A028326 -id:A028326 - cp's OEIS frontend

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.

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).

A227550 A triangle formed like Pascal's triangle, but with factorial(n) on the borders instead of 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 6, 4, 4, 6, 24, 10, 8, 10, 24, 120, 34, 18, 18, 34, 120, 720, 154, 52, 36, 52, 154, 720, 5040, 874, 206, 88, 88, 206, 874, 5040, 40320, 5914, 1080, 294, 176, 294, 1080, 5914, 40320, 362880, 46234, 6994, 1374, 470, 470, 1374, 6994, 46234, 362880, 3628800

Offset: 0

Vincenzo Librandi, Aug 04 2013

A003422 gives the second column (after 0).

			Triangle begins:
       1;
       1,     1;
       2,     2,    2;
       6,     4,    4,    6;
      24,    10,    8,   10,  24;
     120,    34,   18,   18,  34, 120;
     720,   154,   52,   36,  52, 154,  720;
    5040,   874,  206,   88,  88, 206,  874, 5040;
   40320,  5914, 1080,  294, 176, 294, 1080, 5914, 40320;
  362880, 46234, 6994, 1374, 470, 470, 1374, 6994, 46234, 362880;

Vincenzo Librandi, Rows n = 0..70, flattened

Cf. similar triangles with t on the borders: A007318 (t = 1), A028326 (t = 2), A051599 (t = prime(n)), A051601 (t = n), A051666 (t = n^2), A108617 (t = fibonacci(n)), A134636 (t = 2n+1), A137688 (t = 2^n), A227075 (t = 3^n).
Cf. A003422.
Cf. A227791 (central terms), A001563, A074911.

a227550 n k = a227550_tabl !! n !! k
a227550_row n = a227550_tabl !! n
a227550_tabl = map fst $ iterate
   (\(vs, w:ws) -> (zipWith (+) ([w] ++ vs) (vs ++ [w]), ws))
   ([1], a001563_list)
-- Reinhard Zumkeller, Aug 05 2013

function T(n,k)
  if k eq 0 or k eq n then return Factorial(n);
  else return T(n-1,k-1) + T(n-1,k);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 02 2021

t = {}; Do[r = {}; Do[If[k == 0||k == n, m = n!, m = t[[n, k]] + t[[n, k + 1]]]; r = AppendTo[r, m], {k, 0, n}]; AppendTo[t, r], {n, 0, 10}]; t = Flatten[t]

def T(n,k): return factorial(n) if (k==0 or k==n) else T(n-1, k-1) + T(n-1, k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 02 2021

From G. C. Greubel, May 02 2021: (Start)
T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = T(n, n) = n!.
Sum_{k=0..n} T(n, k) = 2^n * (1 +Sum_{j=1..n-1} j*j!/2^j) = A140710(n). (End)

A139524 Triangle T(n,k) read by rows: the coefficient of [x^k] of the polynomial 2*(x+1)^n + 2^n in row n, column k.

Original entry on oeis.org

3, 4, 2, 6, 4, 2, 10, 6, 6, 2, 18, 8, 12, 8, 2, 34, 10, 20, 20, 10, 2, 66, 12, 30, 40, 30, 12, 2, 130, 14, 42, 70, 70, 42, 14, 2, 258, 16, 56, 112, 140, 112, 56, 16, 2, 514, 18, 72, 168, 252, 252, 168, 72, 18, 2, 1026, 20, 90, 240, 420, 504, 420, 240, 90, 20, 2

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jun 09 2008

			Triangle begins as:
     3;
     4,  2;
     6,  4,  2;
    10,  6,  6,   2;
    18,  8, 12,   8,   2;
    34, 10, 20,  20,  10,   2;
    66, 12, 30,  40,  30,  12,   2;
   130, 14, 42,  70,  70,  42,  14,   2;
   258, 16, 56, 112, 140, 112,  56,  16,  2;
   514, 18, 72, 168, 252, 252, 168,  72, 18,  2;
  1026, 20, 90, 240, 420, 504, 420, 240, 90, 20, 2;

Advanced Number Theory, Harvey Cohn, Dover Books, 1963, Pages 88-89

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007283, A028326, A052548.

A139524:= func< n,k | k eq 0 select 2+2^n else 2*Binomial(n,k) >;
[A139524(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 02 2021

(* First program *)
T[n_, k_]:= SeriesCoefficient[Series[2*(1+x)^n + 2^n, {x, 0, 20}], k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, May 02 2021 *)
(* Second program *)
T[n_, k_]:= T[n, k] = If[k==0, 2 + 2^n, 2*Binomial[n, k]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 02 2021 *)

def A139524(n,k): return 2+2^n if (k==0) else 2*binomial(n,k)
flatten([[A139524(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 02 2021

Sum_{k=0..n} T(n,k) = 3*2^n = A007283(n).
From R. J. Mathar, Sep 12 2013: (Start)
T(n,0) = 2 + 2^n = A052548(n).
T(n,k) = 2*binomial(n,k) = A028326(n,k) if k>0. (End)

A317644 Triangle read by rows: multiplicative version of Pascal's triangle except n-th row begins and ends with (n+1)-st prime.

Original entry on oeis.org

2, 3, 3, 5, 9, 5, 7, 45, 45, 7, 11, 315, 2025, 315, 11, 13, 3465, 637875, 637875, 3465, 13, 17, 45045, 2210236875, 406884515625, 2210236875, 45045, 17, 19, 765765, 99560120034375, 899311160300888671875, 899311160300888671875, 99560120034375, 765765, 19, 23, 14549535, 76239655318123171875, 89535527067809533413858673095703125, 808760563041730681160065242862701416015625, 89535527067809533413858673095703125, 76239655318123171875, 14549535, 23

Offset: 0

Philipp O. Tsvetkov, Aug 02 2018

			Triangle begins:
   2;
   3,      3;
   5,      9,      5;
   7,     45,     45,      7;
  11,    315,   2025,    315,     11;
  13,   3465, 637875, 637875,   3465,     13;
  ...
Formatted as a symmetric triangle:
.
                       2
.
                   3       3
.
               5       9       5
.
           7      45      45       7
.
      11      315    2025     315     11
.
  13     3465   637875  637875   3465     13
...

Cf. A000040, A007318, A007949, A028326, A051599, A080046, A112765.

t = {{2}};
Table[AppendTo[
    t, {Prime[i],
      Table[
       t[[i - 1]][[j]]*t[[i - 1]][[j + 1]], {j,
        1, (t[[i - 1]] // Length) - 1}], Prime[i]} // Flatten], {i, 2, 10}] //
   Last // Flatten
t={}; Do[r={}; Do[If[k==0||k==n, m=Prime[n + 1], m=t[[n, k]]t[[n, k + 1]]]; r=AppendTo[r, m], {k, 0, n}]; AppendTo[t, r], {n, 0, 10}]; t (* Vincenzo Librandi, Sep 03 2018 *)

From Rémy Sigrist, Sep 02 2018: (Start)
A007949(T(n+1, k+1)) = A028326(n, k) for any n >= 0 and k = 0..n.
A112765(T(n+1, k+1)) = A007318(n, k) for any n > 0 and k = 0..n.
(End)

cp's OEIS Frontend

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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A227550 A triangle formed like Pascal's triangle, but with factorial(n) on the borders instead of 1.

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A139524 Triangle T(n,k) read by rows: the coefficient of [x^k] of the polynomial 2*(x+1)^n + 2^n in row n, column k.

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A317644 Triangle read by rows: multiplicative version of Pascal's triangle except n-th row begins and ends with (n+1)-st prime.

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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.