A054335 -id:A054335 - cp's OEIS frontend

A026671 Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when on the diagonal, (1,1).

1, 3, 11, 43, 173, 707, 2917, 12111, 50503, 211263, 885831, 3720995, 15652239, 65913927, 277822147, 1171853635, 4945846997, 20884526283, 88224662549, 372827899079, 1576001732485, 6663706588179, 28181895551161, 119208323665543, 504329070986033, 2133944799315027

Offset: 0

Clark Kimberling, Miklos Bona

1, 1, 3, 11, 43, 173, ... is the unique sequence for which both the Hankel transform of the sequence itself and the Hankel transform of its left shift are the powers of 2 (A000079). For example, det[{{1, 1, 3}, {1, 3, 11}, {3, 11, 43}}] = det[{{1, 3, 11}, {3, 11, 43}, {11, 43, 173}}] = 4. - David Callan, Mar 30 2007
From Paul Barry, Jan 25 2009: (Start)
a(n) is the image of F(2n+2) under the Catalan matrix (1,xc(x)) where c(x) is the g.f. of A000108.
The sequence 1,1,3,... is the image of A001519 under (1,xc(x)). This sequence has g.f. given by 1/(1-x-2x^2/(1-3x-x^2/(1-2x-x^2/(1-2x-x^2/(1-... (continued fraction). (End)
Binomial transform of A111961. - Philippe Deléham, Feb 11 2009
From Paul Barry, Nov 03 2010: (Start)
The sequence 1,1,3,... has g.f. 1/(1-x/sqrt(1-4x)), INVERT transform of A000984.
It is an eigensequence of the sequence array for A000984. (End)

L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Jean-Christophe Aval, Adrien Boussicault and Sandrine Dasse-Hartaut, The tree structure in staircase tableaux, arXiv:1109.4907 [math.CO], 2011-2013.
Cyril Banderier, Markus Kuba, and Michael Wallner, Analytic Combinatorics of Composition schemes and phase transitions with mixed Poisson distributions, arXiv:2103.03751 [math.PR], 2021.
Miklos Bona, The permutation classes equinumerous to the smooth class, Electron. J. Combin., 5 (1998), no. 1, Research Paper 31, 12 pp.
David Callan and Toufik Mansour, Five subsets of permutations enumerated as weak sorting permutations, arXiv:1602.05182 [math.CO], 2016.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Huyile Liang, Jeffrey Remmel and Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see page 16.

a(n) = T(2n-1, n-1), T given by A026736.
a(n) = T(2n, n), T given by A026670.
a(n) = T(2n+1, n+1), T given by A026725.
Row sums of triangle A054335.
Cf. A026781, A001076, A176280.

a:=[3,11,43];; for n in [4..30] do a[n]:=(2*(4*n-3)*a[n-1] - 3*(5*n-8)*a[n-2] - 2*(2*n-3)*a[n-3])/n; od; Concatenation([1], a); # G. C. Greubel, Jul 16 2019

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/(Sqrt(1-4*x)-x) )); // G. C. Greubel, Jul 16 2019

Table[SeriesCoefficient[1/(Sqrt[1-4*x]-x),{x,0,n}],{n,0,30}] (* Vaclav Kotesovec, Oct 08 2012 *)

{a(n)= if(n<0, 0, polcoeff( 1/(sqrt(1 -4*x +x*O(x^n)) -x), n))} /* Michael Somos, Apr 20 2007 */

