A064999 -id:A064999 - cp's OEIS frontend

A144328 A002260 preceded by a column of 1's: a (1, 1, 2, 3, 4, 5, ...) crescendo triangle by rows.

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

Offset: 1

Gary W. Adamson, Sep 18 2008

Row sums = A000124.
Eigensequence of the triangle = A000142, the factorials.
The triangle as an infinite lower triangular matrix * [1,2,3,...] = A064999.
Generated from A128227 by rotating each row by one position to the right. - R. J. Mathar, Sep 25 2008
A sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A.  Sequence A144328 is the reluctant sequence of A028310 (1 followed by the natural numbers). - Boris Putievskiy, Dec 12 2012
If offset were changed to 0, a(n) would equal the
Let S_n be the set of partitions of n into distinct parts where the number of parts is maximal for that n. For example, for n=6, the set S_6 consists of just one such partition: S_6={1,2,3}. Similarly, for n=7, S_7={1,2,4}, But for n=8, S_8 will contain two partitions S_8= { {1,2,5}, {1,3,4} }. Then |S(n)| = a(n+1). Cf. A178702. - David S. Newman and Benoit Jubin, Dec 13 2010

			First few rows of the triangle:
  1;
  1, 1;
  1, 1, 2;
  1, 1, 2, 3;
  1, 1, 2, 3, 4;
  1, 1, 2, 3, 4, 5;
  ...

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A000124, A000142, A002260, A010054, A064999, A028310, A156702.

a144328 n k = a144328_tabl !! (n-1) !! (k-1)
a144328_row n = a144328_tabl !! (n-1)
a144328_tabl = [1] : map (\xs@(x:_) -> x : xs) a002260_tabl
-- Reinhard Zumkeller, Apr 29 2015

Flatten[Table[Join[{1},Range[n]],{n,0,11}]] (* Harvey P. Dale, Aug 10 2013 *)

from math import comb, isqrt
def A144328(n): return n-comb((m:=isqrt(k:=n<<1))+(k>m*(m+1)),2)-(comb(isqrt(n-1<<1)+1,2)!=n-1) # Chai Wah Wu, Nov 08 2024

Triangle A002260 (natural numbers crescendo triangle) preceded by a column of 1's, = a (1, 1, 2, 3, 4, 5, ...) crescendo triangle by rows.
a(n) = A028310(m-1), where m = n-t*(t+1)/2, t = floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 13 2012
a(n) = A002260(n)+A010054(n-1)-1. - Chai Wah Wu, Nov 08 2024

A300454 Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial 2(x + 1)^(n + 1) + x^3 + 2x^2 - x - 2.

Original entry on oeis.org

0, 1, 2, 1, 0, 3, 4, 1, 0, 5, 8, 3, 0, 7, 14, 9, 2, 0, 9, 22, 21, 10, 2, 0, 11, 32, 41, 30, 12, 2, 0, 13, 44, 71, 70, 42, 14, 2, 0, 15, 58, 113, 140, 112, 56, 16, 2, 0, 17, 74, 169, 252, 252, 168, 72, 18, 2, 0, 19, 92, 241, 420, 504, 420, 240, 90, 20, 2, 0

Offset: 0

Franck Maminirina Ramaharo, Mar 06 2018

Row sums of column 1,2 and 3 yields {4, 8, 16, 30, 52, ...}, in A046127.
Almost twice Pascal's triangle A028326 (up to horizontal shift), except column 0 to 3.
The polynomial P(n;x) = 2*(x + 1)^(n + 1) + x^3 + 2*x^2 - x - 2 is a simplified version of the bracket polynomial associated with a twist knot of n half twists that is only concerned with the enumeration of the state diagrams. The simplification arises when the twist knot is thought of as a planar diagram with no crossing information at each double point. In this case, P(n;x) = x*(A,B,x), where (A,B,d) denotes the bracket polynomial for the n-twist knot (see links for the definition of the bracket polynomial). For example, the bracket polynomial for the trefoil (n = 2) is A^3*d^1 + 3*BA^2*d^0 + 3*AB^2*d^1 + B^3*d^2, where A and B are the "splitting variables". Then setting A = B = 1 and d = x, we obtain 3 + 4*x + x^2 (also see A299989, row 1).

