 1535 ... = abs(A140683)

...

MATHEMATICA

T[n_, k_] := (2^k - (-1)^k + n*(2^(k + 1) + (-1)^k))/3; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 13 2022 *)

CROSSREFS

Cf. A001477, A000027, A016777, A016885, A017449, A321373.

Antidiagonal sums give A320933(n+1).

KEYWORD

nonn,tabl,easy

AUTHOR

Paul Curtz, Jul 13 2022

STATUS

approved

A358492

Irregular triangle read by rows: T(n,k) is one half of the number of line segments of length 1 in the k-th antidiagonal of the Dyck path described in the n-th row of A237593.

+10
0

1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 3, 1, 1, 3, 2, 1, 1, 1, 3, 2, 1, 1, 1, 3, 3, 1, 1, 1, 1, 4, 2, 1, 1, 1, 4, 2, 2, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 3, 4, 2, 1, 1, 1, 1, 2, 4, 2, 2, 1, 1, 1, 1, 1, 3, 5, 2, 1, 1, 1, 1, 1, 1, 3, 5, 2, 1, 1, 1, 1, 1, 3, 5, 2, 2, 1, 1, 1, 1, 1, 1, 1, 5, 4, 2, 1, 1, 1, 1, 1, 1, 5, 4, 2, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

LINKS

Table of n, a(n) for n=1..109.

EXAMPLE

Triangle begins (first 19 rows):

1;

1, 1;

1, 2;

1, 1, 2;

1, 2, 2;

1, 1, 1, 3;

1, 1, 3, 2;

1, 1, 1, 3, 2;

1, 1, 1, 3, 3;

1, 1, 1, 1, 4, 2;

1, 1, 1, 4, 2, 2;

1, 1, 1, 1, 1, 3, 4;

1, 1, 1, 1, 3, 4, 2;

1, 1, 1, 1, 2, 4, 2, 2;

1, 1, 1, 1, 1, 3, 5, 2;

1, 1, 1, 1, 1, 1, 3, 5, 2;

1, 1, 1, 1, 1, 3, 5, 2, 2;

1, 1, 1, 1, 1, 1, 1, 5, 4, 2;

1, 1, 1, 1, 1, 1, 5, 4, 2, 2;

...

For n = 10 the 10th row of A237593 is [6, 2, 1, 1, 1, 1, 2, 6]. When that row is interpreted as a symmetric Dyck path in the fourth quadrant using 20 line segments of length 1 the Dyck path looks like this:

.

|

_ _|

_|

|

_ _ _ _ _ _|

.

The numbers of line segments of length 1 in the successive antidiagonals are respectively [2, 2, 2, 2, 8, 4] so the 10th row of triangle is [1, 1, 1, 1, 4, 2].

CROSSREFS

Row sums give A000027.

Row n has length A008619(n).

Column 1 gives A000012.

Cf. A196020, A235791, A236104, A237270, A237271, A237591, A237593, A245092, A262626, A339575.

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Nov 19 2022

STATUS

approved

A360033

Table T(n,k), n >= 1 and k >= 0, read by antidiagonals, related to Jacobsthal numbers A001045.

+10
0

1, 2, 1, 3, 3, 3, 4, 5, 7, 5, 5, 7, 11, 13, 11, 6, 9, 15, 21, 27, 21, 7, 11, 19, 29, 43, 53, 43, 8, 13, 23, 37, 59, 85, 107, 85, 9, 15, 27, 45, 75, 117, 171, 213, 171, 10, 17, 31, 53, 91, 149, 235, 341, 427, 341, 11, 19, 35, 61, 107, 181, 299, 469

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..63.

FORMULA

T(n,k) = T(1,k) + (n-1)*2^k.

T(n,k) = 2*T(n, k-1) + (-1)^k.

T(n,k) = T(n-1,k) + 2^k.

T(n,k) = 2^k * n - A001045(k).

T(n,k) = T(n,k-1) +2*T(n,k-2).