my(x='x+O('x^66)); Vec( 1/(sqrt(1-4*x)-x) ) \\ Joerg Arndt, May 04 2013

(1/(sqrt(1-4*x)-x)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 16 2019

From Wolfdieter Lang, Mar 21 2000: (Start)
G.f.: 1/(sqrt(1-4*x)-x).
a(n) = Sum_{i=1..n} a(i-1)*binomial(2*(n-i), n-i) + binomial(2*n, n), n >= 1, a(0)=1. (End)
G.f.: 1/(1 -x -2*x*c(x)) where c(x) = g.f. for Catalan numbers A000108. - Michael Somos, Apr 20 2007
From Paul Barry, Jan 25 2009: (Start)
G.f.: 1/(1 - 3xc(x) + x^2*c(x)^2);
G.f.: 1/(1-3x-2x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-... (continued fraction).
a(0) = 1, a(n) = Sum_{k=0..n} (k/(2n-k))*C(2n-k,n-k)*F(2k+2). (End)
a(n) = Sum_{k=0..n} A039599(n,k) * A000045(k+2). - Philippe Deléham, Feb 11 2009
From Paul Barry, Feb 08 2009: (Start)
G.f.: 1/(1-x/(1-2x/(1-x/(1-x/(1-x/(1-x/(1-x/(1-... (continued fraction);
G.f. of 1,1,3,... is 1/(1-x-2x/(1-x/(1-x/(1-x/(1-... (continued fraction). (End)
From Gary W. Adamson, Jul 14 2011: (Start)
a(n) = the upper left term in M^n, M = the infinite square production matrix:
  3, 2, 0, 0, 0, 0, ...
  1, 1, 1, 0, 0, 0, ...
  1, 1, 1, 1, 0, 0, ...
  1, 1, 1, 1, 1, 0, ...
  1, 1, 1, 1, 1, 1, ...
  ... (End)
From Vaclav Kotesovec, Oct 08 2012: (Start)
D-finite with recurrence: n*a(n) = 2*(4*n-3)*a(n-1) - 3*(5*n-8)*a(n-2) - 2*(2*n-3)*a(n-3).
a(n) ~ (2+sqrt(5))^n/sqrt(5). (End)
a(n) = Sum_{k=0..n+1} 4^(n+1-k) * binomial(n-k/2,n+1-k). - Seiichi Manyama, Mar 30 2025
From Peter Luschny, Mar 30 2025: (Start)
a(n) = 4^n*(binomial(n-1/2, n)*hypergeom([1, (1-n)/2, -n/2], [1/2, 1/2-n], -1/4) + hypergeom([(1-n)/2, 1-n/2], [1-n], -1/4)/4) for n > 0.
a(n) = A001076(n) + A176280(n). (End)

A098615 G.f. A(x) satisfies: A(xG(x)) = G(x), where G(x) is the g.f. for A098614(n) = Fibonacci(n+1)Catalan(n).

Original entry on oeis.org

1, 1, 3, 5, 13, 25, 61, 125, 295, 625, 1447, 3125, 7151, 15625, 35491, 78125, 176597, 390625, 880125, 1953125, 4390901, 9765625, 21920913, 48828125, 109486993, 244140625, 547018941, 1220703125, 2733608905, 6103515625, 13662695645, 30517578125, 68294088535, 152587890625, 341399727335, 762939453125, 1706739347095, 3814697265625, 8532741458075, 19073486328125, 42660172763995, 95367431640625

Offset: 0

Paul D. Hanna, Oct 14 2004

nonn

G.f. satisfies: A(x) = x/(series reversion of x*G098614(x)), where G098614 is the g.f. for A098614 = {1*1, 1*1, 2*2, 3*5, 5*14, 8*42, 13*132, ...}.
Hankel transform is 2^n. Image of F(n+1) under the Riordan array (c(x^2),xc(x^2)), c(x) the g.f. of A000108. The sequence 0,1,1,3,5,... has general term Sum_{k=0..floor(n/2)} (C(n-1,k) - C(n-1,k-1))*F(n-2k). It is the image of the Fibonacci numbers under the transform of generating functions g(x)-> g(xc(x^2)), c(x) the g.f. of A000108. This sequence has Hankel transform -(-4)^((n-1)/2)(1-(-1)^n)/2. - Paul Barry, Oct 01 2007
The sequence of fractions 1, 1/2, 3/4, 5/8, 13/16, 25/32, ... or a(n)/2^n is the image of F(n+1) under the Chebyshev related (rational) Riordan array c((x/2)^2),(x/2)c((x/2)^2)) where c(x) is the g.f. of A000108. The Hankel transform of this fraction sequence is 1/(2^(n^2)). - Paul Barry, Jun 17 2008

Paul D. Hanna, Table of n, a(n) for n = 0..300
Paul Barry and A. Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, J. Int. Seq. 13 (2010) # 10.9.4, example 30.
Cyril Banderier, Markus Kuba, and Michael Wallner, Analytic Combinatorics of Composition schemes and phase transitions with mixed Poisson distributions, arXiv:2103.03751 [math.PR], 2021.

