A284938 -id:A284938 - cp's OEIS frontend

A300453 Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (x + 1)^n + x^2 - 1.

0, 0, 1, 0, 1, 1, 0, 2, 2, 0, 3, 4, 1, 0, 4, 7, 4, 1, 0, 5, 11, 10, 5, 1, 0, 6, 16, 20, 15, 6, 1, 0, 7, 22, 35, 35, 21, 7, 1, 0, 8, 29, 56, 70, 56, 28, 8, 1, 0, 9, 37, 84, 126, 126, 84, 36, 9, 1, 0, 10, 46, 120, 210, 252, 210, 120, 45, 10, 1, 0, 11, 56, 165

Offset: 0

Franck Maminirina Ramaharo, Mar 06 2018

This is essentially the usual Pascal triangle A007318, horizontally shifted and with the first three columns altered.
Let P(n;x) = (x + 1)^n + x^2 - 1. Then P(n;x) = P(n-1;x) + x*(x + 1)^(n - 1), with P(0;x) = x^2.
Let a (2,n)-torus knot be projected on the plane. The resulting projection is a planar diagram with n double points. Then, T(n,k) gives the number of state diagrams having k components that are obtained by deleting each double point, then pasting the edges in one of the two ways as shown below.
         \  /     \_/                   \  /     \   /
   (1)    \/  ==>                 (2)      \/  ==>  | |
          /\       _                     /\       | |
         /  \     /   \                   /  \     /   \
See example for the case n = 2.

			The triangle T(n,k) begins
n\k  0   1   2    3     4     5     6     7     8     9    10   11  12  13 14
0:   0   0   1
1:   0   1   1
2:   0   2   2
3:   0   3   4    1
4:   0   4   7    4     1
5:   0   5  11   10     5     1
6:   0   6  16   20    15     6     1
7:   0   7  22   35    35    21     7     1
8:   0   8  29   56    70    56    28     8     1
9:   0   9  37   84   126   126    84    36     9     1
10:  0  10  46  120   210   252   210   120    45    10     1
11:  0  11  56  165   330   462   462   330   165    55    11    1
12:  0  12  67  220   495   792   924   792   495   220    66   12   1
13:  0  13  79  286   715  1287  1716  1716  1287   715   286   78  13   1
14:  0  14  92  364  1001  2002  3003  3432  3003  2002  1001  364  91  14  1
...
The states of the (2,2)-torus knot (Hopf Link) are the last four diagrams:
                                    ____  ____
                                   /    \/    \
                                  /     /\     \
                                 |     |  |     |
                                 |     |  |     |
                                  \     \/     /
                                   \____/\____/
                           ___    ____         __________
                         /    \  /    \       /    __    \
                        /     /  \     \     /    /  \    \
                       |      |  |      |   |     |  |     |
                       |      |  |      |   |     |  |     |
                        \      \/      /     \     \/     /
                         \_____/\_____/       \____/\____/
      ____    ____        ____    ____        ____________        __________
     /    \  /    \      /    \  /    \      /     __     \      /    __    \
    /     /  \     \    /     /  \     \    /     /  \     \    /    /  \    \
   |     |    |     |  |     |    |     |  |     |    |     |  |    |    |    |
   |     |    |     |  |     |    |     |  |     |    |     |  |    |    |    |
    \     \  /     /    \     \__/     /    \     \  /     /    \    \__/    /
     \____/  \____/      \____________/      \____/  \____/      \__________/
There are 2 diagrams that consist of two components, and 2 diagrams that consist of one component.

Colin Adams, The Knot Book, W. H. Freeman and Company, 1994.
Louis H. Kauffman, Knots and Physics, World Scientific Publishers, 1991.

Michael De Vlieger, Table of n, a(n) for n = 0..11327 (rows 0 <= n <= 150, flattened).
Agnijo Banerjee, Knot theory [Foil knot family].
Allison Henrich, Rebecca Hoberg, Slavik Jablan, Lee Johnson, Elizabeth Minten and Ljiljana Radovic, The Theory of Pseudoknots, arXiv preprint arXiv:1210.6934 [math.GT], 2012.
Abdullah Kopuzlu, Abdulgani Şahin and Tamer Ugur, On polynomials of K(2,n) torus knots, Applied Mathematical Sciences, Vol. 3 (2009), 2899-2910.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Franck Ramaharo, Note on sequences A123192, A137396 and A300453, arXiv:1911.04528 [math.CO], 2019.
Franck Ramaharo, A bracket polynomial for 2-tangle shadows, arXiv:2002.06672 [math.CO], 2020.
Wikipedia, Torus knot.
Xinfei Li, Xin Liu and Yong-Chang Huang, Tackling tangledness of cosmic strings by knot polynomial topological invariants, arxiv preprint arXiv:1602.08804 [hep-th], 2016.