EXAMPLE

The array T(n,k), for n <= 1 and k >= 0, begins:

n = 1: 1, 1, 3, 5, 11, 21, 43, ... -> A001045(k+1)

n = 2: 2, 3, 7, 13, 27, 53, 107, ... -> A048573(k)

n = 3: 3, 5, 11, 21, 43, 85, 171, ... -> A001045(k+3)

n = 4: 4, 7, 15, 29, 59, 117, 235, ... -> ?

n = 5: 5, 9, 19, 37, 75, 149, 299, ... -> A062092(k+1)

n = 6: 6, 11, 23, 45, 91, 181, 363, ... -> ?

n = 7: 7, 13, 27, 53, 107, 213, 427, ... -> A048573(k+2)

CROSSREFS

Cf. A001045, A048573, A062092.

Columns: A000027, A005408, A004767, A004770, A106839 for k = 0, 1, 2, 3, 4.

KEYWORD

nonn,tabl,easy

AUTHOR

Philippe Deléham, Jan 22 2023

STATUS

approved

A360145

Triangle read by rows where row n is the largest (or middle or n-th) column of the reverse pyramid summation of order n described in A359087.

+10
0

1, 2, 4, 3, 7, 19, 4, 10, 28, 78, 5, 13, 37, 105, 301, 6, 16, 46, 132, 382, 1108, 7, 19, 55, 159, 463, 1351, 3951, 8, 22, 64, 186, 544, 1594, 4680, 13758, 9, 25, 73, 213, 625, 1837, 5409, 15945, 47049, 10, 28, 82, 240, 706, 2080, 6138, 18132, 53610, 158616, 11, 31, 91, 267, 787, 2323, 6867, 20319, 60171, 178299, 528619

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The integer that is at the k-th row of the middle column of this pyramid of order n will be noted T(n,k).

Each row has n terms.

LINKS

Table of n, a(n) for n=1..66.

FORMULA

T(n,1) = n.

T(n,2) = 3n - 2.

T(n,3) = 9n - 8.

T(n,4) = 27n - 30.

T(n,5) = 81n - 104.

T(n,n) = A359087(n).

T(n,k) = 3^(k-1)*n - 2*A132894(k-1) for 1 <= k <= n (conjectured).

EXAMPLE

Triangle begins:

n=1: 1;

n=2: 2, 4;

n=3: 3, 7, 19;

n=4: 4, 10, 28, 78;

n=5: 5, 13, 37, 105, 301;

n=6: 6, 16, 46, 132, 382, 1108;

...

For n=5, the reverse pyramid summation is as follows and row 5 here is the middle column 5,13,37,...

1 2 3 4 5 4 3 2 1

6 9 12 13 12 9 6

27 34 37 34 27

98 105 98

301

PROG

(PARI) f(v) = if (#v == 1, v, vector(#v-2, i, v[i]+v[i+1]+v[i+2]));

row(n) = my(u = concat([1..n], Vecrev([1..n-1])), v=u, w = vector(n)); for (i=1, n, w[i] = v[#v\2+1]; v = f(v); ); w; \\ Michel Marcus, Jan 30 2023

CROSSREFS

Cf. A132894, A359087 (right diagonal).

Columns k=1..3: A000027, A016777, A017173.

KEYWORD

nonn,tabl

AUTHOR

Bernard Schott, Jan 27 2023

STATUS

approved

A360607

a(n) = (n + 1/3) * (3*n + 3)! / ((n + 1)!)^3.

+10
0

2, 120, 3920, 115500, 3279276, 91483392, 2527462080, 69413752980, 1898945262500, 51809303385840, 1410778953118080, 38360293308573600, 1041896542898685600, 28274302212143040000, 766761376923703054080, 20781963787558897929060, 563007692048242597348500, 15246741943087253044446000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..17.

MAPLE

a := n -> (n + 1/3)*(3*n + 3)!/(n + 1)!^3: seq(a(n), n = 0..17);

CROSSREFS

Family: A000027 (m=1), A257589 (m=2), this sequence (m=3).

KEYWORD

nonn

AUTHOR

Peter Luschny, Feb 15 2023

STATUS

approved

A364842

Table read by antidiagonals: row n gives the Euler transform of the sequence (2,...,2,0,0,...) that contains n 2s followed by 0s.

