A055450 -id:A055450 - cp's OEIS frontend

A055452 a(n) = T(n,n-2), array T as in A055450.

1, 3, 5, 19, 75, 305, 1270, 5390, 23236, 101480, 448085, 1997115, 8973255, 40602093, 184853055, 846206025, 3892585325, 17984308775, 83417287855, 388297304825, 1813341109825, 8493372326675, 39889629750600

Offset: 2

Clark Kimberling, May 18 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 2..500

Cf. A055450, A055451, A055453, A055454, A055455.

[n le 3 select 2*n-3 else Round(5*HypergeometricSeries2F1(4-n,7/2,5,-4)): n in [2..40]]; // G. C. Greubel, Jan 29 2024

Table[If[n<4, 2*n-3, 5*Hypergeometric2F1[4-n,7/2,5,-4]], {n,2,40}] (* G. C. Greubel, Jan 29 2024 *)

def A055452(n): return 2*n-3 if n<4 else 5*hypergeometric([4-n,7/2],[5],-4).simplify()
[A055452(n) for n in range(2,41)] # G. C. Greubel, Jan 29 2024

a(n) = 5*hypergeometric([4-n,7/2],[5],-4), for n>3. - Peter Luschny, Aug 15 2012

A055455 a(n) = A055450(n, n-5).

Original entry on oeis.org

1, 6, 20, 48, 90, 132, 561, 2420, 10571, 46672, 207963, 934064, 4224685, 19225588, 87969426, 404479884, 1867924322, 8660317820, 40295911480, 188105782260, 880716750140, 4134823623820, 19461238795225, 91810738725036, 434062054130187, 2056265327125528

Offset: 5

Clark Kimberling, May 18 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 5..500

Cf. A055450, A055451, A055452, A055453, A055454.

[1,6,20,48,90] cat [Floor(132*HypergeometricSeries2F1(10-n,13/2,8,-4)): n in [10..40]]; // G. C. Greubel, Jan 30 2024

Join[{1,6,20,48,90}, Table[132*Hypergeometric2F1[10-n,13/2,8,-4], {n, 10, 40}]] (* G. C. Greubel, Jan 30 2024 *)

def A055455(n): return (1,6,20,48,90)[n-5] if n<10 else 132*hypergeometric([10-n,13/2],[8],-4).simplify()
[A055455(n) for n in range(5,41)] # G. C. Greubel, Jan 30 2024

a(n) = A055450(n, n-5), n >= 5.

a(n) = 132*Hypergeometric2F1([10-n, 13/2], [8], -4), for n >= 10. - G. C. Greubel, Jan 30 2024

a(13) corrected and more terms from Sean A. Irvine, Mar 21 2022

A055451 Row sums of array in A055450.

Original entry on oeis.org

1, 4, 13, 47, 173, 678, 2735, 11378, 48279, 208410, 911571, 4031919, 17999628, 81000573, 367040404, 1673295419, 7669312343, 35319197637, 163350479756, 758406642839, 3533447414030, 16514820417166, 77412170863861

Offset: 0

Clark Kimberling, May 18 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A055450, A055452, A055453, A055454, A055455.

B:=Binomial; G:=Gamma; F:=Factorial;
p:= func< n,k,j | B(n-2*k+j-1, j)*G(n-k+j+3/2)/(F(j)*G(n-k+3/2)*B(n-k+j+2, j)) >;
f:= func< n,k | (n-k+1)*Binomial(n+k, k)/(n+1) >;
function T(n,k) // T = A055450
  if k lt n/2 then return f(n-k+1, k);
  else return Round(Catalan(n-k+1)*(&+[p(n,k,j)*(-4)^j: j in [0..n]]));
  end if;
end function;
A055451:= func< n | (&+[T(n,k): k in [0..n]]) >;
[A055451(n): n in [0..40]]; // G. C. Greubel, Jan 29 2024

T[n_, 0]:= 1; T[n_, k_]:= T[n, k]= If[1<=kA055451[n_]:= A055451[n]= Sum[T[n,k], {k,0,n}];
Table[A055451[n], {n,0,40}] (* G. C. Greubel, Jan 29 2024 *)

def f(n,k): return (n-k+1)*binomial(n+k, k)/(n+1)
def T(n,k): # T = A055450
    if kA055451(n): return sum(T(n,k) for k in range(n+1))
[A055451(n) for n in range(41)] # G. C. Greubel, Jan 30 2024

a(n) = Sum_{k=0..n} A055450(n, k). - G. C. Greubel, Jan 29 2024

A055453 a(n) = T(n,n-3), array T as in A055450.

