A152254 -id:A152254 - cp's OEIS frontend

A084773 Coefficients of 1/sqrt(1-12x+4x^2); also, a(n) is the central coefficient of (1+6x+8x^2)^n.

1, 6, 52, 504, 5136, 53856, 575296, 6225792, 68026624, 748832256, 8291791872, 92255680512, 1030537089024, 11550176206848, 129824329777152, 1462841567576064, 16518691986407424, 186887008999047168, 2117944490818011136, 24038305911245635584, 273199990096465494016

Offset: 0

Paul D. Hanna, Jun 10 2003

nonn

Number of Delannoy paths from (0,0) to (n,n) with steps U(0,1), H(1,0) and D(1,1) where H and D can choose from two colors each (or where one step is monochrome and the other two are bicolored). - Paul Barry, May 30 2005
2^n*P_n(3), where P_n is the n-th Legendre polynomial. 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 30 2005
Diagonal of rational functions 1/(1 - x - 2*y - 2*x*y), 1/(1 - x - 2*y*z - 2*x*y*z), 1/(1 - 2*x - y*z - 2*x*y*z). - Gheorghe Coserea, Jul 07 2018

			G.f.: 1/sqrt(1-2*b*x+(b^2-4*c)*x^2) yields central coefficients of (1+b*x+c*x^2)^n.

Harvey P. Dale, Table of n, a(n) for n = 0..939
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.

Sequences of the form 2^n*LegendreP(n, 2*m+1): A000079 (m=0), this sequence (m=1), A098270 (m=2).
Cf. A001850, A052141, A059474.
See A152254 for another interpretation.

[2^n*Evaluate(LegendrePolynomial(n), 3): n in [0..40]]; // G. C. Greubel, May 21 2023

a := n -> 2^n*hypergeom([-n, -n], [1], 2):
seq(simplify(a(n)), n=0..20); # Peter Luschny, May 20 2015

CoefficientList[Series[1/Sqrt[(1-12x+4x^2)],{x,0,20}],x] (* Harvey P. Dale, Dec 13 2017 *)
Table[2^n*LegendreP[n,3], {n,0,40}] (* G. C. Greubel, May 21 2023 *)

for(n=0,30,t=polcoeff((1+6*x+8*x^2)^n,n,x); print1(t","))

[2^n*gen_legendre_P(n,0,3) for n in range(41)] # G. C. Greubel, May 21 2023

a(n) = 2*A052141(n) = 2^n * A001850(n), n>0.
From Paul Barry, May 30 2005: (Start)
E.g.f.: exp(6*x)*Bessel_I(0, 2*sqrt(8)*x).
a(n) = Sum_{k=0..floor(n/2)} C(n, k)*C(2(n-k), n)*(-1)^k*3^(n-2*k). (End)
D-finite with recurrence: n*a(n) + 6*(-2*n+1)*a(n-1) + 4*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 30 2012
G.f.: G(0)/2, where G(k) = 1 + 1/( 1 - x*(6-2*x)*(2*k+1)/(x*(6-2*x)*(2*k+1) + (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 17 2013
a(n) = 2^n*hyper2F1(-n, -n, 1, 2). - Peter Luschny, May 20 2015
a(n) = A059474(n,n). - Alois P. Heinz, Oct 05 2017
a(n) ~ 2^(n - 5/4) * (1 + sqrt(2))^(2*n + 1) / sqrt(Pi*n). - Vaclav Kotesovec, Jul 11 2018

A326237 Number of non-nesting digraphs with vertices {1..n}, where two edges (a,b), (c,d) are nesting if a < c and b > d or a > c and b < d.

Original entry on oeis.org

1, 2, 12, 104, 1008, 10272, 107712, 1150592

Offset: 0

Gus Wiseman, Jun 19 2019

These are digraphs with the property that, if the edges are listed in lexicographic order, the sequence of targets is weakly increasing. For example, the digraph with lexicographically ordered edge set {(1,2),(2,1),(3,1),(3,2)} is nesting because the targets are (2,1,1,2), a sequence that is not weakly increasing.
Also the number of non-semicrossing digraphs with vertices {1..n}, where two edges (a,b), (c,d) are semicrossing if a < c and b < d or a > c and b > d. For example, the a(2) = 4 non-semicrossing digraph edge-sets are:
  {}
  {11}
  {12}
  {21}
  {22}
  {11,12}
  {11,21}
  {12,21}
  {12,22}
  {21,22}
  {11,12,21}
  {12,21,22}
Apparently a duplicate of A152254. - R. J. Mathar, Jul 12 2019

			The a(2) = 12 non-nesting digraph edge-sets:
  {}
  {11}
  {12}
  {21}
  {22}
  {11,12}
  {11,21}
  {11,22}
  {12,22}
  {21,22}
  {11,12,22}
  {11,21,22}

Nesting digraphs are A326209.
Non-nesting set partitions are A000108.
Non-capturing set partitions are A054391.
Cf. A002416, A002720, A054391, A058681, A122880, A229865.
Cf. A326249, A326250, A326251, A326256, A326257.