Cf. A000045, A000108, A046748, A053121, A054335, A098614.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (x+Sqrt(1-4*x^2))/(1-5*x^2) )); // G. C. Greubel, Jul 31 2024

Array[Sum[Binomial[(# - 1)/2, (# - k)/2]*2^(# - k - 1)*((-1)^(# - k) + 1), {k, 0, #}] &, 42, 0] (* or *)
CoefficientList[Series[(Sqrt[1 - 4 x^2] + x)/(1 - 5 x^2), {x, 0, 41}], x] (* Michael De Vlieger, May 20 2021 *)

a(n):=sum(binomial((n-1)/2,(n-k)/2)*2^(n-k-1)*((-1)^(n-k)+1),k,0,n); /* Vladimir Kruchinin, Apr 16 2011 */

{ a(n) = polcoeff((sqrt(1-4*x^2+x^2*O(x^n))+x)/(1-5*x^2),n) }
for(n=0,50,print1(a(n),", "))

def A098615_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (x+sqrt(1-4*x^2))/(1-5*x^2) ).list()
A098615_list(30) # G. C. Greubel, Jul 31 2024

G.f.: (x + sqrt(1-4*x^2)) / (1-5*x^2).
G.f. satisfies: A(x) = sqrt(1 + 2*x*A(x) + 5*x^2*A(x)^2). - Paul D. Hanna, Nov 18 2014
a(2*n) = A046748(n).
a(2*n+1) = 5^n.
a(n) = Sum_{k=0..floor((n+1)/2)} (C(n,k) - C(n,k-1))*Fibonacci(n-2k+1). - Paul Barry, Oct 01 2007
G.f.: 1/(1-x-2x^2/(1-x^2/(1-x^2/(1-x^2/(1-x^2/(1-.... (continued fraction). - Paul Barry, Feb 09 2009
a(n) = Sum_{k=0..n} binomial((n-1)/2,(n-k)/2)*2^(n-k-1)*(1+(-1)^(n-k)). - Vladimir Kruchinin, Apr 16 2011
From Gary W. Adamson, Sep 22 2011: (Start)
a(n) is the upper left term in M^n, M = an infinite square production matrix as follows:
   1, 1, 1, 0, 0, 0, ...
   1, 0, 0, 1, 0, 0, ...
   1, 0, 0, 0, 1, 0, ...
   0, 1, 0, 0, 0, 1, ...
   0, 0, 1, 0, 0, 0, ...
   0, 0, 0, 1, 0, 0, ...
   0, 0, 0, 0, 1, 0, ...
   0, 0, 0, 0, 0, 1, ...
   ... (End)
a(n) = Sum_{k=0..floor(n/2)} A054335(n-k,n-2k). - Philippe Deléham, Feb 01 2012
a(n) = Sum_{k=0..n} A053121(n,k)*A000045(k+1). - Philippe Deléham, Feb 03 2012
n*a(n) +(n-1)*a(n-1) -3*(3*n-4)*a(n-2) -3*(3*n-7)*a(n-3) +20*(n-3)*a(n-4) +20*(n-4)*a(n-5) = 0. - R. J. Mathar, Jul 21 2017

A054336 A convolution triangle of numbers based on A001405 (central binomial coefficients).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 6, 10, 9, 4, 1, 10, 22, 22, 14, 5, 1, 20, 44, 54, 40, 20, 6, 1, 35, 93, 123, 109, 65, 27, 7, 1, 70, 186, 281, 276, 195, 98, 35, 8, 1, 126, 386, 618, 682, 541, 321, 140, 44, 9, 1, 252, 772, 1362, 1624, 1440, 966, 497, 192, 54, 10, 1

Offset: 0

Wolfdieter Lang, Mar 13 2000

T(n,k) is the number of 2-Motzkin paths (i.e., Motzkin paths with blue and red level steps) with no level steps at positive height and having k blue level steps. Example: T(4,2)=9 because, denoting U=(1,1), D=(1,-1), B=blue (1,0), R=red (1,0), we have BBRR, BRBR, BRRB, RBBR, RBRB, RRBB, BBUD, BUDB, and UDBB. - Emeric Deutsch, Jun 07 2011
In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.
The g.f. for the row polynomials p(n,x) (increasing powers of x) is 1/(1-(1+x)*z-z^2*c(z^2)), with c(x) the g.f. for Catalan numbers A000108.
Column sequences: A001405, A045621.
Riordan array (f(x), x*f(x)), f(x) the g.f. of A001405. - Philippe Deléham, Dec 08 2009
From Paul Barry, Oct 21 2010: (Start)
Riordan array ((sqrt(1+2x) - sqrt(1-2x))/(2x*sqrt(1-2x)), (sqrt(1+2x)-sqrt(1-2x))/(2*sqrt(1-2x))),
inverse of Riordan array ((1+x)/(1+2x+2x^2), x(1+x)/(1+2x+2x^2)) (A181472). (End)

