A225345 T(n,k) = Number of n X k {-1,1}-arrays such that the sum over i=1..n,j=1..k of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute k-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

0, 1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 3, 0, 1, 0, 3, 6, 7, 0, 0, 1, 0, 9, 0, 15, 0, 1, 0, 3, 12, 31, 0, 33, 8, 0, 1, 0, 17, 0, 107, 0, 77, 0, 1, 0, 5, 22, 81, 0, 395, 410, 181, 0, 0, 1, 0, 27, 0, 397, 0, 1525, 0, 443, 0, 1, 0, 5, 34, 171, 0, 2073, 4508, 6095, 0, 1113, 58, 0, 1, 0, 41, 0, 1081, 0

Offset

1,10

Comments

Table starts

.0...1...0.....1....0......1.....0.......1.....0........1......0........1

.0...1...0.....1....0......1.....0.......1.....0........1......0........1

.0...1...0.....3....0......3.....0.......5.....0........5......0........7

.2...3...6.....9...12.....17....22......27....34.......41.....48.......57

.0...7...0....31....0.....81.....0.....171.....0......309......0......509

.0..15...0...107....0....397.....0....1081.....0.....2399......0.....4675

.0..33...0...395....0...2073.....0....7261.....0....19709......0....45385

.8..77.410..1525.4508..11291.25056...50659.95130...168289.283338...457627

.0.181...0..6095....0..63121.....0..364051.....0..1478059......0..4749875

.0.443...0.24893....0.360909.....0.2676331.....0.13280209......0.50435657

Links

R. H. Hardin, Table of n, a(n) for n = 1..3132

Formula

Empirical for row n:

n=1: a(n) = a(n-2);

n=2: a(n) = a(n-2);

n=3: a(n) = a(n-2) +a(n-4) -a(n-6);

n=4: a(n) = 2*a(n-1) -a(n-2) +a(n-3) -2*a(n-4) +a(n-5);

n=5: a(n) = 3*a(n-2) -2*a(n-4) -2*a(n-6) +3*a(n-8) -a(n-10);

n=6: [order 26, even n];

n=7: [order 42, even n];

n=8: [order 28];

n=9: [order 58, even n];

n=10: [order 90, even n];

n=11: [order 102, even n];

n=12: [order 66].

Example

Some solutions for n=4, k=4
.-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....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...-1..1..1..1...-1..1..1..1...-1.-1.-1.-1...-1.-1..1..1

Crossrefs

Column 1 is A063074(n/4).

Row 3 is A063196(n/2+1).

Row 4 is A008810(n+1).

Row 5 is A202254(n/2).

Keyword

nonn,tabl

Author

R. H. Hardin, May 05 2013

Status

approved

A108263 Triangle read by rows: T(n,k) is the number of short bushes with n edges and k branchnodes (i.e., nodes of outdegree at least two). A short bush is an ordered tree with no nodes of outdegree 1.

1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 5, 0, 1, 9, 5, 0, 1, 14, 21, 0, 1, 20, 56, 14, 0, 1, 27, 120, 84, 0, 1, 35, 225, 300, 42, 0, 1, 44, 385, 825, 330, 0, 1, 54, 616, 1925, 1485, 132, 0, 1, 65, 936, 4004, 5005, 1287, 0, 1, 77, 1365, 7644, 14014, 7007, 429, 0, 1, 90, 1925, 13650, 34398

Offset

0,9

Comments

Row n has 1+floor(n/2) terms. Row sums are the Riordan numbers (A005043). Column 3 yields A033275; column 4 yields A033276.

Related to the number of certain non-crossing partitions for the root system A_n. Cf. p. 12, Athanasiadis and Savvidou. Diagonals are A033282/A086810. Also see A132081 and A100754.- Tom Copeland, Oct 19 2014

Links

Indranil Ghosh, Rows 0..100, flattened

C. Athanasiadis and C. Savvidou, The local h-vector of the cluster subdivision of a simplex, arXiv preprint arXiv:1204.0362, 2012

F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112.

Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 10.

Formula

G.f. G=G(t, z) satisfies z*(1+t*z)*G^2 - (1+z)*G + 1 = 0.

T(n, k) = A086810(n-k, k). - Philippe Deléham, May 30 2005

Example

T(6,3)=5 because the only short bushes with 6 edges and 3 branchnodes are the five full binary trees with 6 edges.
Triangle begins:
1;
0;
0,1;
0,1;
0,1,2;
0,1,5;
0,1,9,5

Maple

G:=(1+z-sqrt((1-z)^2-4*t*z^2))/2/z/(1+t*z): Gser:=simplify(series(G, z=0, 18)): P[0]:=1: for n from 1 to 16 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 16 do seq(coeff(t*P[n], t^k), k=1..1+floor(n/2)) od; # yields sequence in triangular form
A108263 := (n, k) -> binomial(n-k-1, n-2*k)*binomial(n, k)/(n-k+1);
seq(print(seq(A108263(n, k), k=0..ceil((n-1)/2))), n=0..8); # Peter Luschny, Sep 25 2014

Mathematica

T[n_, k_]:=Binomial[n-k-1, n-2k]*Binomial[n, k]/(n-k+1); Flatten[Table[T[n, k], {n, 0, 11}, {k, 0, Ceiling[(n-1)/2]}]] (* Indranil Ghosh, Feb 20 2017 *)

Crossrefs

Cf. A005043, A033275, A033276.

Columns: A000007, A000012, A000096, A033275, A033276, A033277, A033278, A033279; A005043 (row sums), A033282, A086810

Keyword

nonn,tabf

Author

Emeric Deutsch, May 29 2005

Status

approved

A243067 Integers from 0 to A000120(n)-1 followed by integers from 0 to A000120(n+1)-1 and so on, starting with n=1.

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

Offset

1,12

Links

Table of n, a(n) for n=1..120.

Formula

a(n) = n - (1 + A000788(A100922(n-1)-1)).

Example

For n=1, also 1 in binary notation, so the count of its 1-bits is 1 (A000120(1)=1), we list numbers from 0 to 0, thus just 0.
For n=2, 10 in binary, thus A000120(2)=1, we list numbers from 0 to 0, thus just 0.
For n=3, 11 in binary, thus A000120(3)=2, we list numbers from 0 to 1, and so we have the first four terms of the sequence: 0; 0; 0, 1;

Prog

(Scheme) (define (A243067 n) (- n (+ 1 (A000788 (- (A100922 (- n 1)) 1)))))

Crossrefs

Cf. A000788, A100922, A163510, A163511, A227740.

Keyword

nonn

Author

Antti Karttunen, Jun 19 2014

Status

approved

A288424 Partial sums of A288384.

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

Offset

0,10

Comments

It appears that the number of zeros is infinite.

Observation: for at least the first 110 terms the largest distance between two zeros that are between nonzero terms is 3.

Question: are there distances > 3?

From Hartmut F. W. Hoft, Jun 13 2017: (Start)

Yes: a(346...351) = {0,1,2,3,4,0).

Conjecture: a(n) >= 0 for all n >= 0, and a(n) is unbounded.

First occurrences: 3 = a(337) occurring 27 times; 4 = a(350) occurring 8 times; 5 = a(830) occurring 5 times; all through n=2500. (End)

Links

Table of n, a(n) for n=0..104.

Mathematica

(* function a288384[] is defined in A288384 *)
a288424[n_] := Accumulate[a288384[n]]
a288424[104] (* data *) (* Hartmut F. W. Hoft, Jun 13 2017 *)

Crossrefs

Cf. A274650, A286294, A288384.

Keyword

nonn

Author

Omar E. Pol, Jun 09 2017

Extensions

Signs reversed at the suggestion of Hartmut F. W. Hoft by Omar E. Pol, Jun 13 2017

Status

approved

A331981 Number of compositions (ordered partitions) of n into distinct odd primes.