			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   1    2    1
1:   0   3    4    1
2:   0   5    8    3
3:   0   7   14    9     2
4:   0   9   22   21    10     2
5:   0  11   32   41    30    12     2
6:   0  13   44   71    70    42    14     2
7:   0  15   58  113   140   112    56    16     2
8:   0  17   74  169   252   252   168    72    18     2
9:   0  19   92  241   420   504   420   240    90    20     2
10:  0  21  112  331   660   924   924   660   330   110    22    2
11:  0  23  134  441   990  1584  1848  1584   990   440   132   24    2
12:  0  25  158  573  1430  2574  3432  3432  2574  1430   572  156   26   2
13:  0  27  184  729  2002  4004  6006  6864  6006  4004  2002  728  182  28  2

Inga Johnson and Allison K. Henrich, An Interactive Introduction to Knot Theory, Dover Publications, Inc., 2017.

Agnijo Banerjee, Knot theory.
Răzvan Gelca and Fumikazu Nagasato,Some results about the kauffman bracket skein module of the twist knot exterior, J. Knot Theory Ramifications 15 (2006), 1095-1106.
L. H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.
Kelsey Lafferty, The three-variable bracket polynomial for reduced, alternating links, Rose-Hulman Undergraduate Mathematics Journal, Vol. 14: Iss. 2, Article 7 (2013).
Franck Ramaharo, Enumerating the states of the twist knot, arXiv preprint arXiv:1712.06543 [math.CO], 2017.
Franck Ramaharo, A one-variable bracket polynomial for some Turk's head knots, arXiv:1807.05256 [math.CO], 2018.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Alexander Stoimenow, Generating functions, Fibonacci numbers and rational knots, Journal of Algebra, Volume 310, Issue 2 (2007), 491-525.
Eric Weisstein's World of Mathematics, Bracket Polynomial.
Wikipedia, Twist knot.

Row sums: A020707(Pisot sequences).
Triangles related to the regular projection of some knots: A299989 (connected summed trefoils); A300184 (chain links); A300453 ((2,n)-torus knot).
Cf. A002061, A005408, A007318, A014206, A028326, A028326, A046127, A046127, A046127, A064999, A155753, A299989, A300454, A300454.

P(n, x) := 2*(x + 1)^(n + 1) + x^3 + 2*x^2 - x - 2$
T : []$
for i:0 thru 20 do
  T : append(T, makelist(ratcoef(P(i, x), x, n), n, 0, max(3, i + 1)))$
T;

row(n) = Vecrev(2*(x + 1)^(n + 1) + x^3 + 2*x^2 - x - 2);
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Mar 12 2018

T(n,1) = A005408(n).
T(n,2) = A014206(n).
T(n,3) = A064999(n+1).
T(n,1) + T(n,2) = A002061(n+2).
T(n,1) + T(n,3) = A046127(n+1).
T(n,2) + T(n,3) = A155753(n+1).
T(n,1) + T(n,2) + T(n,3) = A046127(n+2).
T(n,k) = A028326(n,k-1), k >= 4 and n >= k - 1.
T(n,k) = A300454(n,k-1) + 2*A300454(n,k) + A007318(n,k-1), with T(n,0) = 0.
G.f: (2*x + 2)/(1 - y*(x + 1)) + (x^3 + 2*x^2 - x - 2)/(1 - y).

A210864 Triangle of coefficients of polynomials u(n,x) jointly generated with A210865; see the Formula section.