			Fourth row polynomial (n=3): p(3,x)= 3 + 5*x + 3*x^2 + x^3.
From _Paul Barry_, Oct 21 2010: (Start)
Triangle begins
   1;
   1,  1;
   2,  2,   1;
   3,  5,   3,   1;
   6, 10,   9,   4,  1;
  10, 22,  22,  14,  5,  1;
  20, 44,  54,  40, 20,  6, 1;
  35, 93, 123, 109, 65, 27, 7, 1;
Production matrix is
   1,  1;
   1,  1,  1;
  -1,  1,  1,  1;
   1, -1,  1,  1,  1;
  -1,  1, -1,  1,  1,  1;
   1, -1,  1, -1,  1,  1,  1;
  -1,  1, -1,  1, -1,  1,  1, 1;
   1, -1,  1, -1,  1, -1,  1, 1, 1;
  -1,  1, -1,  1, -1,  1, -1, 1, 1, 1; (End)

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A001405, A035324, A054335.

Row sums: A054341.

A053121:= function(n,k)
    if ((n-k+1) mod 2)=0 then return 0;
    else return (k+1)*Binomial(n+1, Int((n-k)/2))/(n+1);
    fi;
  end;
T:= function(n,k)
    return Sum([k..n], j-> Binomial(j,k)*A053121(n,j));
  end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Jul 21 2019

A053121:= func< n,k | ((n-k+1) mod 2) eq 0 select 0 else (k+1)*Binomial(n+1, Floor((n-k)/2))/(n+1) >;
T:= func< n,k | (&+[Binomial(j,k)*A053121(n,j): j in [k..n]]) >;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jul 21 2019

c[n_, j_] /; n < j || OddQ[n - j] = 0; c[n_, j_] = (j + 1) Binomial[n + 1, (n - j)/2]/(n + 1); t[n_, k_] := Sum[c[n, j]*Binomial[j, k], {j, 0, n}]; Flatten[Table[t[n, k], {n, 0, 10}, {k, 0, n}]][[;; 66]] (* Jean-François Alcover, Jul 13 2011, after Philippe Deléham *)

A053121(n,k) = if((n-k+1)%2==0, 0, (k+1)*binomial(n+1, (n-k)\2)/(n+1) );
T(n,k) = sum(j=k,n, A053121(n,j)*binomial(j,k));
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jul 21 2019

def A053121(n, k):
    if (n-k+1) % 2==0: return 0
    else: return (k+1)*binomial(n+1, ((n-k)//2))/(n+1)
def T(n,k): return sum(binomial(j,k)*A053121(n,j) for j in (k..n))
[[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jul 21 2019

G.f. for column m: cbi(x)*(x*cbi(x))^m, with cbi(x) := (1+x*c(x^2))/sqrt(1-4*x^2) = 1/(1-x-x^2*c(x^2)), where c(x) is the g.f. for Catalan numbers A000108.
T(n,k) = Sum_{j>=0} A053121(n,j)*binomial(j,k). - Philippe Deléham, Mar 30 2007
T(n,k) = T(n-1,k-1) + T(n-1,l) + Sum_{j>=0} T(n-1,k+1+j)*(-1)^j. - Philippe Deléham, Feb 23 2012

A054337 7-fold convolution of A000302 (powers of 4).

Original entry on oeis.org

1, 28, 448, 5376, 53760, 473088, 3784704, 28114944, 196804608, 1312030720, 8396996608, 51908706304, 311452237824, 1820797698048, 10404558274560, 58265526337536, 320460394856448, 1734256254517248, 9249366690758656, 48680877319782400, 253140562062868480

Offset: 0

Wolfdieter Lang, Mar 13 2000

With a different offset, number of n-permutations (n>=6) of 5 objects: u, v, z, x, y with repetition allowed, containing exactly six (6) u's. Example: a(1)=28 because we have uuuuuuv, uuuuuvu, uuuuvuu, uuuvuuu, uuvuuuu, uvuuuuu, vuuuuuu, uuuuuuz, uuuuuzu, uuuuzuu, uuuzuuu, uuzuuuu, uzuuuuu, zuuuuuu, uuuuuux, uuuuuxu, uuuuxuu, uuuxuuu, uuxuuuu, uxuuuuu, xuuuuuu, uuuuuuy, uuuuuyu, uuuuyuu, uuuyuuu, uuyuuuu, uyuuuuu, yuuuuuu. - Zerinvary Lajos, Jun 16 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A000302, A054335.

