A059474 -id:A059474 - cp's OEIS frontend

A007070 a(n) = 4a(n-1) - 2a(n-2) with a(0) = 1, a(1) = 4.

1, 4, 14, 48, 164, 560, 1912, 6528, 22288, 76096, 259808, 887040, 3028544, 10340096, 35303296, 120532992, 411525376, 1405035520, 4797091328, 16378294272, 55918994432, 190919389184, 651839567872, 2225519493120, 7598398836736, 25942556360704, 88573427769344, 302408598355968

Offset: 0

N. J. A. Sloane, Mira Bernstein, Simon Plouffe

Joe Keane (jgk(AT)jgk.org) observes that this sequence (beginning at 4) is "size of raises in pot-limit poker, one blind, maximum raising."
It appears that this sequence is the BinomialMean transform of A002315 - see A075271. - John W. Layman, Oct 02 2002
Number of (s(0), s(1), ..., s(2n+3)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+3, s(0) = 1, s(2n+3) = 4. - Herbert Kociemba, Jun 11 2004
a(n) = number of distinct matrix products in (A+B+C+D)^n where commutators [A,B]=[C,D]=0 but neither A nor B commutes with C or D. - Paul D. Hanna and Joshua Zucker, Feb 01 2006
The n-th term of the sequence is the entry (1,2) in the n-th power of the matrix M=[1,-1;-1,3]. - Simone Severini, Feb 15 2006
Hankel transform of this sequence is [1,-2,0,0,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007
A204089 convolved with A000225, e.g., a(4) = 164 = (1*31 + 1*15 + 4*7 + 14*3 + 48*1) = (31 + 15 + 28 + 42 + 48). - Gary W. Adamson, Dec 23 2008
Equals INVERT transform of A000225: (1, 3, 7, 15, 31, ...). - Gary W. Adamson, May 03 2009
For n>=1, a(n-1) is the number of generalized compositions of n when there are 2^i-1 different types of the part i, (i=1,2,...). - Milan Janjic, Sep 24 2010
Binomial transform of A078057. - R. J. Mathar, Mar 28 2011
Pisano period lengths:  1, 1, 8, 1, 24, 8, 6, 1, 24, 24, 120, 8, 168, 6, 24, 1, 8, 24, 360, 24, ... . - R. J. Mathar, Aug 10 2012
a(n) is the diagonal of array A228405. - Richard R. Forberg, Sep 02 2013
From Wolfdieter Lang, Oct 01 2013: (Start)
a(n) appears together with A106731, both interspersed with zeros, in the representation of nonnegative powers of the algebraic number rho(8) = 2*cos(Pi/8) = A179260 of degree 4, which is the length ratio of the smallest diagonal and the side in the regular octagon.
The minimal polynomial for rho(8) is C(8,x) = x^4 - 4*x^2 + 2, hence rho(8)^n = A(n+1)*1 + A(n)*rho(8) + B(n+1)*rho(8)^2 + B(n)*rho(8)^3, n >= 0, with A(2*k) = 0, k >= 0, A(1) = 1, A(2*k+1) = A106731(k-1), k >= 1, and  B(2*k) = 0, k >= 0, B(1) = 0, B(2*k+1) = a(k-1), k >= 1. See also the P. Steinbach reference given under A049310. (End)
The ratio a(n)/A006012(n) converges to 1+sqrt(2). - Karl V. Keller, Jr., May 16 2015
From Tom Copeland, Dec 04 2015: (Start)
An aerated version of this sequence is given by the o.g.f. = 1 / (1 - 4 x^2 + 2 x^4) =  1 / [x^4 a_4(1/x)] = 1 / determinant(I - x M) = exp[-log(1 -4 x + 2 x^4)], where M is the adjacency matrix for the simple Lie algebra B_4 given in A265185 with the characteristic polynomial a_4(x) = x^4 - 4 x^2 + 2 = 2 T_4(x/2) = A127672(4,x), where T denotes a Chebyshev polynomial of the first kind.
A133314 relates a(n) to the reciprocal of the e.g.f. 1 - 4 x + 4 x^2/2!. (End)
a(n) is the number of vertices of the Minkowski sum of n simplices with vertices e_(2*i+1), e_(2*i+2), e_(2*i+3), e_(2*i+4) for i=0,...,n-1, where e_i is a standard basis vector. - Alejandro H. Morales, Oct 03 2022

