A045868 -id:A045868 - 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)

A371521 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^6.

Original entry on oeis.org

1, 6, 57, 614, 7158, 88002, 1123689, 14760024, 198172050, 2707560544, 37522666803, 526190125308, 7452866846847, 106465245105972, 1532129408941797, 22191180837313808, 323243244688652943, 4732225866305323686, 69591395772704207770, 1027547992261749954798

Offset: 0

Seiichi Manyama, Mar 26 2024

nonn

Cf. A349333, A371379, A371519, A371522, A371523.

Cf. A045868, A371516, A371517, A371520.

a(n) = 6*sum(k=0, n, binomial(n-1, n-k)*binomial(6*k+5, k)/(5*k+6));

a(n) = 6 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+5,k)/(5*k+6) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+6,k)/(k+1).

G.f.: A(x) = B(x)^6 where B(x) is the g.f. of A349333.

A371517 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^4.

Original entry on oeis.org

1, 4, 26, 188, 1459, 11892, 100444, 871528, 7722557, 69590628, 635807180, 5876094308, 54836925779, 516029817620, 4891147100886, 46653935716492, 447490869463145, 4313492172957396, 41763413498670702, 405968522259130636, 3960526930400038404

Offset: 0

Seiichi Manyama, Mar 26 2024

nonn

Cf. A045868, A371516, A371520, A371521.
Cf. A349331, A371483, A371518.
Cf. A371486.

a(n) = 4*sum(k=0, n, binomial(n-1, n-k)*binomial(4*k+3, k)/(3*k+4));

a(n) = 4 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(4*k+3,k)/(3*k+4) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(4*k+4,k)/(k+1).

G.f.: A(x) = B(x)^4 where B(x) is the g.f. of A349331.

A371516 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^3.

Original entry on oeis.org

1, 3, 15, 82, 477, 2901, 18235, 117555, 773085, 5166478, 34987170, 239570655, 1655933060, 11538839130, 80971109712, 571702698185, 4058556404958, 28951715755830, 207424064434502, 1491898838023884, 10768487956456506, 77977009814421534, 566310026687320290

Offset: 0

Seiichi Manyama, Mar 26 2024

nonn

Cf. A045868, A371517, A371520, A371521.
Cf. A270386, A307678.
Cf. A371483.

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

a(n) = 3 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(3*k+2,k)/(2*k+3) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(3*k+3,k)/(k+1).

G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A307678.

A371520 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^5.

Original entry on oeis.org

1, 5, 40, 360, 3495, 35726, 378965, 4133080, 46059020, 522196465, 6004261226, 69849651025, 820651943130, 9723556336780, 116056250171385, 1394082307995626, 16840510019954835, 204453614350921540, 2493311080293185200, 30528431677508637205, 375155454309681439001

Offset: 0

Seiichi Manyama, Mar 26 2024

nonn

Cf. A045868, A371516, A371517, A371521.

Cf. A349332, A371486.

a(n) = 5*sum(k=0, n, binomial(n-1, n-k)*binomial(5*k+4, k)/(4*k+5));

a(n) = 5 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(5*k+4,k)/(4*k+5) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(5*k+5,k)/(k+1).

G.f.: A(x) = B(x)^5 where B(x) is the g.f. of A349332.

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

A045902 Catafusenes (see reference for precise definition).

Original entry on oeis.org

1, 4, 18, 80, 355, 1580, 7066, 31772, 143645, 652860, 2981910, 13682328, 63046776, 291646860, 1353967250, 6306552800, 29464361530, 138045441260, 648449195350, 3053348997200, 14409512770575, 68143962854836, 322886537205062, 1532716400556220, 7288075248828605, 34710221395625380

Offset: 0

N. J. A. Sloane

nonn

4-fold convolution of A002212. Convolution of A045868 with itself. - Emeric Deutsch, Mar 13 2004