Row sums: A000079 (powers of 2).
Triangles related to the regular projection of some knots: A299989 (connected summed trefoils); A300184 (chain links); A300454 (twist knot).
When n = 3 (trefoil), the corresponding 4-tuplet (0,3,4,1) appears in several triangles: A030528 (row 5), A256130 (row 3), A128908 (row 3), A186084 (row 6), A284938 (row 7), A101603 (row 3), A155112 (row 3), A257566 (row 3).
Cf. A001477, A007318, A007318, A152947.

f[n_] := CoefficientList[ Expand[(x + 1)^n + x^2 - 1], x]; Array[f, 12, 0] // Flatten (* or *)
CoefficientList[ CoefficientList[ Series[(x^2 + y*x/(1 - y*(x + 1)))/(1 - y), {y, 0, 11}, {x, 0, 11}], y], x] // Flatten (* Robert G. Wilson v, Mar 08 2018 *)

P(n, x) := (x + 1)^n + x^2 - 1$
T : []$
for i:0 thru 20 do
  T : append(T, makelist(ratcoef(P(i, x), x, n), n, 0, max(2, i)))$
T;

row(n) = Vecrev((x + 1)^n + x^2 - 1);
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Mar 12 2018

T(n,1) = A001477(n).
T(n,2) = A152947(n).
T(n,k) = A007318(n,k-1), k >= 1.
T(n,0) = 0, T(0,1) = 1, T(0,2) = 1 and T(n,k) = T(n-1,k) + A007318(n-1,k-1).
G.f.: (x^2 + y*x/(1 - y*(x + 1)))/(1 - y).

A284945 Positions of 0 in A284944; complement of A284946.

Original entry on oeis.org

1, 6, 10, 14, 18, 19, 24, 28, 32, 33, 38, 42, 46, 47, 52, 56, 60, 61, 63, 68, 72, 76, 80, 81, 86, 90, 94, 95, 100, 104, 108, 109, 111, 116, 120, 124, 128, 129, 134, 138, 142, 143, 148, 152, 156, 157, 159, 164, 168, 172, 176, 177, 182, 186, 190, 191, 196, 200

Offset: 1

Clark Kimberling, Apr 18 2017

Conjecture: -4 < n*r - a(n) < 3 for n >= 1, where r = 2 + sqrt(2).

			As a word, A284938 = 011110..., in which 0 is in positions 1,6,10,14,...

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A284944, A284946.

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 1, 1, 0}}] &, {0}, 6] (* A284944 *)
Flatten[Position[s, 0]]  (* A284945 *)
Flatten[Position[s, 1]]  (* A284946 *)

A284946 Positions of 1 in A284944; complement of A284945.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 20, 21, 22, 23, 25, 26, 27, 29, 30, 31, 34, 35, 36, 37, 39, 40, 41, 43, 44, 45, 48, 49, 50, 51, 53, 54, 55, 57, 58, 59, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 84, 85, 87, 88, 89, 91, 92

Offset: 1

Clark Kimberling, Apr 18 2017

Conjecture: -1 < n*r - a(n) < 3 for n >= 1, where r = 2 + sqrt(2).

			As a word, A284938 = 011110..., in which 1 is in positions 12,3,4,5,7,...

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A284945, A284946.

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 1, 1, 0}}] &, {0}, 6] (* A284944 *)
Flatten[Position[s, 0]]  (* A284945 *)
Flatten[Position[s, 1]]  (* A284946 *)

A057094 Coefficient triangle for certain polynomials (rising powers).

Original entry on oeis.org

0, 0, -1, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 2, -1, 0, 0, 0, -1, 3, -1, 0, 0, 0, 0, -3, 4, -1, 0, 0, 0, 0, 1, -6, 5, -1, 0, 0, 0, 0, 0, 4, -10, 6, -1, 0, 0, 0, 0, 0, -1, 10, -15, 7, -1, 0, 0, 0, 0, 0, 0, -5, 20, -21, 8, -1, 0, 0, 0, 0, 0, 0, 1, -15, 35, -28, 9, -1, 0, 0, 0, 0, 0, 0, 0, 6, -35, 56, -36, 10, -1, 0, 0, 0, 0, 0, 0, 0, -1, 21, -70, 84

Offset: 0

Wolfdieter Lang, Aug 11 2000

The row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n) are negative scaled Chebyshev U-polynomials: p(n,x)= -U(n-1,sqrt(x)/2)*(sqrt(x))^(n+1), n >= 1. p(0,x)=0. p(n-1,1/x) appears in the n-th power of the g.f. of Catalan's numbers A000108, c(x): (c(x))^n = p(n-1,1/x)*1 -p(n,1/x)*x*c(x). Cf. Lang reference eqs.(1) and (2).

Signed version of A284938. - Eric W. Weisstein, Apr 06 2017

			Triangle begins:
0;
0, -1;
0, 0, -1;
0, 0, 1, -1;
0, 0, 0, 2, -1;
0, 0, 0, -1, 3, -1;
...

T. Copeland, Addendum to Elliptic Lie Triad
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Note 1 and Table.
Index entries for sequences related to Chebyshev polynomials.

Cf. A284938 (unsigned version).

Prepend[CoefficientList[Table[I^n x^(n/2) Fibonacci[n - 1, -I Sqrt[x]], {n, 2, 14}], x], {0}] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
Prepend[CoefficientList[Table[-x^(n/2) ChebyshevU[n - 2, Sqrt[x]/2], {n, 2, 14}], x], {0}] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)

tabl(nn) = {for (n=0, nn, for (k=0, n, if ((n==0) || (k < n\2+1), v = 0, v = (-1)^(n-k+1)*binomial(k-1, n-k)); print1(v, ", ");); print(););} \\ Michel Marcus, Jan 14 2016

a(n, m)=0 if n= 1 and n >= m >=floor(n/2)+1; else 0.

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A300453 Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (x + 1)^n + x^2 - 1.

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A284945 Positions of 0 in A284944; complement of A284946.

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A284946 Positions of 1 in A284944; complement of A284945.

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A057094 Coefficient triangle for certain polynomials (rising powers).

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