			a(3) = 48 = 3 * 4 + 4 + 1 + 1 = 3*a(2) + a(1) + a(0) + 1.
Example for the octagon rho(8) powers: rho(8)^4  = 2 + sqrt(2) = -2*1 + 4*rho(8)^2  = A(5)*1 + A(4)*rho(8) + B(5)*rho(8)^2 + B(4)*rho(8)^3, with a(5) = A106731(1) = -2, B(5) = a(1) = 4, A(4) = 0, B(4) = 0. - _Wolfdieter Lang_, Oct 01 2013

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), Article 12.7.8.
A. Bernini, F. Disanto, R. Pinzani, and S. Rinaldi, Permutations defining convex permutominoes, J. Int. Seq. 10 (2007) # 07.9.7.
A. Burstein, S. Kitaev, and T. Mansour, Independent sets in certain classes of (almost) regular graphs, arXiv:math/0310379 [math.CO], 2003.
Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607. (See Cor. 3.7(e).)
Tomislav Doslic and I. Zubac, Counting maximal matchings in linear polymers, Ars Mathematica Contemporanea 11 (2016) 255-276.
G. Dresden and Y. Li, Periodic Weighted Sums of Binomial Coefficients, arXiv:2210.04322 [math.NT], 2022.
L. Escobar, P. Gallardo, J. González-Anaya, J. L. González, G. Montúfar, and A. H. Morales, Enumeration of max-pooling responses with generalized permutohedra, arXiv:2209.14978 [math.CO], 2022. (See Ex. 5.1)
S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
Pamela Fleischmann, Jonas Höfer, Annika Huch, and Dirk Nowotka, alpha-beta-Factorization and the Binary Case of Simon's Congruence, arXiv:2306.14192 [math.CO], 2023.
A. S. Fraenkel and C. Kimberling, Generalized Wythoff arrays, shuffles and interspersions, Discr. Math. 126 (1-3) (1994) 137-149.
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 8.
Sela Fried, Toufik Mansour, and Mark Shattuck, Counting k-ary words by number of adjacency differences of a prescribed size, arXiv:2504.03013 [math.CO], 2025. See p. 7.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 440
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.
Index entries for sequences related to poker
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Row sums of A059474. - David W. Wilson, Aug 14 2006
Cf. A000129, A000225, A002307, A002315, A006012 (same recurrence), A007052, A075271, A078057, A106731, A179260, A204089, A228405.
Equals 2 * A003480, n>0.
Row sums of A140071.
Cf. A127672, A265185, A133314.

a007070 n = a007070_list !! n
a007070_list = 1 : 4 : (map (* 2) $ zipWith (-)
   (tail $ map (* 2) a007070_list) a007070_list)
-- Reinhard Zumkeller, Jan 16 2012

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-8); S:=[ ((4+r)^(1+n)-(4-r)^(1+n))/((2^(1+n))*r): n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Vincenzo Librandi, Mar 27 2011

[n le 2 select 3*n-2 else 4*Self(n-1)-2*Self(n-2): n in [1..23]];  // Bruno Berselli, Mar 28 2011

A007070 :=proc(n) option remember; if n=0 then 1 elif n=1 then 4 else 4*procname(n-1)-2*procname(n-2); fi; end:
seq(A007070(n), n=0..30); # Wesley Ivan Hurt, Dec 06 2015

LinearRecurrence[{4,-2}, {1,4}, 30] (* Harvey P. Dale, Sep 16 2014 *)

a(n)=polcoeff(1/(1-4*x+2*x^2)+x*O(x^n),n)

a(n)=if(n<1,1,ceil((2+sqrt(2))*a(n-1)))