Original entry on oeis.org

1, 2, 1, 3, 3, 2, 4, 9, 9, 3, 5, 21, 30, 21, 5, 6, 41, 91, 96, 47, 8, 7, 71, 242, 358, 278, 101, 13, 8, 113, 565, 1187, 1303, 757, 209, 21, 9, 169, 1182, 3517, 5238, 4364, 1951, 422, 34, 10, 241, 2263, 9332, 18816, 21213, 13674, 4802, 833, 55, 11, 331

Offset: 1

Clark Kimberling, Mar 28 2012

Row n starts with n and ends with F(n), where F=A000045 (Fibonacci numbers).
Column 2:  A064999
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
2...1
3...3....2
4...9....9....3
5...21...30...21...5
First three polynomials u(n,x): 1, 2 + x, 3 + 3x + 2x^2.

Cf. A210865, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := (x + n)*u[n - 1, x] + x*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A210864 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210865 *)

u(n,x)=u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=(x+n)*u(n-1,x)+x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.

A014112 a(n) = n(n-1) + (n-2)(n-3) + ... + 10 + 1 for n odd; otherwise, a(n) = n(n-1) + (n-2)(n-3) + ... + 21.

Original entry on oeis.org

1, 2, 7, 14, 27, 44, 69, 100, 141, 190, 251, 322, 407, 504, 617, 744, 889, 1050, 1231, 1430, 1651, 1892, 2157, 2444, 2757, 3094, 3459, 3850, 4271, 4720, 5201, 5712, 6257, 6834, 7447, 8094, 8779, 9500, 10261, 11060, 11901, 12782, 13707, 14674, 15687

Offset: 1

Jon Wild, Jul 14 1997

			From _Bruno Berselli_, Mar 12 2018: (Start)
n=1: 1;
n=2: 1*2;
n=3: 1 + 0*1 + 2*3 = 7;
n=4: 1*2 + 3*4 = 14;
n=5: 1 + 0*1 + 2*3 + 4*5 = 27;
n=6: 1*2 + 3*4 + 5*6 = 44;
n=7: 1 + 0*1 + 2*3 + 4*5 + 6*7 = 69, etc.
(End)

Delbert L. Johnson, Table of n, a(n) for n = 1..20000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A064999, A178218 (first differences).

[n le 2 select n else Self(n-2)+n*(n-1):n in [1..50]]; // Vincenzo Librandi, Feb 28 2016
(C#) public BigInteger a(BigInteger n) => (n * (n + 2) * (2 * n - 1) + 9) / 12; // Delbert L. Johnson, Mar 19 2023

LinearRecurrence[{3, -2, -2, 3, -1}, {1, 2, 7, 14, 27}, 50] (* Vincenzo Librandi, Feb 28 2016 *)

a(n) = a(n-2) + n*(n-1) for n > 2, a(1)=1, a(2)=2.
G.f.: x*(1 - x^3 + 3*x^2 - x)/((x + 1)*(x - 1)^4). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n) + a(n+1) = A064999(n). - R. J. Mathar, Feb 27 2016
a(n) = n*(n + 2)*(2*n - 1)/12 + 3*(1 - (-1)^n)/8. - Bruno Berselli, Mar 12 2018

More terms from Erich Friedman

A128137 A002260 * A128132.

Original entry on oeis.org

1, -1, 4, -1, 1, 9, -1, 1, 5, 16, -1, 1, 5, 11, 25, -1, 1, 5, 11, 19, 36, -1, 1, 5, 11, 19, 29, 49, -1, 1, 5, 11, 19, 29, 41, 64, -1, 1, 5, 11, 19, 29, 41, 55, 81, -1, 1, 5, 11, 19, 29, 41, 55, 71, 100

Offset: 1

Gary W. Adamson, Feb 16 2007

Deleting the right and left borders gives terms of A028387: (1, 5, 11, 29, 29, 41, 55, 71, ...). Row sums = A064999: (1, 3, 9, 21, 41, ...).

			First few rows of the triangle:
   1;
  -1, 4;
  -1, 1, 9;
  -1, 1, 5, 16;
  -1, 1, 5, 11, 25;
  -1, 1, 5, 11, 19, 36;
  -1, 1, 5, 11, 19, 29, 49;
  ...