1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 2, 1, 2, 1, 2, 6, 4, 1, 4, 7, 4, 12, 4, 13, 6, 12, 28, 18, 28, 19, 6, 25, 52, 24, 54, 30, 56, 31, 98, 156, 102, 37, 104, 157, 150, 276, 150, 175, 154, 288, 200, 528, 246, 307, 226, 666, 990, 780, 1038, 679, 348, 799, 1828, 1272, 1162, 1164

Offset

0,9

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Index entries for sequences related to compositions

Example

a(16) = 4 because we have [13, 3], [11, 5], [5, 11] and [3, 13].

Maple

s:= proc(n) option remember; `if`(n<1, 0, ithprime(n+1)+s(n-1)) end:
b:= proc(n, i, t) option remember; `if`(s(i)<n, 0, `if`(n=0, t!, (p
      ->`if`(p>n, 0, b(n-p, i-1, t+1)))(ithprime(i+1))+b(n, i-1, t)))
    end:
a:= n-> b(n, numtheory[pi](n), 0):
seq(a(n), n=0..72);  # Alois P. Heinz, Feb 03 2020

Mathematica

s[n_] := s[n] = If[n < 1, 0, Prime[n + 1] + s[n - 1]];
b[n_, i_, t_] := b[n, i, t] = If[s[i] < n, 0, If[n == 0, t!, If[# > n, 0, b[n - #, i - 1, t + 1]]&[Prime[i + 1]] + b[n, i - 1, t]]];
a[n_] := b[n, PrimePi[n], 0];
a /@ Range[0, 72] (* Jean-François Alcover, Nov 09 2020, after Alois P. Heinz *)

Crossrefs

Cf. A002124, A023360, A024939, A065091, A099773, A219107, A331926, A331982.

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 03 2020

Status

approved

A321761 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in s(u), where H is Heinz number, m is monomial symmetric functions, and s is Schur functions.

1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 2, 3, 4, 0, 0, 1, 2, 1, 3, 5, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

1,12

Comments

Row n has length A000041(A056239(n)).

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Links

Table of n, a(n) for n=1..87.

Wikipedia, Symmetric polynomial

Formula

If s(y) = Sum_{|z| = |y|} c(y,z) * m(z), then Sum_{|z| = |y|} c(y,z) * P(z) = A296188(H(y)), where P(y) is the number of distinct permutations of y.

Example

Triangle begins:
   1
   1
   1   1
   0   1
   1   1   1
   0   1   2
   1   1   1   1   1
   0   0   1
   0   1   0   1   2
   0   1   1   2   3
   1   1   1   1   1   1   1
   0   0   0   1   3
   1   1   1   1   1   1   1   1   1   1   1
   0   1   1   2   2   3   4
   0   0   1   2   1   3   5
   0   0   0   0   1
   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
   0   0   0   1   0   2   5
For example, row 15 gives: s(32) = m(32) + 2m(221) + m(311) + 3m(2111) + 5m(11111).

Crossrefs

Row sums are A321762.

Cf. A000085, A008480, A056239, A082733, A124795, A153452, A296188, A300121, A321742-A321765.

Keyword

nonn,tabf

Author

Gus Wiseman, Nov 20 2018

Status

approved

A332033 Number of compositions (ordered partitions) of n into distinct twin primes.

1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 2, 1, 2, 1, 2, 6, 4, 1, 4, 7, 4, 12, 4, 12, 6, 12, 26, 18, 26, 19, 4, 19, 52, 18, 52, 24, 54, 24, 74, 144, 98, 25, 76, 145, 100, 258, 102, 150, 104, 156, 124, 396, 146, 282, 148, 396, 890, 510, 890, 403, 198, 403, 940, 636, 988, 642

Offset

0,9

Links

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

Eric Weisstein's World of Mathematics, Twin Primes

Index entries for sequences related to compositions

Example

a(15) = 6 because we have [7, 5, 3], [7, 3, 5], [5, 7, 3], [5, 3, 7], [3, 7, 5] and [3, 5, 7].

