A152842 -id:A152842 - cp's OEIS frontend

A299989 Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2k+1) = A152842(2n,2(n-k)) and T(n,2k) = A152842(2n,2(n-k)+1) for n >= k > 0.

0, 1, 0, 3, 4, 1, 0, 9, 24, 22, 8, 1, 0, 27, 108, 171, 136, 57, 12, 1, 0, 81, 432, 972, 1200, 886, 400, 108, 16, 1, 0, 243, 1620, 4725, 7920, 8430, 5944, 2810, 880, 175, 20, 1, 0, 729, 5832, 20898, 44280, 61695, 59472, 40636, 19824, 6855, 1640, 258, 24, 1

Offset: 0

Franck Maminirina Ramaharo, Feb 26 2018

T(n,k) is the number of state diagrams having k components of n connected summed trefoil knots.

Row sums gives A001018.

			The triangle T(n, k) begins:
n\k 0     1      2      3       4       5       6      7        8       9
0:  0     1
1:  0     3      4      1
2:  0     9     24     22       8       1
3:  0    27    108    171     136      57      12       1
4:  0    81    432    972    1200     886     400     108      16       1

V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994.

Michael De Vlieger, Table of n, a(n) for n = 0..10301 (rows 0 <= n <= 100, flattened.)
Ryo Hanaki, Pseudo diagrams of knots, links and spatial graphs, Osaka Journal of Mathematics, Vol. 47 (2010), 863-883.
Louis H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.
Carolina Medina, Jorge Ramírez-Alfonsín and Gelasio Salazar, On the number of unknot diagrams, arXiv:1710.06470 [math.CO], 2017.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Franck Ramaharo, A bracket polynomial for 2-tangle shadows, arXiv:2002.06672 [math.CO], 2020.

Row 2: row 5 of A158454.
Row 3: row 2 of A220665.
Row 4: row 5 of A219234.

row[n_] := CoefficientList[x*(x^2 + 4*x + 3)^n, x]; Array[row, 7, 0] // Flatten (* Jean-François Alcover, Mar 16 2018 *)

g(x, y) := taylor(x/(1 - y*(x^2 + 4*x + 3)), y, 0, 10)$
a : makelist(ratcoef(g(x, y), y, n), n, 0, 10)$
T : []$
for i:1 thru 11 do
  T : append(T, makelist(ratcoef(a[i], x, n), n, 0, 2*i - 1))$
T;

T(n, k) = polcoeff(x*(x^2 + 4*x + 3)^n, k);
tabf(nn) = for (n=0, nn, for (k=0, 2*n+1, print1(T(n, k), ", ")); print); \\ Michel Marcus, Mar 03 2018

T(n,k) = coefficients of x*(x^2 + 4*x + 3)^n.
T(n,k) = T(n-1,k-2) + 4*T(n-1,k-1) + 3*T(n-1,k), with T(n,0) = 0, T(n,1) = 3^n and  T(n,2) =  4*n*3^(n-1).
T(n,n+k+1) = A152842(2*n,n+k) and T(n,n-k) = A152842(2*n,n+k+1), for n >= k >= 0.
T(n,1) =  A000244(n).
T(n,2) =  A120908(n).
T(n,n+1) = A069835(n).
T(n,2*n-1) = A139272(n).
T(n,2*n) = A008586(n).
T(n,2*n-2) = A140138(4*n) = A185872(2n,2) for n >= 1.
G.f.: x/(1 - y*(x^2 + 4*x + 3)).

Typo in row 6 corrected by Jean-François Alcover, Mar 16 2018

A069835 Define an array as follows: b(i,0)=b(0,j)=1, b(i,j) = 2*b(i-1,j-1) + b(i-1,j) + b(i,j-1). Then a(n) = b(n,n).

Original entry on oeis.org

1, 4, 22, 136, 886, 5944, 40636, 281488, 1968934, 13875544, 98365972, 700701808, 5011371964, 35961808432, 258805997752, 1867175631136, 13500088649734, 97794850668952, 709626281415076, 5157024231645616, 37528209137458516, 273431636191026064, 1994448720786816712

Offset: 0

Benoit Cloitre, May 03 2002

2^n*LegendreP(n,k) yields the central coefficients of (1 + 2*k*x + (k^2-1)*x^2)^n, with g.f. 1/sqrt(1 - 4*k*x + 4*x^2) and e.g.f. exp(2*k*x)BesselI(0, 2*sqrt(k^2-1)*x). - Paul Barry, May 25 2005
Number of Delannoy paths from (0,0) to (n,n) with steps U(0,1), H(1,0) and D(1,1) where D can have two colors. - Paul Barry, May 25 2005
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U steps can have three colors and H steps can have four colors. - N-E. Fahssi, Mar 31 2008
Number of lattice paths from (0,0) to (n,n) using steps (1,0), (0,1), and two kinds of steps (1,1). - Joerg Arndt, Jul 01 2011
Hankel transform is 2^n*3^C(n+1,2) = (-1)^C(n+1,2)*A127946(n). - Paul Barry, Jan 24 2011
Central terms of triangle A152842. - Reinhard Zumkeller, May 01 2014
Diagonal of rational functions 1/(1 - x - y - 2*x*y), 1/(1 - x - y*z - 2*x*y*z). - Gheorghe Coserea, Jul 06 2018
The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. - Peter Bala, Jan 07 2022

			The array b is a rewriting of A081577:
  1,  1,  1,   1,   1,    1,    1,    1,     1,     1,     1, ...
  1,  4,  7,  10,  13,   16,   19,   22,    25,    28,    31, ...
  1,  7, 22,  46,  79,  121,  172,  232,   301,   379,   466, ...
  1, 10, 46, 136, 307,  586, 1000, 1576,  2341,  3322,  4546, ...
  1, 13, 79, 307, 886, 2086, 4258, 7834, 13327, 21331, 32521, ...

Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..250
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - _N. J. A. Sloane_, Oct 08 2012
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.

Cf. A001850.

List([0..25],n->Sum([0..n],k->Binomial(n,k)^2*3^k)); # Muniru A Asiru, Jul 29 2018

a069835 n = a081577 (2 * n) n -- Reinhard Zumkeller, Mar 16 2014

Table[Hypergeometric2F1[-n, -n, 1, 3], {n, 0, 21}] (* Arkadiusz Wesolowski, Aug 13 2012 *)

PARI
```
a(n)=sum(k=0,n,binomial(n,k)^2*3^k)
```

a(n)=if(n<0, 0, polcoeff((1+4*x+3*x^2)^n, n))

/* as lattice paths: same as in A092566 but use */
steps=[[1,0], [0,1], [1,1], [1,1]]; /* note the double [1,1] */
\\ Joerg Arndt, Jul 01 2011

a(n)=pollegendre(n,2)<Charles R Greathouse IV, Mar 18 2017

From Vladeta Jovovic, May 13 2003: (Start)
a(n) = 2^n*LegendreP(n, 2) = 2^n*hypergeom([ -n, n+1], [1], -1/2) = 2^n*GegenbauerC(n, 1/2, 2) = Sum_{k=0..n} 3^k*binomial(n, k)^2.
D-finite with recurrence: a(n) = 4*(2*n-1)/n*a(n-1) - 4*(n-1)/n*a(n-2).
G.f.: 1/sqrt(1 - 8*x + 4*x^2). (End)
a(n) equals the central coefficient of (1 + 4*x + 3*x^2)^n. - Paul D. Hanna, Jun 03 2003
E.g.f.: exp(4*x)*Bessel_I(0, 2*sqrt(3)*x). - Paul Barry, Sep 20 2004
a(n) = Sum_{k=0..floor(n/2)} C(n, k)*C(2*(n-k), n)*(-1)^k*2^(n-2*k). - Paul Barry, May 25 2005
a(n) = Sum_{k=0..n} C(n, k)*C(n+k, k)*2^(n-k). - Paul Barry, May 25 2005
a(n) = Sum_{k=0..n} C(n, k)^2*3^k. - Paul Barry, Oct 15 2005
G.f.: 1/(1-4x-6x^2/(1-4x-3x^2/(1-4x-3x^2/(1-4x-3x^2/(1-... (continued fraction). - Paul Barry Jan 24 2011
Asymptotic: a(n) ~ sqrt(1/2 + 1/sqrt(3))*(1+sqrt(3))^(2*n)/sqrt(Pi*n). - Vaclav Kotesovec, Sep 11 2012
0 = a(n)*(16*a(n+1) - 48*a(n+2) + 8*a(n+3)) + a(n+1)*(-16*a(n+1) + 64*a(n+2) - 12*a(n+3)) + a(n+2)*(-4*a(n+2) + a(n+3)) for all n in Z. - Michael Somos, Apr 21 2020

A094015 Expansion of (1+4x)/(1-8x^2).

Original entry on oeis.org

1, 4, 8, 32, 64, 256, 512, 2048, 4096, 16384, 32768, 131072, 262144, 1048576, 2097152, 8388608, 16777216, 67108864, 134217728, 536870912, 1073741824, 4294967296, 8589934592, 34359738368, 68719476736, 274877906944

Offset: 0

Paul Barry, Apr 21 2004

Row sums of triangle A135838. - Gary W. Adamson, Dec 01 2007

Row sums of triangle A152842. - Reinhard Zumkeller, May 01 2014

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,8).

Cf. A010685, A094014, A135838, A152842.

a094015 = sum . a152842_row  -- Reinhard Zumkeller, May 01 2014

[2*8^Floor((n-1)/2)*(3+(-1)^n): n in [0..30]]; // G. C. Greubel, Nov 22 2021

a:=n->mul(3-(-1)^j,j=1..n):seq(a(n),n=0..25); # Zerinvary Lajos, Dec 13 2008

Table[8^Floor[n/2]*Mod[4^n, 5], {n, 0, 30}] (* G. C. Greubel, Nov 22 2021 *)

[8^(n//2)*(4^n%5) for n in (0..30)] # G. C. Greubel, Nov 22 2021

a(n) = 2^(3*n/2)*(1 + sqrt(2) + (-1)^n*(1 - sqrt(2)))/2.
a(n) = (1/4)*(3 + (-1)^n)*8^floor((n+1)/2). - Paul Barry, Jul 14 2004
a(n) = (1 + sqrt(2))*(2*sqrt(2))^n/2 + (1 - sqrt(2))*(-2*sqrt(2))^n/2. Third binomial transform is A002315 (NSW numbers). - Paul Barry, Jul 17 2004
a(n) = 2^A007494(n). - Paul Barry, Aug 18 2007
a(n) = A016116(n+1)*A000079(n). - R. J. Mathar, Jul 08 2009
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = 8^floor(n/2)*A010685(n). - G. C. Greubel, Nov 22 2021

cp's OEIS Frontend

A299989 Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) for n >= k > 0.

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A069835 Define an array as follows: b(i,0)=b(0,j)=1, b(i,j) = 2*b(i-1,j-1) + b(i-1,j) + b(i,j-1). Then a(n) = b(n,n).

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A094015 Expansion of (1+4*x)/(1-8*x^2).

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A299989 Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2k+1) = A152842(2n,2(n-k)) and T(n,2k) = A152842(2n,2(n-k)+1) for n >= k > 0.

A094015 Expansion of (1+4x)/(1-8x^2).