[lucas_number1(n,4,2) for n in range(1, 24)]# Zerinvary Lajos, Apr 22 2009

G.f.: 1/(1 - 4*x + 2*x^2).
Preceded by 0, this is the binomial transform of the Pell numbers A000129. Its e.g.f. is then exp(2*x)*sinh(sqrt(2)*x)/sqrt(2). - Paul Barry, May 09 2003
a(n) = ((2+sqrt(2))^(n+1) - (2-sqrt(2))^(n+1))/sqrt(8). - Al Hakanson (hawkuu(AT)gmail.com), Dec 27 2008, corrected Mar 28 2011
a(n) = (2 - sqrt(2))^n*(1/2 - sqrt(2)/2) + (2 + sqrt(2))^n*(1/2 + sqrt(2)/2). - Paul Barry, May 09 2003
a(n) = ceiling((2 + sqrt(2))*a(n-1)). - Benoit Cloitre, Aug 15 2003
a(n) = U(n, sqrt(2))*sqrt(2)^n. - Paul Barry, Nov 19 2003
a(n) = (1/4)*Sum_{r=1..7} sin(r*Pi/8)*sin(r*Pi/2)*(2*cos(r*Pi/8))^(2*n+3). - Herbert Kociemba, Jun 11 2004
a(n) = center term in M^n * [1 1 1], where M = the 3 X 3 matrix [1 1 1 / 1 2 1 / 1 1 1]. M^n * [1 1 1] = [A007052(n) a(n) A007052(n)]. E.g., a(3) = 48 since M^3 * [1 1 1] = [34 48 34], where 34 = A007052(3). - Gary W. Adamson, Dec 18 2004
This is the binomial mean transform of A002307. See Spivey and Steil (2006). - Michael Z. Spivey (mspivey(AT)ups.edu), Feb 26 2006
a(2n) = Sum_{r=0..n} 2^(2n-1-r)*(4*binomial(2n-1,2r) + 3*binomial(2n-1,2r+1)) a(2n-1) = Sum_{r=0..n} 2^(2n-2-r)*(4*binomial(2n-2,2r) + 3*binomial(2n-2,2r+1)). - Jeffrey Liese, Oct 12 2006
a(n) = 3*a(n - 1) + a(n - 2) + a(n - 3) + ... + a(0) + 1. - Gary W. Adamson, Feb 18 2011
G.f.: 1/(1 - 4*x + 2*x^2) = 1/( x*(1 + U(0)) ) - 1/x  where U(k)= 1 - 2^k/(1 - x/(x - 2^k/U(k+1) )); (continued fraction 3rd kind, 3-step). - Sergei N. Gladkovskii, Dec 05 2012
G.f.: A(x) = G(0)/(1-2*x) where G(k) = 1 + 2*x/(1 - 2*x - x*(1-2*x)/(x + (1-2*x)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 04 2013
G.f.: G(0)/(2*x) - 1/x, where G(k) = 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - (1-x)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
a(n-1) = Sum_{k=0..n} binomial(2*n, n+k)*(k|8) where (k|8) is the Kronecker symbol. - Greg Dresden, Oct 11 2022
E.g.f.: exp(2*x)*(cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x)). - Stefano Spezia, May 20 2024

A059576 Summatory Pascal triangle T(n,k) (0 <= k <= n) read by rows. Top entry is 1. Each entry is the sum of the parallelogram above it.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 4, 8, 8, 4, 8, 20, 26, 20, 8, 16, 48, 76, 76, 48, 16, 32, 112, 208, 252, 208, 112, 32, 64, 256, 544, 768, 768, 544, 256, 64, 128, 576, 1376, 2208, 2568, 2208, 1376, 576, 128, 256, 1280, 3392, 6080, 8016, 8016, 6080, 3392, 1280, 256