+10
0

1, 1, 2, 1, 2, 3, 1, 2, 5, 4, 1, 2, 5, 8, 5, 1, 2, 5, 10, 14, 6, 1, 2, 5, 10, 18, 20, 7, 1, 2, 5, 10, 20, 30, 30, 8, 1, 2, 5, 10, 20, 34, 49, 40, 9, 1, 2, 5, 10, 20, 36, 59, 74, 55, 10, 1, 2, 5, 10, 20, 36, 63, 94, 110, 70, 11, 1, 2, 5, 10, 20, 36, 65, 104, 149, 158, 91, 12

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..78.

EXAMPLE

Table begins:

| 0 1 2 3 4 5 6 7 8 9 10

--+----------------------------------

1 | 1 2 3 4 5 6 7 8 9 10 11

2 | 1 2 5 8 14 20 30 40 55 70 91

3 | 1 2 5 10 18 30 49 74 110 158 221

4 | 1 2 5 10 20 34 59 94 149 224 334

5 | 1 2 5 10 20 36 63 104 169 264 405

6 | 1 2 5 10 20 36 65 108 179 284 445

7 | 1 2 5 10 20 36 65 110 183 294 465

8 | 1 2 5 10 20 36 65 110 185 298 475

9 | 1 2 5 10 20 36 65 110 185 300 479

MATHEMATICA

Seed[i_, n_] := ConstantArray[2, i]~Join~ConstantArray[0, n - i];

A364842Table[n_] := Table[Seed[i, n] // EulerTransform, {i, 1, n}]

(*EulerTransform is defined in A005195*)

CROSSREFS

Cf. A000027 (row 1), A006918 (row 2), A117485 (row 3), A117486 ( row 4), A117487 (row 5), A160647 (row 6), A000712 (main diagonal).

Analogous for initial 1s sequence A008284.

Cf. A115994.

KEYWORD

nonn,tabl

AUTHOR

Peter Kagey, Nov 09 2023

STATUS

approved

A365763

a(n) = number of polynomials of degree 4 in a regular Groebner basis (graded reverse lexicographic order) of n quadratic polynomials in n variables.

+10
0

0, 0, 1, 3, 5, 10, 14, 22, 29, 39, 50, 60, 76, 91, 105, 126, 146, 165, 189, 215, 240, 264, 297, 329, 360, 390, 430, 469, 507, 544, 588, 635, 681, 726, 770, 826, 881, 935

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Table of n, a(n) for n=1..38.

EXAMPLE

For n=3, the leading monomial is x3^4, so a(3) = 1.

For n=4, the 3 leading monomials are x1x4^3, x2x4^3, x3x4^3, so a(4) = 3.

PROG

(Magma)

function a(n);

F:=GF(251);

P<[x]>:=PolynomialRing(F, n, "grevlex");

M2:=[ &*[P| x[i] : i in s] : s in Multisets({1..n}, 2) ];

A:=[ &+[Random(F)*m : m in M2] : i in [1..n]];

G:=GroebnerBasis(A, 4);

return #[ g : g in G | TotalDegree(g) eq 4 ];

end function;

CROSSREFS

Cf. A000027 (degree 2), A006463 (degree 3).

KEYWORD

nonn,more

AUTHOR

Gilles Macario-Rat, Sep 18 2023

STATUS

approved

A367198

T(n, k) = Sum_{m = 0..n-1} Stirling1(m+1, k)*binomial(n, m)*(-1)^(n + k), where "Stirling1" are the signed Stirling numbers of the first kind.

+10
0

1, 1, 2, 4, 6, 3, 15, 30, 18, 4, 76, 165, 125, 40, 5, 455, 1075, 930, 380, 75, 6, 3186, 8015, 7679, 3675, 945, 126, 7, 25487, 67536, 70042, 37688, 11550, 2044, 196, 8, 229384, 634935, 702372, 414078, 144417, 30870, 3990, 288, 9, 2293839, 6591943, 7696245, 4886390, 1885065, 463092, 73080, 7200, 405, 10

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

To use the unsigned Stirling numbers rewrite the formula as: T(n, k) = Sum_{m = 0..n-1} abs(Stirling1(m+1, k))*binomial(n, m)*(-1)^(1+m+n). Replacing in this formula Stirling1 (A008275) by Stirling2 (A048993) one obtains a shifted version of A321331.