Cf. A038231.

List([0..30], n-> 4^n*Binomial(n+6,6)); # G. C. Greubel, Jul 21 2019

[4^n*Binomial(n+6, 6): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq(seq(binomial(i, j)*4^(i-6), j =i-6), i=6..36); # Zerinvary Lajos, Dec 03 2007
seq(binomial(n+6,6)*4^n,n=0..30); # Zerinvary Lajos, Jun 16 2008

Table[4^n*Binomial[n+6,6], {n,0,30}] (* G. C. Greubel, Jul 21 2019 *)

vector(30, n, n--; 4^n*binomial(n+6,6) ) \\ G. C. Greubel, Jul 21 2019

[lucas_number2(n, 4, 0)*binomial(n,6)/2^12 for n in range(6, 36)] # Zerinvary Lajos, Mar 11 2009

a(n) = binomial(n+6, 6)*4^n.
G.f.: 1/(1 - 4*x)^7.
a(n) = A054335(n+13, 13).
E.g.f.: (45 + 1080*x + 5400*x^2 + 9600*x^3 + 7200*x^4 + 2304*x^5 + 256*x^6)*exp(4*x)/45. - G. C. Greubel, Jul 21 2019
From Amiram Eldar, Mar 25 2022: (Start)
Sum_{n>=0} 1/a(n) = 8394/5 - 5832*log(4/3).
Sum_{n>=0} (-1)^n/a(n) = 75000*log(5/4) - 83674/5. (End)

A104624 Expansion of exp( arcsinh( -2*x ) ) in powers of x.

Original entry on oeis.org

1, -2, 2, 0, -2, 0, 4, 0, -10, 0, 28, 0, -84, 0, 264, 0, -858, 0, 2860, 0, -9724, 0, 33592, 0, -117572, 0, 416024, 0, -1485800, 0, 5348880, 0, -19389690, 0, 70715340, 0, -259289580, 0, 955277400, 0, -3534526380, 0, 13128240840, 0, -48932534040, 0, 182965127280, 0, -686119227300, 0

Offset: 0

Paul Barry, Mar 17 2005

First column in inverse of A054335.

With offset 1 the coefficient sequence of series reversion of A000984 (binomial(2n,n)) also with offset 1. - Michael Somos, Jan 14 2011

			G.f. = 1 - 2*x + 2*x^2 - 2*x^4 + 4*x^6 - 10*x^8 + 28*x^10 - 84*x^12 + 264*x^14 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics, Vol. 342, Issue 11 (2019), 3079-3097. See page 3091.
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Exp(Argsinh(-2*x)))); // G. C. Greubel, Aug 12 2018

s := proc(n) option remember; `if`(n<2, n+1, 4*(n-2)*s(n-2)/(n+1)) end: A104624 := n -> `if`(n<2, (-1)^n*(n+1), (-1)^(n/2-1)*s(n-1)); seq(A104624(n), n=0..47); # Peter Luschny, Sep 23 2014

CoefficientList[ Series[ Exp[ ArcSinh[ -2x]], {x, 0, 49}], x]
Table[(-1)^n 2 HypergeometricPFQ[{-n+1, 2-n}, {2}, -1], {n, 0, 46}] (* Peter Luschny, Sep 23 2014 *)

{a(n) = if( n<0, 0, polcoeff( sqrt( 1 + 4*x^2 + x*O(x^n) ) - 2*x, n ) )}; /* Michael Somos, Jan 14 2011 */

def A104624(n):
    if n < 2: return (-1)^n*(n+1)
    if n % 2 == 1: return 0
    return (-1)^(n/2+1)*binomial(n,n/2)/(n-1)
[A104624(n) for n in range(47)] # Peter Luschny, Sep 23 2014