Offset: 0

Floor van Lamoen, Jan 23 2001

We may also relabel the entries as U(0,0), U(1,0), U(0,1), U(2,0), U(1,1), U(0,2), U(3,0), ... [That is, T(n,k) = U(n-k, k) for 0 <= k <= n and U(m,s) = T(m+s, s) for m,s >= 0.]
From Petros Hadjicostas, Jul 16 2020: (Start)
We explain the parallelogram definition of T(n,k).
    T(0,0) *
           |\
           | \
           |  * T(k,k)
  T(n-k,0) *  |
            \ |
             \|
              * T(n,k)
The definition implies that T(n,k) is the sum of all T(i,j) such that (i,j) has integer coordinates over the set
{(i,j): a(1,0) + b(1,1), 0 <= a <= n-k, 0 <= b <= k} - {(n,k)}.
The parallelogram can sometimes be degenerate; e.g., when k = 0 or n = k. (End)
T(n,k) is the number of 2-compositions of n having sum of the entries of the first row equal to k (0 <= k <= n). A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n. - Emeric Deutsch, Oct 12 2010
From Michel Marcus and Petros Hadjicostas, Jul 16 2020: (Start)
Robeva and Sun (2020) let A(m,n) = U(m-1, n-1) be the number of subdivisions of a 2-row grid with m points on the top and n points at the bottom (and such that the lower left point is the origin).
The authors proved that A(m,n) = 2*(A(m,n-1) + A(m-1,n) - A(m-1,n-1)) for m, n >= 2 (with (m,n) <> (2,2)), which is equivalent to a similar recurrence for U(n,k) given in the Formula section below. (They did not explicitly specify the value of A(1,1) = U(0,0) because they did not care about the number of subdivisions of a degenerate polygon with only one side.)
They also proved that, for (m,n) <> (1,1), A(m,n) = (2^(m-2)/(n-1)!) * Q_n(m) =
= (2^(m-2)/(n-1)!) * Sum_{k=1..n} A336244(n,k) * m^(n-k), where Q_n(m) is a polynomial in m of degree n-1. (End)
With the square array notation of Petros Hadjicostas, Jul 16 2020 below, U(i,j) is the number of lattice paths from (0,0) to (i,j) whose steps move north or east or have positive slope. For example, representing a path by its successive lattice points rather than its steps, U(1,2) = 8 counts {(0,0),(1,2)}, {(0,0),(0,1),(1,2)}, {(0,0),(0,2),(1,2)}, {(0,0),(1,0),(1,2)}, {(0,0),(1,1),(1,2)}, {(0,0),(0,1),(0,2),(1,2)}, {(0,0),(0,1),(1,1),(1,2)}, {(0,0),(1,0),(1,1),(1,2)}. If north (vertical) steps are excluded, the resulting paths are counted by A049600.  - David Callan, Nov 25 2021

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins
[0]   1;
[1]   1,   1;
[2]   2,   3,   2;
[3]   4,   8,   8,   4;
[4]   8,  20,  26,  20,   8;
[5]  16,  48,  76,  76,  48,  16;
[6]  32, 112, 208, 252, 208, 112, 32;
  ...
T(5,2) = 76 is the sum of the elements above it in the parallelogram bordered by T(0,0), T(5-2,0) = T(3,0), T(2,2) and T(5,2). We of course exclude T(5,2) from the summation. Thus
T(5,2) = Sum_{a=0..5-2, b=0..2, (a,b) <> (5-2,2)} T(a(1,0) + b(1,1)) =
= (1 + 1 + 2) + (1 + 3 + 8) + (2 + 8 + 26) + (4 + 20) = 76. [Edited by _Petros Hadjicostas_, Jul 16 2020]
From _Petros Hadjicostas_, Jul 16 2020: (Start)
Square array U(n,k) (with rows n >= 0 and columns k >= 0) begins
   1,   1,   2,    4,    8, ...
   1,   3,   8,   20,   48, ...
   2,   8,  26,   76,  208, ...
   4,  20,  76,  252,  768, ...
   8,  48, 208,  768, 2568, ...
  16, 112, 544, 2208, 8016, ...
  ...
Consider the following 2-row grid with n = 3 points at the top and k = 2 points at the bottom:
   A  B  C
   *--*--*
   |    /
   |   /
   *--*
   D  E
The sets of the dividing internal lines of the A(3,2) = U(3-1, 2-1) = 8 subdivisions of the above 2-row grid are as follows: { }, {DC}, {DB}, {EB}, {EA}, {DB, DC}, {DB, EB}, and {EA, EB}. See Robeva and Sun (2020).
These are the 2-compositions of n = 3 with sum of first row entries equal to k = 1:
[1; 2], [0,1; 2,0], [0,1; 1,1], [1,0; 0,2], [1,0; 1,1], [0,0,1; 1,1,0], [0,1,0; 1,0,1], and [1,0,0; 0,1,1]. We have T(3,2) = 8 such matrices. See _Emeric Deutsch_'s contribution above. See also Section 2 in Castiglione et al. (2007). (End)