Original entry on oeis.org

1, 4, 9, 14, 56, 230, 965, 4120, 17846, 78244, 346605, 1549030, 6976140, 31628838, 144250962, 661352970, 3046379300, 14091723450, 65432979080, 304880016970, 1425043805000, 6680031216850, 31396257423925, 147923222356036

Offset: 3

Clark Kimberling, May 18 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 3..500

Cf. A055450, A055451, A055452, A055454, A055455.

[n le 6 select (n-2)^2 else Round(14*HypergeometricSeries2F1(6-n,9/2,6,-4)): n in [3..40]]; // G. C. Greubel, Jan 30 2024

Table[If[n<6, (n-2)^2, 14*Hypergeometric2F1[6-n,9/2,6,-4]], {n,3,40}] (* G. C. Greubel, Jan 30 2024 *)

def A055453(n): return (n-2)^2 if n<6 else 14*hypergeometric([6-n,9/2],[6],-4).simplify()
[A055453(n) for n in range(3,41)] # G. C. Greubel, Jan 30 2024

a(n) = 14*Hypergeometric2F1([6-n, 9/2], [6], -4), for n >= 6 and a(n) = (n-2)^2 for 3 <= n <= 5. - G. C. Greubel, Jan 30 2024

A055454 a(n) = A055450(n, n-4).

Original entry on oeis.org

1, 5, 14, 28, 42, 174, 735, 3155, 13726, 60398, 268361, 1202425, 5427110, 24652698, 112622124, 517102008, 2385026330, 11045344150, 51341255630, 239447037890, 1120163788030, 5254987411850, 24716226207075

Offset: 4

Clark Kimberling, May 18 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 4..500

Cf. A055450, A055451, A055452, A055453, A055455.

[1,5,14,28] cat [Floor(42*HypergeometricSeries2F1(8-n,11/2,7,-4)): n in [8..40]]; // G. C. Greubel, Jan 30 2024

Join[{1,5,14,28}, Table[42*Hypergeometric2F1[8-n,11/2,7,-4], {n,8,40}]] (* G. C. Greubel, Jan 30 2024 *)

def A055454(n): return (1,5,14,28)[n-4] if n<8 else 42*hypergeometric([8-n,11/2],[7],-4).simplify()
[A055454(n) for n in range(4,41)] # G. C. Greubel, Jan 30 2024

a(n) = A055450(n, n-4).
a(n) = 42*Hypergeometric2F1([8-n, 11/2], [7], -4) for n >= 8. - G. C. Greubel, Jan 30 2024
a(n) ~ 128 * 5^(n - 13/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 09 2025

A133380 A055450 + A030111 - A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 4, 5, 1, 1, 7, 6, 7, 1, 1, 10, 13, 8, 9, 1, 1, 16, 17, 21, 10, 11, 1, 24, 30, 26, 31, 12, 13, 1, 1, 38, 42, 52, 37, 43, 14, 15, 1, 1, 59, 69, 69, 84, 50, 57, 16, 17, 1, 1

Offset: 1

Gary W. Adamson, Oct 28 2007

Row sums = A134508

			First few rows of the triangle are:
1;
1, 1;
3, 1, 1;
4, 5, 1, 1;
7, 6, 7, 1, 1;
10, 13, 8, 9, 1, 1;
16, 17, 21, 10, 11, 1, 1;
24, 30, 26, 31, 12, 13, 1, 1;
38, 42, 52, 37, 43, 14, 15, 1, 1;
...

Cf. A055450, A030111, A134508.

A045868 Expansion of g.f.: ((1 - x - sqrt(1-6x+5x^2))/(2*x))^2.

Original entry on oeis.org

1, 2, 7, 26, 101, 406, 1676, 7066, 30302, 131782, 579867, 2576982, 11550237, 52152330, 237005385, 1083211410, 4975796735, 22960105510, 106377393365, 494674698190, 2308015808015, 10801388134690, 50691017885290, 238503869991926, 1124828963516896, 5316520644648026, 25179670936870021

Offset: 0

N. J. A. Sloane

Convolution of A002212 with itself.
Number of skew Dyck paths of semilength n+1 starting with UU. A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. Example: a(2)=7 because we have UUDDUD, UUDUDD, UUDUDL, UUUDDD, UUUDDL, UUUDLD and UUUDLL. - Emeric Deutsch, May 11 2007
a(n) is also the number of path-pairs (u,v) having the following six properties: 1) the lengths of u and v sum up to 2n, 2) u and v both start at (0,0), 3) (0,0) is the only vertex that u and v have in common, 4) the steps that u can make are (1,0), (0,1) and (0,-1), 5) the steps that v can make are (1,0), (-1,0) and (0,1), 6) if A and B are the termini of u and v, respectively, then B=A+(1,-1). - Svjetlan Feretic, Jun 09 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. J. Cyvin et al., Enumeration and classification of certain polygonal systems representing polycyclic conjugated hydrocarbons: annelated catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.
Emeric Deutsch, Emanuele Munarini, and Simone Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 99.

Cf. A055450.

Essentially the first differences of A002212 and A025238.

[n le 2 select n else (2*(3*n-2)*Self(n-1) - 5*(n-3)*Self(n-2))/(n+1): n in [1..30]]; // G. C. Greubel, Jan 12 2024

a := n->(2/n)*sum(binomial(n,j)*binomial(2*j+1,j-1),j=1..n): 1,seq(a(n),n=1..22);

a[n_] := 2*Hypergeometric2F1[ 5/2, 1-n, 4, -4]; a[0] = 1; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Apr 30 2012, after Maple *)

a(n)=polcoeff((1-x-sqrt(1-6*x+5*x^2+x^2*O(x^n)))^2/4,n+2)

my(x='x+O('x^66)); Vec(((1-x-sqrt(1-6*x+5*x^2))/(2*x))^2) \\ Joerg Arndt, May 04 2013

def A045868(n): return 1 if n==0 else (2/n)*sum( binomial(n,j)*binomial(2*j+1,j-1) for j in range(1,n+1))
[A045868(n) for n in range(31)] # G. C. Greubel, Jan 12 2024

a(n) = (2/n)*Sum_{j=1..n} binomial(n, j)*binomial(2j+1, j-1) for n >= 1.
a(n) = A055450(n, n-1).
D-finite with recurrence: (n+2)*a(n) = (6*n+2)*a(n-1) - (5*n-10)*a(n-2). - Vladeta Jovovic, Jul 16 2004
a(n) = 2*Hypergeometric2F1(5/2, 1-n, 4, -4). - Jean-François Alcover, Apr 30 2012
a(n) ~ 2*5^(n+1/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012
G.f.: 1 - 1/x + Q(0)*(1-x)/x, where Q(k) = 1 + (4*k+1)*x/((1-x)*(k+1) - x*(1-x)*(2*k+2)*(4*k+3)/(x*(8*k+6)+(2*k+3)*(1-x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013
G.f.: 1/x - 1 - 2*(1-x)/x/( G(0) + 1), where G(k) = 1 + 2*x*(4*k+1)/( (2*k+1)*(1-x) - x*(1-x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1-x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 24 2013

More terms from Emeric Deutsch, May 11 2007

cp's OEIS Frontend

A055452 a(n) = T(n,n-2), array T as in A055450.

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A055455 a(n) = A055450(n, n-5).

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A055451 Row sums of array in A055450.

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A055453 a(n) = T(n,n-3), array T as in A055450.

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A055454 a(n) = A055450(n, n-4).

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A133380 A055450 + A030111 - A000012 as infinite lower triangular matrices.

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A045868 Expansion of g.f.: ((1 - x - sqrt(1-6*x+5*x^2))/(2*x))^2.

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A045868 Expansion of g.f.: ((1 - x - sqrt(1-6x+5x^2))/(2*x))^2.