G.f.: sqrt( 1 + 4*x^2 ) - 2*x = exp( asinh( -2*x ) ). - Michael Somos, Jan 14 2011
The positive sequence 1,2,2,0,2,... has g.f. 2(1+x)-sqrt(1-4x^2). - Paul Barry, Oct 10 2007
From Vladimir Kruchinin, Jan 16 2011: (Start)
The o.g.f. A(x) satisfies A(x)=x*sqrt(1-4*A(x)),
a(n) = 1/(n*(n+1)) * sum(j=0...n+1, j * 2^(j) * binomial(2*n-j-1,n-1) * binomial(n+1,j) * (-1)^(n-j)). (End)
Conjecture: n*a(n) + (n-1)*a(n-1) + 4*(n-3)*a(n-2) + 4*(n-4)*a(n-3) = 0. - R. J. Mathar, Nov 15 2012
If n is even, a(n) ~ (-1)^(1+n/2) * 2^(n+1) * n^(n-1) / exp(n). - Vaclav Kotesovec, Oct 23 2013
G.f.: 2*S(0) -1-2*x-4*x^2, where S(k) = 2*k+1 + x^2*(2*k+3)/(1 + x^2*(2*k+1)/S(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 23 2013
a(n) = (-1)^n*2*hypergeom([-n+1, 2-n], [2], -1). - Peter Luschny, Sep 23 2014

A111959 Renewal array for aerated central binomial coefficients.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 4, 0, 1, 6, 0, 6, 0, 1, 0, 16, 0, 8, 0, 1, 20, 0, 30, 0, 10, 0, 1, 0, 64, 0, 48, 0, 12, 0, 1, 70, 0, 140, 0, 70, 0, 14, 0, 1, 0, 256, 0, 256, 0, 96, 0, 16, 0, 1, 252, 0, 630, 0, 420, 0, 126, 0, 18, 0, 1, 0, 1024, 0, 1280, 0, 640, 0, 160, 0, 20, 0, 1, 924, 0, 2772, 0

Offset: 0

Paul Barry, Aug 23 2005

Row sums are A098615.
Binomial transform (product with C(n,k)) is A111960.
Diagonal sums are A026671 (with interpolated zeros).
Inverse is (1/sqrt(1+4x^2),x/sqrt(1+4x^2)), or (sqrt(-1))^(n-k)*T(n,k). [corrected by Peter Bala, Aug 13 2021]
The Riordan array (1,x/sqrt(1-4*x^2)) is the same array with an additional column of zeros (besides the top element 1) added to the left. - Vladimir Kruchinin, Feb 17 2011

			From _Peter Bala_, Aug 13 2021: (Start)
Triangle begins
  1;
  0,  1;
  2,  0, 1;
  0,  4, 0, 1;
  6,  0, 6, 0, 1;
  0, 16, 0, 8, 0, 1;
Infinitesimal generator begins
  0;
  0, 0;
  2, 0, 0;
  0, 4, 0, 0;
  0, 0, 6, 0, 0;
  0, 0, 0, 8, 0, 0; (End)

Paul Barry, On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths, arXiv:2101.10218 [math.CO], 2021.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Cf. A054335, A038231, A046521.

Riordan array (1/sqrt(1-4x^2), x/sqrt(1-4x^2)); number triangle T(n, k)=(1+(-1)^(n-k))*binomial((n-1)/2, (n-k)/2)*2^(n-k)/2.
G.f.: 1/(1-xy-2x^2/(1-x^2/(1-x^2/(1-x^2/(1-.... (continued fraction). - Paul Barry, Jan 28 2009
From Peter Bala, Aug 13 2021: (Start)
T(2*n,2*k) = A046521(n,k); T(2*n+1,2*k+1) = A038231(n,k).
The row entries, read from right to left, are the coefficients in the n-th order Taylor polynomial of (sqrt(1 + 4*x^2))^((n-1)/2) at x = 0.
The infinitesimal generator of this array has the sequence [2, 4, 6, 8, 10, ...] on the second subdiagonal below the main diagonal and zeros elsewhere.
The m-th power of the array is the Riordan array (1/sqrt(1 - 4*m*x^2), x/sqrt(1 - 4*m*x^2)) with entries given by sqrt(m)^(n-k)*T(n,k). (End)

A054338 8-fold convolution of A000302 (powers of 4).