Table[Length[Select[Subsets[Tuples[Range[n],2]],OrderedQ[Last/@#]&]],{n,4}]

A002416(n) = a(n) + A326209(n).

A100631 Triangle read by rows: T(n,k) = 2*(T(n-1,k-1) - T(n-2,k-1) + T(n-1,k)) for 0 < k < n, T(n,0) = T(n,n) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 12, 8, 1, 1, 16, 32, 32, 16, 1, 1, 32, 80, 104, 80, 32, 1, 1, 64, 192, 304, 304, 192, 64, 1, 1, 128, 448, 832, 1008, 832, 448, 128, 1, 1, 256, 1024, 2176, 3072, 3072, 2176, 1024, 256, 1, 1, 512, 2304, 5504, 8832, 10272, 8832, 5504, 2304, 512, 1

Offset: 0

Reinhard Zumkeller, Dec 03 2004

From Petros Hadjicostas, Feb 09 2021: (Start)
The rectangular version (R(n,k): n,k >= 0) of this symmetric triangular array (T(n,k): 0 <= k <= n) is given by R(n,k) = T(n+k,k) for n,k >= 0. Conversely, T(n,k) = R(n-k, k) for 0 <= k <= n.
Note that [o.g.f of R](x,y) = [o.g.f. of T](x, y/x) and [o.g.f of T](x,y) = [o.g.f of R](x,x*y). (End)
From Petros Hadjicostas, Feb 10 2021: (Start)
All the conjectures below are true because one has to prove only one of them, and the rest follow from the proved one.
As Peter Luschny pointed out, one has to show only that the function S(n,k) = 2^n*hypergeom([-k + 1, n], [1], -1) satisfies the recurrence S(n,k) = 2*(S(n,k-1) - S(n-1,k-1) + S(n-1,k)) for n, k > 0 and the initial conditions S(n,0) = S(0,n) = 1 for n >= 0.
This is quite easy to achieve because S(n,k) = 2^n*Sum_{s=0}^{k-1} binomial(k-1,s)*binomial(n+s-1,s) for n >= 0 and k >= 1. The proof of the recurrence relies on the identity binomial(m,n) = binomial(m-1, n) + binomial(m-1,n-1).
Note that without the 2^n in the formula R(n,k) = 2^n*hypergeom([-k + 1, n], [1], -1), we essentially get array A049600.
In addition, note that without the 2^(n-k-1) in the formula T(n,k+1) = 2^(n-k-1)*hypergeom([-k, n-k+1], [1], -1), we essentially get A208341 (without the first column and the main diagonal of T). (End)

			From _Petros Hadjicostas_, Feb 09 2021: (Start)
Triangle T(n,k) (with rows n >= 0 and columns 0 <= k <= n) begins:
  1,
  1,   1,
  1,   2,    1,
  1,   4,    4,    1,
  1,   8,   12,    8,    1,
  1,  16,   32,   32,   16,    1,
  1,  32,   80,  104,   80,   32,    1,
  1,  64,  192,  304,  304,  192,   64,    1,
  1, 128,  448,  832, 1008,  832,  448,  128,   1,
  1, 256, 1024, 2176, 3072, 3072, 2176, 1024, 256, 1,
  ...
Rectangular array R(n,k) (with rows n >= 0 and columns k >= 0) begins:
  1,   1,    1,    1,     1,     1,      1,       1, ...
  1,   2,    4,    8,    16,    32,     64,     128, ...
  1,   4,   12,   32,    80,   192,    448,    1024, ...
  1,   8,   32,  104,   304,   832,   2176,    5504, ...
  1,  16,   80,  304,  1008,  3072,   8832,   24320, ...
  1,  32,  192,  832,  3072, 10272,  32064,   95104, ...
  1,  64,  448, 2176,  8832, 32064, 107712,  341504, ...
  1, 128, 1024, 5504, 24320, 95104, 341504, 1150592, ...
  ... (End)

Index entries for triangles and arrays related to Pascal's triangle

Cf. A000079, A001787, A049600, A084773, A087161, A100312, A152254, A208341, A341344.