LINKS

Table of n, a(n) for n=1..55.

FORMULA

T(n+1, n) = n^2*(n+1)/2 = A002411(n).

T(n, n-2) = 6*T(n-1, n-3) - 15*T(n-2, n-4) + 20*T(n-3, n-5) - 15*T(n-4, n-6) + 6*T(n-5, n-7) - T(n-6, n-8), for n > 8.

T(n, n-k) = (-1)^k*Sum_{m=0..n-1} Stirling1(m+1, n-k)*binomial(n, m).

EXAMPLE

Triangle begins:

1;

1, 2;

4, 6, 3;

15, 30, 18, 4;

76, 165, 125, 40, 5;

455, 1075, 930, 380, 75, 6;

MAPLE

T := (n, k) -> local m; add(Stirling1(m+1, k)*binomial(n, m)*(-1)^(n + k), m = 0..n-1): seq(seq(T(n, k), k = 1..n), n = 1..9); # Peter Luschny, Nov 10 2023

PROG

(PARI) T(n, k) = sum(m=0, n-1, stirling(m+1, k)*binomial(n, m)*(-1)^(n+k))

CROSSREFS

Cf. A002411, A002467 (first column), A000027 (main diagonal), A008275.

Cf. A180191(n+1) (row sums), A321331 (variant with Stirling2).

KEYWORD

nonn,tabl

AUTHOR

Thomas Scheuerle, Nov 10 2023

STATUS

approved

A367408

a(n) = n for n a power of 2; otherwise let 2^r be the greatest power of 2 which does not exceed n, then a(n) = the least novel m*a(k) where k = n - 2^r, and m is not a prior term.

+10
0

1, 2, 3, 4, 5, 12, 18, 8, 6, 14, 21, 28, 35, 84, 126, 16, 7, 20, 27, 36, 45, 108, 162, 72, 54, 140, 189, 252, 315, 756, 1134, 32, 9, 22, 30, 40, 50, 120, 180, 80, 60, 154, 210, 280, 350, 840, 1260, 160, 70, 200, 270, 360, 450, 1080, 1620, 720, 540, 1400, 1890, 2520, 3150, 7560, 11340, 64, 10

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Based on a recursion similar to that which produces the Doudna sequence, A005940, using the same definition of k, the "distance" from the greatest power of 2 less than n (compare with A365436).

Sequence is conjectured to be a permutation of the positive integers, A000027.

LINKS

Table of n, a(n) for n=1..65.

FORMULA

a(2^k) = 2^k for all k >= 0.

a(2^k + 1) = m, the least unused term up to a(2^k), where multiples (other than 1) of m have been used to generate terms between a(2^(k-1)) and a(2^k) except for those which have occurred earlier; see Example.

EXAMPLE

a(3) = 3 since k = 1, a(1) = 1, and 3 is the smallest number which has not already occurred.

a(7) = 18, since k = 3, a(3) = 3, m = 6 is the least unused number and 18 has not already occurred.

For n = 18, k = 2, a(2) = 2, m = 9 is the least unused number, so we should expect a(18) = 2*9 = 18, but 18 has already occurred at a(7). Therefore we increment to m = 10, the next smallest unused number, and find a(18) = 20 (which has not occurred previously).

PROG

(PARI) lista(nn) = my(va=vector(nn)); for (n=1, nn, my(p=2^logint(n, 2)); if (p == n, va[n] = n, my(k=n-p, m=1); while (#select(x->(x==m), va) || #select(x->(x==m*va[k]), va), m++); va[n] = m*va[k]; ); ); va; \\ Michel Marcus, Dec 17 2023

CROSSREFS

Cf. A005940, A053644, A365436.

KEYWORD

nonn

AUTHOR

David James Sycamore, Nov 17 2023

STATUS

approved