Original entry on oeis.org

1, 32, 576, 7680, 84480, 811008, 7028736, 56229888, 421724160, 2998927360, 20392706048, 133479530496, 845370359808, 5202279137280, 31213674823680, 183120225632256, 1052941297385472, 5946021444059136, 33033452466995200, 180814687187763200, 976399310813921280

Offset: 0

Wolfdieter Lang, Mar 13 2000

With a different offset, number of n-permutations (n>=7) of 5 objects: u, v, z, x, y with repetition allowed, containing exactly seven (7) u's. - Zerinvary Lajos, Jun 23 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (32,-448,3584,-17920,57344,-114688,131072,-65536).

Cf. A000302, A054335.

Cf. A038231.

List([0..20], n-> 4^n*Binomial(n+7,7) ); # G. C. Greubel, Jul 21 2019

[4^n*Binomial(n+7, 7): n in [0..20]]; // Vincenzo Librandi, Oct 15 2011

seq(binomial(n+7,7)*4^n,n=0..20); # Zerinvary Lajos, Jun 23 2008

Table[4^n*Binomial[n+7,7], {n,0,20}] (* G. C. Greubel, Jul 21 2019 *)
LinearRecurrence[{32,-448,3584,-17920,57344,-114688,131072,-65536},{1,32,576,7680,84480,811008,7028736,56229888},30] (* Harvey P. Dale, Jun 08 2025 *)

vector(20, n, n--; 4^n*binomial(n+7,7)) \\ G. C. Greubel, Jul 21 2019

a(n) = binomial(n+7, 7)*4^n.
G.f.: 1/(1-4*x)^8.
a(n) = A054335(n+15, 15).
E.g.f.: (315 + 8820*x + 52920*x^2 + 117600*x^3 + 117600*x^4 + 56448*x^5 + 12544*x^6 + 1024*x^7)*exp(4*x)/315. - G. C. Greubel, Jul 21 2019
From Amiram Eldar, Mar 27 2022: (Start)
Sum_{n>=0} 1/a(n) = 20412*log(4/3) - 88067/15.
Sum_{n>=0} (-1)^n/a(n) = 437500*log(5/4) - 292873/3. (End)

A054339 9-fold convolution of A000302 (powers of 4).

Original entry on oeis.org

1, 36, 720, 10560, 126720, 1317888, 12300288, 105431040, 843448320, 6372720640, 45883588608, 317013884928, 2113425899520, 13655982735360, 85837605765120, 526470648692736, 3158823892156416, 18581317012684800, 107358720517734400, 610249569258700800

Offset: 0

Wolfdieter Lang, Mar 13 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (36,-576,5376,-32256,129024,-344064,589824,-589824,262144).

Cf. A000302, A054335.

Cf. A038231.

List([0..20], n-> 4^n*Binomial(n+8, 8)); # G. C. Greubel, Jul 21 2019

[Binomial(n+8, 8)*4^n: n in [0..20]]; // Vincenzo Librandi, May 31 2011

seq(binomial(n+8,8)*4^n,n=0..20); # Zerinvary Lajos, Jun 23 2008

Table[Binomial[n+8,8]4^n,{n,0,20}] (* or *) LinearRecurrence[ {36,-576,5376,-32256,129024,-344064,589824,-589824,262144},{1,36,720,10560,126720,1317888,12300288,105431040,843448320},20]

vector(20, n, n--; 4^n*binomial(n+8, 8)) \\ G. C. Greubel, Jul 21 2019

[4^n*binomial(n+8, 8) for n in (0..20)] # G. C. Greubel, Jul 21 2019

a(n) = binomial(n+8, 8)*4^n.
G.f.: 1/(1-4*x)^9.
a(n) = A054335(n+17, 17).
a(n) = 36*a(n-1) - 576*a(n-2) + 5376*a(n-3) - 32256*a(n-4) + 129024*a(n-5) - 344064*a(n-6) + 589824*a(n-7) - 589824*a(n-8) + 262144*a(n-9). - Harvey P. Dale, Aug 30 2013
E.g.f.: (16/7!)*(315 + 10080*x + 70560*x^2 + 188160*x^3 + 235200*x^4 + 150528*x^5 + 50176*x^6 + 8192*x^7 + 512*x^8)*exp(4*x). - G. C. Greubel, Jul 21 2019
From Amiram Eldar, Mar 27 2022: (Start)
Sum_{n>=0} 1/a(n) = 704696/35 - 69984*log(4/3).
Sum_{n>=0} (-1)^n/a(n) = 2500000*log(5/4) - 11715016/21. (End)

A054340 10-fold convolution of A000302 (powers of 4).