T[, 0] = T[n, n_] = 1;
T[n_, k_] /; 0, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 03 2021 *)

From Petros Hadjicostas, Feb 09 2021: (Start)
Formulas for the triangular array (T(n,k): 0 <= k <= n):
T(n,k) = T(n,n-k) for 0 <= k <= n.
Sum_{k=0..n} T(n,k) = A087161(n+1).
T(n,1) = T(n,n-1) = 2^(n-1) = A000079(n-1) for n >= 1.
T(n,2) = T(n,n-2) = (n-1)*2^(n-2) = A001787(n-1) for n >= 2.
T(n,3) = T(n,n-3) = (n^2-n-4)*2^(n-4) = A100312(n-3) for n >= 3.
T(n,floor(n/2)) = T(n,ceiling(n/2)) = A341344(n).
Bivariate o.g.f.: Sum_{n,k >= 0} T(n,k)*x^n*y^k = (3*x^2*y - 2*x*y - 2*x + 1)/((1 - x)*(-x*y + 1)*(2*x^2*y - 2*x*y - 2*x + 1)).
Conjecture based on Peter Luschny's formulas in other sequences: T(n,k) = 2^(n-k)*hypergeom([-k + 1, n-k], [1], -1) = 2^k*hypergeom([-(n-k) + 1, k], [1], -1).
Formulas for the rectangular array (R(n,k): n,k >= 0):
R(n,k) = 2*(R(n,k-1) - R(n-1,k-1) + R(n-1,k)) for n,k > 0 with R(n,0) = R(0,n) = 1 for n >= 0.
R(n,k) = R(k, n) for n,k >= 0.
R(1,n) = R(n,1) = 2^n = A000079(n).
R(2,n) = R(n,2) = (n+1)*2^n = A001787(n+1).
R(3,n) = R(n,3) = (n^2+5*n+2)*2^(n-1) = A100312(n).
R(n,n) = A152254(n-1) = 2*A084773(n-1) for n >= 1.
Bivariate o.g.f.: Sum_{n,k >= 0} R(n,k)*x^n*y^k = (3*x*y - 2*x - 2*y - 1)/((1 - x)*(1 - y)*(2*x*y - 2*x - 2*y - 1)).
Conjecture based on Peter Luschny's formulas in other sequences: R(n,k) = 2^n*hypergeom([-k + 1, n], [1], -1) = 2^k*hypergeom([-n + 1, k], [1], -1). (End)
From Petros Hadjicostas, Feb 10 2021: (Start)
The above conjecture is true (see the comments).
R(n,k) = 2^k*Sum_{s=0}^{n-1} binomial(n-1,s)*binomial(k+s-1,s) = 2^n*Sum_{s=0}^{k-1} binomial(k-1,s)*binomial(n+s-1,s) for n, k >= 1.
To get two binomial formulas for T(n,k), use the equation T(n,k) = R(n-k, k) for 1 <= k <= n and the above formulas for R(n,k). (End)

Offset changed by Petros Hadjicostas, Feb 09 2021

A341344 a(n) = A100631(n, floor(n/2)).

Original entry on oeis.org

1, 1, 2, 4, 12, 32, 104, 304, 1008, 3072, 10272, 32064, 107712, 341504, 1150592, 3688192, 12451584, 40239104, 136053248, 442442752, 1497664512, 4894728192, 16583583744, 54419632128, 184511361024, 607524225024, 2061074178048, 6805625192448, 23100352413696, 76462341095424, 259648659554304

Offset: 0

Petros Hadjicostas, Feb 09 2021

nonn

"Middle" diagonal of Reinhard Zumkeller's symmetric (Pascal-like) triangular array A100631.

Cf. A084773, A100631, A152254.

a(n) = {my(m=matrix(n+1, n+1)); for (i=1, n+1, for (j=1, n+1, if ((j==1) || (j==i), m[i, j] = 1, if (j<=n, m[i,j] = 2*(if (i>1, m[i-1,j-1] + m[i-1,j], 0) - if (i>2, m[i-2,j-1], 0) ))););); m[n+1, (n+2)\2];} \\ Michel Marcus, Feb 10 2021

a(n) = A100631(n, floor(n/2)) = A100631(n, ceiling(n/2)).
a(2*n) = A152254(n-1) = 2*A084773(n-1) for n >= 1.
a(n) = 2^ceiling(n/2)*hypergeom([-floor(n/2) + 1, ceiling(n/2)], [1], -1); see the comments for A100631. - Petros Hadjicostas, Feb 10 2021

cp's OEIS Frontend

A084773 Coefficients of 1/sqrt(1-12*x+4*x^2); also, a(n) is the central coefficient of (1+6*x+8*x^2)^n.

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A326237 Number of non-nesting digraphs with vertices {1..n}, where two edges (a,b), (c,d) are nesting if a < c and b > d or a > c and b < d.

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A100631 Triangle read by rows: T(n,k) = 2*(T(n-1,k-1) - T(n-2,k-1) + T(n-1,k)) for 0 < k < n, T(n,0) = T(n,n) = 1.

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A341344 a(n) = A100631(n, floor(n/2)).

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A084773 Coefficients of 1/sqrt(1-12x+4x^2); also, a(n) is the central coefficient of (1+6x+8x^2)^n.