A055452 -id:A055452 - cp's OEIS frontend

A055450 Path-counting array T; each step of a path is (1 right) or (1 up) to a point below line y=x, else (1 right and 1 up) or (1 up) to a point on the line y=x, else (1 left) or (1 up) to a point above line y=x. T(i,j)=number of paths to point (i-j,j), for 1<=j<=i, i >= 1.

1, 1, 3, 1, 2, 10, 1, 3, 7, 36, 1, 4, 5, 26, 137, 1, 5, 9, 19, 101, 543, 1, 6, 14, 14, 75, 406, 2219, 1, 7, 20, 28, 56, 305, 1676, 9285, 1, 8, 27, 48, 42, 230, 1270, 7066, 39587, 1, 9, 35, 75, 90, 174, 965, 5390, 30302, 171369, 1, 10, 44, 110, 165, 132, 735, 4120, 23236, 131782, 751236

Offset: 0

Clark Kimberling, May 18 2000

			Triangle begins as:
  1;
  1, 3;
  1, 2, 10;
  1, 3,  7, 36;
  1, 4,  5, 26, 137;
  1, 5,  9, 19, 101, 543;
  1, 6, 14, 14,  75, 406, 2219;
  1, 7, 20, 28,  56, 305, 1676, 9285;
  1, 8, 27, 48,  42, 230, 1270, 7066, 39587;
  ...
T(4,4) defined as T(5,4)+T(3,3) when k=4, T(5,4) already defined when k=3.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000108, A002212, A030237, A045868,

Cf. A055451, 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)) >;
A030237:= 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 A030237(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;
[T(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Jan 29 2024

T[n_, 0]:= 1; T[n_, k_]:= T[n, k]= If[1<=kG. C. Greubel, Jan 29 2024 *)
T[n_, k_]:= If[kG. C. Greubel, Jan 29 2024 *)

def A030237(n,k): return (n-k+1)*binomial(n+k, k)/(n+1)
def T(n,k): # T = A055450
    if kA030237(n-k+1,k)
    else: return round(catalan_number(n-k+1)*hypergeometric([n-2*k, (3+2*(n-k))/2], [3+n-k], -4))
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 29 2024

Initial values: T(i, 0)=1 for i >= 0. Recurrence: if 1 <= j < i/2, then T(i, j) = T(i-1, j-1) + T(i-1, j), if j = i/2 then T(2j, j) = T(2j-2, j-1) + T(2j-1, j-1), otherwise T(2j-k, j) = T(2j-k+1, j) + T(2j-k-1, j-1) for j=k, k+1, k+2, ..., for k=1, 2, 3, ...
T(2n, n) = A000108(n) for n >= 0 (Catalan numbers).
T(n, n) = A002212(n+1).
T(n, n-1) = A045868(n).
T(n, k) = A030237(n-k+1, k) for n >= 1, 0 <= k < n/2.
From G. C. Greubel, Jan 29 2024: (Start)
T(n, k) = (n-2*k+2)*binomial(n+1, k)/(n-k+2) for 0 <= k < n/2, otherwise Catalan(n-k +1)*Hypergeometric2F1([n-2*k, (3+2*(n-k))/2], [3+n-k], -4).
Sum_{k=0..n} T(n, k) = A055451(n). (End)

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

A106534 Sum array of Catalan numbers (A000108) read by upward antidiagonals.

Original entry on oeis.org

1, 2, 1, 5, 3, 2, 15, 10, 7, 5, 51, 36, 26, 19, 14, 188, 137, 101, 75, 56, 42, 731, 543, 406, 305, 230, 174, 132, 2950, 2219, 1676, 1270, 965, 735, 561, 429, 12235, 9285, 7066, 5390, 4120, 3155, 2420, 1859, 1430, 51822, 39587, 30302, 23236, 17846, 13726, 10571, 8151, 6292, 4862

Offset: 0

Philippe Deléham, May 30 2005

The underlying array A is A(n, k) = Sum_{j=0..n} binomial(n, j)*A000108(k+j), n >= 0, k>= 0. See the example section. - Wolfdieter Lang, Oct 04 2019

			From _Wolfdieter Lang_, Oct 04 2019: (Start)
The triangle T(n, k) begins:
n\k      0      1      2      3     4     5     6     7     8     9    10 ...
0:       1
1:       2      1
2:       5      3      2
3:      15     10      7      5
4:      51     36     26     19    14
5:     188    137    101     75    56    42
6:     731    543    406    305   230   174   132
7:    2950   2219   1676   1270   965   735   561   429
8:   12235   9285   7066   5390  4120  3155  2420  1859  1430
9:   51822  39587  30302  23236 17846 13726 10571  8151  6292  4862
10: 223191 171369 131782 101480 78244 60398 46672 36101 27950 21658 16796
... reformatted and extended.
-------------------------------------------------------------------------
The array A(n, k) begins:
n\k  0   1    2    3     4     5      6 ...
-------------------------------------------
0:   1   1    2    5    14    42    132 ... A000108
1    2   3    7   19    56   174    561 ... A005807
2:   5  10   26   75   230   735   2420 ...
3:  15  36  101  305   965  3155  10571 ...
4:  51 137  406 1270  4120 13726  46672 ...
5: 188 543 1676 5390 17846 60398 207963 ...
... (End)

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Paul Barry and A. Hennessy, The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010) # 10.8.2, page 5.

