A055454 -id:A055454 - 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)

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

Original entry on oeis.org

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

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.

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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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