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European Journal of Combinatorics, 28(6) (2007), 1724-1741; see Fig. 5, p. 1729.
Fang, E., Jenkins, J., Lee, Z., Li, D., Lu, E., Miller, S. J., ... & Siktar, J. (2019). Central Limit Theorems for Compound Paths on the 2-Dimensional Lattice, arXiv preprint arXiv:1906.10645. Also Fib. Q., 58:1 (2020), 208-225.
Elina Robeva and Melinda Sun, Bimonotone Subdivisions of Point Configurations in the Plane, arXiv:2007.00877 [math.CO], 2020.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A000079, A001792, A003480 (row sums), A052141 (main diagonal).
Cf. A059474, A059473, A008288, A035002, A059226, A059283, A181292, A336244.
Cf. A011782, A049611, A049612, A055589, A055852, A055853, A181306.

a059576 n k = a059576_tabl !! n !! k
a059576_row n = a059576_tabl !! n
a059576_tabl = [1] : map fst (iterate f ([1,1], [2,3,2])) where
   f (us, vs) = (vs, map (* 2) ws) where
     ws = zipWith (-) (zipWith (+) ([0] ++ vs) (vs ++ [0]))
                      ([0] ++ us ++ [0])
-- Reinhard Zumkeller, Dec 03 2012

A011782:= func< n | n eq 0 select 1 else 2^(n-1) >;
function T(n,k) // T = A059576
  if k eq 0 or k eq n then return A011782(n);
  else return 2*T(n-1, k-1) + 2*T(n-1, k) - (2 - 0^(n-2))*T(n-2, k-1);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 02 2022

A059576 := proc(n,k) local b,t1; t1 := min(n+k-2,n,k); add( (-1)^b * 2^(n+k-b-2) * (n+k-b-2)! * (1/(b! * (n-b)! * (k-b)!)) * (-2 * n-2 * k+2 * k^2+b^2-3 * k * b+2 * n^2+5 * n * k-3 * n * b), b=0..t1); end;
T := proc (n, k) if k <= n then add((-1)^j*2^(n-j-1)*binomial(k, j)*binomial(n-j, k), j = 0 .. min(k, n-k)) fi end proc: 1; for n to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form # Emeric Deutsch, Oct 12 2010
T := (n, k) -> `if`(n=0, 1, 2^(n-1)*binomial(n, k)*hypergeom([-k, k - n], [-n], 1/2)): seq(seq(simplify(T(n, k)), k=0..n), n=0..10); # Peter Luschny, Nov 26 2021

T[0, 0] = 1; T[n_, k_] := 2^(n-k-1)*n!*Hypergeometric2F1[ -k, -k, -n, -1 ] / (k!*(n-k)!); Flatten[ Table[ T[n, k], {n, 0, 9}, {k, 0, n}]] (* Jean-François Alcover, Feb 01 2012, after Robert Israel *)

def T(n,k): # T = A059576
    if (k==0 or k==n): return 1 if (n==0) else 2^(n-1) # A011782
    else: return 2*T(n-1, k-1) + 2*T(n-1, k) - (2 - 0^(n-2))*T(n-2, k-1)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 02 2022