Crossrefs

Cf. A001097, A077608, A219107, A283875, A283876, A331926, A331981.

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 05 2020

Status

approved

A344392 T(n, k) = k!*Stirling2(n - k, k), for n >= 0 and 0 <= k <= floor(n/2). Triangle read by rows.

1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 6, 0, 1, 14, 6, 0, 1, 30, 36, 0, 1, 62, 150, 24, 0, 1, 126, 540, 240, 0, 1, 254, 1806, 1560, 120, 0, 1, 510, 5796, 8400, 1800, 0, 1, 1022, 18150, 40824, 16800, 720, 0, 1, 2046, 55980, 186480, 126000, 15120

Offset

0,9

Comments

The antidiagonal representation of the Fubini numbers (A131689).

Links

Table of n, a(n) for n=0..55.

Example

Triangle starts:
[ 0] [1]
[ 1] [0]
[ 2] [0, 1]
[ 3] [0, 1]
[ 4] [0, 1, 2]
[ 5] [0, 1, 6]
[ 6] [0, 1, 14, 6]
[ 7] [0, 1, 30, 36]
[ 8] [0, 1, 62, 150, 24]
[ 9] [0, 1, 126, 540, 240]
[10] [0, 1, 254, 1806, 1560, 120]
[11] [0, 1, 510, 5796, 8400, 1800]

Maple

T := (n, k) -> k!*Stirling2(n - k, k):
seq(seq(T(n, k), k=0..n/2), n = 0..11);

Crossrefs

Cf. A105795 (row sums).

Cf. A019538, A131689, A048993.

Keyword

nonn,tabf

Author

Peter Luschny, May 17 2021

Status

approved

A014944 Inverse of 935th cyclotomic polynomial.

1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 1, 0, 0, -1, 0, 1, 0, 1, -2, 0, 1, 0, 1, -1, -1, 1, 0, 1, 0, -1, 0, 0, 1, 0, 0, -1, 0, 1, 0, 1, -1, -1, 1, 0, 1, 0, -1, 0, 0, 1, 0, 0, -1, 0, 1, 0, 1, -1, -1, 1, 0, 1, 0, -1, 0, 0, 1, 0, 0, -1, 0, 1, 0, 1, -1, -1

Offset

0,24

Comments

Periodic with period length 935. - Ray Chandler, Apr 07 2017

Links

Ray Chandler, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, order 640.

Index to sequences related to inverse of cyclotomic polynomials

Maple

with(numtheory, cyclotomic); c := n->series(1/cyclotomic(n, x), x, 80);

Mathematica

CoefficientList[Series[1/Cyclotomic[935, x], {x, 0, 120}], x] (* Harvey P. Dale, Jul 15 2017 *)

Keyword

sign

Author

Simon Plouffe

Status

approved

A015879 Inverse of 1870th cyclotomic polynomial.

1, 1, 0, 0, 0, -1, -1, 0, 0, 0, 1, 0, -1, 0, 0, -1, 0, 0, -1, 0, 1, 0, 1, 2, 0, -1, 0, -1, -1, 1, 1, 0, 1, 0, -1, 0, 0, -1, 0, 0, -1, 0, 1, 0, 1, 1, -1, -1, 0, -1, 0, 1, 0, 0, 1, 0, 0, 1, 0, -1, 0, -1, -1, 1, 1, 0, 1, 0, -1, 0, 0, -1, 0, 0, -1, 0, 1, 0, 1, 1, -1

Offset

0,24

Comments

Periodic with period length 1870. - Ray Chandler, Apr 07 2017

Links

Ray Chandler, Table of n, a(n) for n = 0..2000

Index entries for linear recurrences with constant coefficients, order 640.

Index to sequences related to inverse of cyclotomic polynomials

Maple

with(numtheory, cyclotomic); c := n->series(1/cyclotomic(n, x), x, 80);

Keyword

sign

Author

Simon Plouffe

Status

approved