Original entry on oeis.org

1, 40, 880, 14080, 183040, 2050048, 20500480, 187432960, 1593180160, 12745441280, 96865353728, 704475299840, 4931327098880, 33381291130880, 219362770288640, 1403921729847296, 8774510811545600, 53679360258867200, 322076161553203200, 1898554215471513600

Offset: 0

Wolfdieter Lang, Mar 13 2000

With a different offset, number of n-permutations (n>=9) of 5 objects: u, v, z, x, y with repetition allowed, containing exactly nine (9) u's. - Zerinvary Lajos, Jul 02 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (40,-720,7680,-53760,258048,-860160,1966080,-2949120,2621440,-1048576).

Cf. A000302, A054335.

List([0..20], n-> 4^n*Binomial(n+9, 9)); # G. C. Greubel, Jul 21 2019

[4^n*Binomial(n+9, 9): n in [0..20]]; // Vincenzo Librandi, Oct 15 2011

seq(binomial(n+9,9)*4^n,n=0..20); # Zerinvary Lajos, Jul 02 2008

Table[4^n*Binomial[n+9,9], {n,0,20}] (* G. C. Greubel, Jul 21 2019 *)

vector(20, n, n--; 4^n*binomial(n+9, 9)) \\ G. C. Greubel, Jul 21 2019

a(n) = binomial(n+9, 9)*4^n.
G.f.: 1/(1 - 4*x)^10.
a(n) = A054335(n+19, 19).
E.g.f.: (2^7/9!)*(2835 + 102060*x + 816480*x^2 + 2540160*x^3 + 3810240*x^4 + 3048192*x^5 + 1354752*x^6 + 331776*x^7 + 41472*x^8 + 2048*x^9)*exp(4*x).
From Amiram Eldar, Mar 27 2022: (Start)
Sum_{n>=0} 1/a(n) = 236196*log(4/3) - 4756383/70.
Sum_{n>=0} (-1)^n/a(n) = 14062500*log(5/4) - 43931373/14. (End)

A104625 Expansion of 1/(sqrt(1-4*x) - x^2).

Original entry on oeis.org

1, 2, 7, 24, 87, 322, 1211, 4604, 17645, 68042, 263655, 1025632, 4002601, 15662422, 61427543, 241386924, 950160607, 3745589510, 14784496003, 58424093536, 231112008371, 915065382154, 3626113490579, 14379912928572, 57064644495359

Offset: 0

Paul Barry, Mar 17 2005

Diagonal sums of convolution triangle of central binomial coefficients A054335.

Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when on the diagonal, (2,2). - Alois P. Heinz, Sep 14 2016

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000984, A026671, A054335.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/(sqrt(1-4*x) - x^2))); // G. C. Greubel, Aug 12 2018

CoefficientList[Series[1/(Sqrt[1-4*x] -x^2), {x, 0, 50}], x] (* G. C. Greubel, Aug 12 2018 *)

x='x+O('x^50); Vec(1/(sqrt(1-4*x) - x^2)) \\ G. C. Greubel, Aug 12 2018

Conjecture: n*a(n) + (n-3)*a(n-1) + 2*(-28*n+51)*a(n-2) + 72*(2*n-5)*a(n-3) - n*a(n-4) + (-5*n+3)*a(n-5) + 18*(2*n-5)*a(n-6) = 0. - R. J. Mathar, Feb 20 2015

a(n) = Sum_{k=0..floor(n+2)/2} 4^(n+2-2*k) * binomial(n+1-3*k/2,n+2-2*k). - Seiichi Manyama, Feb 06 2024

cp's OEIS Frontend

A026671 Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when on the diagonal, (1,1).

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A098615 G.f. A(x) satisfies: A(x*G(x)) = G(x), where G(x) is the g.f. for A098614(n) = Fibonacci(n+1)*Catalan(n).

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A054336 A convolution triangle of numbers based on A001405 (central binomial coefficients).

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A054337 7-fold convolution of A000302 (powers of 4).

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A104624 Expansion of exp( arcsinh( -2*x ) ) in powers of x.

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A111959 Renewal array for aerated central binomial coefficients.

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A098615 G.f. A(x) satisfies: A(xG(x)) = G(x), where G(x) is the g.f. for A098614(n) = Fibonacci(n+1)Catalan(n).