T(n, n-1) = A001792(n-1).
T(2*n, n) = A052141(n).
Sum_{k=0..n} T(n, k) = A003480(n).
G.f.: U(z, w) = Sum_{n >= 0, k >= 0} U(n, k)*z^n*w^k = Sum{n >= 0, k >= 0} T(n, k)*z^(n-k)*w^k = (1-z)*(1-w)/(1 - 2*w - 2*z + 2*z*w).
Maple code gives another explicit formula for U(n, k).
From Jon Stadler (jstadler(AT)capital.edu), Apr 30 2003: (Start)
U(n,k) is the number of ways of writing the vector (n,k) as an ordered sum of vectors, equivalently, the number of paths from (0,0) to (n,k) in which steps may be taken from (i,j) to (p,q) provided (p,q) is to the right or above (i,j).
2*U(n,k) = Sum_{i <= n, j <= k} U(i,j).
U(n,k) = 2*U(n-1,k) + Sum_{i < k} U(n,i).
U(n,k) = Sum_{j=0..n+k} C(n,j-k+1)*C(k,j-n+1)*2^j. (End)
T(n, k) = 2*(T(n-1, k-1) + T(n-1, k)) - (2 - 0^(n-2))*T(n-2, k-1) for n > 1 and 1 < k < n; T(n, 0) = T(n, n) = 2*T(n-1, 0) for n > 0; and T(0, 0) = 1. - Reinhard Zumkeller, Dec 03 2004
From Emeric Deutsch, Oct 12 2010: (Start)
Sum_{k=0..n} k*T(n,k) = A181292(n).
T(n,k) = Sum_{j=0..min(k, n-k)} (-1)^j*2^(n-j-1)*binomial(k, j)*binomial(n-j, k) for (n,k) != (0,0).
G.f.: G(t,z) = (1-z)*(1-t*z)/(1 - 2*z - 2*t*z + 2*t*z^2). (End)
U(n,k) = 0 if k < 0; else U(k,n) if k > n; else 1 if n <= 1; else 3 if n = 2 and k = 1; else 2*U(n,k-1) + 2*U(n-1,k) - 2*U(n-1,k-1). - David W. Wilson; corrected in the case k > n by Robert Israel, Jun 15 2011 [Corrected by Petros Hadjicostas, Jul 16 2020]
U(n,k) = binomial(n,k) * 2^(n-1) * hypergeom([-k,-k], [n+1-k], 2) if n >= k >= 0 with (n,k) <> (0,0). - Robert Israel, Jun 15 2011 [Corrected by Petros Hadjicostas, Jul 16 2020]
U(n,k) = Sum_{0 <= i+j <= n+k-1} (-1)^j*C(i+j+1, j)*C(n+i, n)*C(k+i, k). - Masato Maruoka, Dec 10 2019
T(n, k) = 2^(n - 1)*binomial(n, k)*hypergeom([-k, k - n], [-n], 1/2) = A059474(n, k)/2 for n >= 1. - Peter Luschny, Nov 26 2021
From G. C. Greubel, Sep 02 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 0) = T(n, n) = A011782(n).
T(n, n-2) =  2*A049611(n-1), n >= 2.
T(n, n-3) =  4*A049612(n-2), n >= 3.
T(n, n-4) =  8*A055589(n-3), n >= 4.
T(n, n-5) = 16*A055852(n-4), n >= 5.
T(n, n-6) = 32*A055853(n-5), n >= 6.
Sum_{k=0..floor(n/2)} T(n, k) = A181306(n). (End)

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

Original entry on oeis.org

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

cp's OEIS Frontend

A007070 a(n) = 4*a(n-1) - 2*a(n-2) with a(0) = 1, a(1) = 4.

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A059576 Summatory Pascal triangle T(n,k) (0 <= k <= n) read by rows. Top entry is 1. Each entry is the sum of the parallelogram above it.

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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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A007070 a(n) = 4a(n-1) - 2a(n-2) with a(0) = 1, a(1) = 4.

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