Cf. A002260, A128132, A028387, A128136, A064999.

A002260 * A128132 as infinite lower triangular matrices.

A144330 Triangle read by rows, A144328 * A000012.

Original entry on oeis.org

1, 2, 1, 4, 3, 2, 7, 6, 5, 3, 11, 10, 9, 7, 4, 16, 15, 14, 12, 9, 5, 22, 21, 20, 18, 15, 11, 6, 29, 28, 27, 25, 22, 18, 13, 7, 37, 36, 35, 33, 30, 26, 21, 15, 8, 46, 45, 44, 42, 39, 35, 30, 24, 17, 9, 56, 55, 54, 52, 49, 45, 40, 34, 27, 19, 10

Offset: 1

Gary W. Adamson, Sep 18 2008

Row sums = A064999: (1, 3, 9, 21, 41, 113,...).

Left border = A000124: (1, 2, 4, 7, 11, 16, 22, 29,...).

			First few rows of the triangle =
1;
2, 1
4, 3, 2
7, 6, 5, 3;
11, 10, 9, 7, 4;
16, 15, 14, 12, 9, 5;
22, 21, 20, 18, 15, 11, 6;
29, 28, 27, 25, 22, 18, 13, 7;
...
Row 4 = (7, 6, 5, 3) = partial sums from the right of triangle A144328, row 4: (1, 1, 2, 3).

A144328, Cf. A064999, A000124

Triangle read by rows, partial sums of A144328 starting from the right.

A363685 Irregular triangle read by rows: T(n,k) is the number of descending plane partitions of order n with the sum of the parts equal to k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 4, 2, 3, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 3, 4, 4, 7, 7, 10, 11, 14, 14, 18, 18, 21, 21, 23, 22, 25, 22, 23, 21, 21, 18, 18, 14, 14, 11, 10, 7, 7, 4, 4, 3, 2, 1, 1, 0, 1

Offset: 1

Ludovic Schwob, Jun 15 2023

			Rows 1 through 5 are
  1;
  1, 0, 1;
  1, 0, 1, 1, 1, 1, 1, 0, 1;
  1, 0, 1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 4, 2, 3, 2, 2, 1, 1, 0, 1;
  1, 0, 1, 1, 2, 3, 4, 4, 7, 7, 10, 11, 14, 14, 18, 18, 21, 21, 23, 22, 25, 22, 23, 21, 21, 18, 18, 14, 14, 11, 10, 7, 7, 4, 4, 3, 2, 1, 1, 0, 1.

Row sums give A005130.

Cf. A198890, A005130, A064999.

T(0, 0) = 1, T(1, 0) = 1, T(n, k) = 0 for k < 0 or k > (1/3)*(n+1)*n*(n-1).

cp's OEIS Frontend

A144328 A002260 preceded by a column of 1's: a (1, 1, 2, 3, 4, 5, ...) crescendo triangle by rows.

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A300454 Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial 2*(x + 1)^(n + 1) + x^3 + 2*x^2 - x - 2.

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A210864 Triangle of coefficients of polynomials u(n,x) jointly generated with A210865; see the Formula section.

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A014112 a(n) = n*(n-1) + (n-2)*(n-3) + ... + 1*0 + 1 for n odd; otherwise, a(n) = n*(n-1) + (n-2)*(n-3) + ... + 2*1.

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A128137 A002260 * A128132.

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A144330 Triangle read by rows, A144328 * A000012.

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A363685 Irregular triangle read by rows: T(n,k) is the number of descending plane partitions of order n with the sum of the parts equal to k.

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A300454 Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial 2(x + 1)^(n + 1) + x^3 + 2x^2 - x - 2.

A014112 a(n) = n(n-1) + (n-2)(n-3) + ... + 10 + 1 for n odd; otherwise, a(n) = n(n-1) + (n-2)(n-3) + ... + 21.