S. J. Cyvin et al., Enumeration and classification of certain polygonal systems... : annelated catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A002212, A045868.

a := n->(4/n)*sum(binomial(n,j)*binomial(2*j+3,j-1),j=1..n): 1,seq(a(n),n=1..22);

Table[SeriesCoefficient[(1-x-Sqrt[1-6*x+5*x^2])^4/(16*x^4),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2012 *)

x='x+O('x^66); Vec((1-x-sqrt(1-6*x+5*x^2))^4/(16*x^4)) \\ Joerg Arndt, May 04 2013

G.f.: (1 - x - sqrt(1-6*x+5*x^2))^4/(16*x^4). - Emeric Deutsch, Mar 13 2004
a(n) = (4/n)*Sum_{j=1..n} binomial(n, j)*binomial(2j+3, j-1) for n >= 1. - Emeric Deutsch, Mar 25 2004
Recurrence: (n+1)*(n+4)*a(n) = (6*n^2+19*n+19)*a(n-1) - 5*(n-2)*(n+2)*a(n-2). - Vaclav Kotesovec, Oct 08 2012
a(n) ~ 16*5^(n+1/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012

More terms from Emeric Deutsch, Mar 13 2004

A371583 G.f. satisfies A(x) = ( 1 + x*A(x)^(5/2) / (1 - x) )^2.

Original entry on oeis.org

1, 2, 13, 104, 940, 9166, 94044, 1000602, 10939780, 122161128, 1387361151, 15974899766, 186069556707, 2188416960148, 25953579753464, 310022550197360, 3726709235290628, 45047517497268968, 547217895030263028, 6676784544374859088, 81789906534091716353

Offset: 0

Seiichi Manyama, Mar 28 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..896

Cf. A045868, A270386, A371518, A371523.
Cf. A349332, A371486, A371520.
Cf. A371578, A371585.

a(n, r=2, s=1, t=5, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

a(n) = 2 * Sum_{k=0..n} binomial(5*k+2,k) * binomial(n-1,n-k)/(5*k+2).

G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A349332.

A073149 Triangle of numbers arising in recursive computation of A002212.

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 10, 13, 16, 26, 36, 46, 55, 65, 101, 137, 173, 203, 233, 269, 406, 543, 680, 788, 888, 996, 1133, 1676, 2219, 2762, 3173, 3533, 3893, 4304, 4847, 7066, 9285, 11504, 13133, 14503, 15799, 17169, 18798, 21017, 30302, 39587, 48872, 55529

Offset: 0

Paul D. Hanna, Jul 18 2002

Related to restricted hexagonal polyominoes with n cells (A002212) and catafusenes (A045868).

Only T(n,k) for 0<=k<=n are listed since T(n,k)=T(n,n) if k>n.

			T(5,3)=T(5,2)+T(3,0)T(5-2,0)=203+10*3=233.
{1}, {1,2}, {3,4,7}, {10,13,16,26}, {36,46,55,65,101},...

T(n, 0)=A002212(n). T(n, n)=A045868(n).

T(n,k)=if(k<0 || n<0,0,if(n==0,1,if(k==0,T(n-1,0)+if(n>1,T(n-1,n-1)),T(n,k-1)+T(k,0)*T(n-k,0))))

G.f.: Sum_{n>=0, k>=0} T(n, k)*y^k*x^n = A(x)*A(xy)/(1-y) where A(x) is g.f. of A002212.

T(0, k)=T(1, 0)=1. T(n+1, 0)=T(n, 0)+T(n, n), n>0. T(n, k)=T(n, k-1)+T(k, 0)T(n-k, 0), k>0. T(n, k)=T(n, n), k>n.

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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A371521 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^6.

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A371517 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^4.

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A371516 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^3.

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A371520 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^5.

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

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A045902 Catafusenes (see reference for precise definition).

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A371583 G.f. satisfies A(x) = ( 1 + x*A(x)^(5/2) / (1 - x) )^2.

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