Columns: A007317, A002212, see also A045868, A055452-A055455.
Diagonals: A000108, A005807.
Cf. A059346 (Catalan difference array as triangle).

function T(n,k)
  if k gt n then return 0;
  elif k eq n then return Catalan(n);
  else return T(n-1, k) + T(n, k+1);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 18 2021

# Uses floating point, precision might have to be adjusted.
C := n -> binomial(2*n,n)/(n+1);
H := (n,k) -> hypergeom([k-n,k+1/2],[k+2],-4);
T := (n,k) -> C(k)*H(n,k);
seq(print(seq(round(evalf(T(n,k),32)),k=0..n)),n=0..7);
# Peter Luschny, Aug 16 2012

T[n_, n_] := CatalanNumber[n]; T[n_, k_] /; 0 <= k < n := T[n-1, k] + T[n, k+1]; T[, ] = 0; Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 11 2019 *)

def T(n, k) :
    if k > n : return 0
    if n == k : return binomial(2*n, n)/(n+1)
    return T(n-1, k) + T(n, k+1)
A106534 = lambda n,k: T(n, k)
for n in (0..5): [A106534(n,k) for k in (0..n)] # Peter Luschny, Aug 16 2012

T(n, k) = 0 if k > n; T(n, n) = A000108(n); T(n, k) = T(n-1, k) + T(n, k+1) if 0 <= k < n.
T(n, k) = binomial(2*k,k)/(k+1)*hypergeometric([k-n, k+1/2], [k+2], -4). - Peter Luschny, Aug 16 2012
T(n, k) = A(n-k, k) = Sum_{j=0..n-k} binomial(n-k, j)*A000108(k+j), n >= 0, k = 0..n. - Wolfdieter Lang, Oct 03 2019
G.f.: (sqrt(1-4*x*y)-sqrt((5*x-1)/(x-1)))/(2*x*(x*y-y+1)). - Vladimir Kruchinin, Jan 12 2024

A277969 a(n) = Sum_{k=0..n} binomial(n-3,n-k)*Catalan(k).

Original entry on oeis.org

1, -1, 2, 5, 19, 75, 305, 1270, 5390, 23236, 101480, 448085, 1997115, 8973255, 40602093, 184853055, 846206025, 3892585325, 17984308775, 83417287855, 388297304825, 1813341109825, 8493372326675, 39889629750600, 187812852106636

Offset: 0

Vladimir Kruchinin, Nov 06 2016

sign

Robert Israel, Table of n, a(n) for n = 0..1430

Cf. A000108, A055452.

f:= gfun:-rectoproc({(5*n-10)*a(n)+(-7-6*n)*a(n+1)+(n+3)*a(n+2),a(0) = 1, a(1) = -1, a(2) = 2, a(3) = 5},a(n),remember):
map(f, [$0..30]); # Robert Israel, Nov 21 2016

CoefficientList[Series[((1 - x)^3 (1 - Sqrt[(5 x - 1) / (x - 1)])) / (2 x), {x, 0, 25}], x] (* Vincenzo Librandi, Nov 07 2016 *)

a(n):=sum((binomial(2*k,k)*binomial(n-3,n-k))/(k+1),k,0,n);

x='x+O('x^50); Vec(((1-x)^3*(1-sqrt((5*x-1)/(x-1))))/(2*x)) \\ G. C. Greubel, Apr 09 2017

G.f.: ((1-x)^3*(1-sqrt((5*x-1)/(x-1))))/(2*x).
a(n) ~ 8*5^(n-3/2) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Nov 07 2016
(5*n-10)*a(n)-(7+6*n)*a(n+1)+(n+3)*a(n+2)=0 for n >= 2. - Robert Israel, Nov 21 2016
a(n) = A055452(n+1) for n > 2. - Georg Fischer, Oct 23 2018

cp's OEIS Frontend

A055450 Path-counting array T; each step of a path is (1 right) or (1 up) to a point below line y=x, else (1 right and 1 up) or (1 up) to a point on the line y=x, else (1 left) or (1 up) to a point above line y=x. T(i,j)=number of paths to point (i-j,j), for 1<=j<=i, i >= 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A055451 Row sums of array in A055450.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A106534 Sum array of Catalan numbers (A000108) read by upward antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

A277969 a(n) = Sum_{k=0..n} binomial(n-3,n-k)*Catalan(k).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple