A038629 -id:A038629 - cp's OEIS frontend

A078817 Table by antidiagonals giving variants on Catalan sequence: T(n,k)=C(2n,n)C(2k,k)(2k+1)/(n+k+1).

1, 3, 1, 10, 4, 2, 35, 15, 9, 5, 126, 56, 36, 24, 14, 462, 210, 140, 100, 70, 42, 1716, 792, 540, 400, 300, 216, 132, 6435, 3003, 2079, 1575, 1225, 945, 693, 429, 24310, 11440, 8008, 6160, 4900, 3920, 3080, 2288, 1430, 92378, 43758, 30888, 24024, 19404

Offset: 0

Henry Bottomley, Dec 07 2002

			Rows start:
     1,     3,    10,    35,   126,   462,  1716,
     1,     4,    15,    56,   210,   792,  3003,
     2,     9,    36,   140,   540,  2079,  8008,
     5,    24,   100,   400,  1575,  6160, 24024,
    14,    70,   300,  1225,  4900, 19404, 76440,
    42,   216,   945,  3920, 15876, 63504,252252,
   132,   693,  3080, 12936, 52920,213444,853776,
etc.

Ira Gessel, Super ballot numbers, J. Symbolic Computation 14 (1992), 179-194.
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
Jovan Mikic, A Note on the Gessel Numbers, arXiv:2203.12931 [math.CO], 2022.

Columns include A000108 (catalan), A038629, A078818 and A078819. Rows include A001700, A001791, A007946 and A078820. Diagonals include A002894 and A060150.

Essentially a reflected version of A033820.

A078817 := proc(n,k)
    binomial(2*n,n)*binomial(2*k,k)*(2*k+1)/(n+k+1) ;
end proc: # R. J. Mathar, Dec 06 2018

T(n, k) = A000984(n)*A002457(k)/(n+k+1) = T(k, n)*(2k+1)/(2n+1).

A078818 a(n) = 30*binomial(2n,n)/(n+3).

Original entry on oeis.org

10, 15, 36, 100, 300, 945, 3080, 10296, 35100, 121550, 426360, 1511640, 5408312, 19501125, 70794000, 258529200, 949074300, 3500409330, 12964479000, 48198087000, 179799820200, 672822343050, 2524918756464, 9500112378000, 35830670759000, 135439935469020

Offset: 0

Henry Bottomley, Dec 07 2002

			a(5) = 30*binomial(10,5)/8 = 945.

Muniru A Asiru, Table of n, a(n) for n = 0..300

Cf. A000108, A000984, A001622, A007946, A038629, A078817, A078819.

List([0..30],n->30*Binomial(2*n,n)/(n+3)); # Muniru A Asiru, Aug 09 2018

[30*Binomial(2*n,n)/(n+3): n in [0..30]]; // Vincenzo Librandi, Aug 11 2018

Table[(30 Binomial[2 n, n] / (n + 3)), {n, 0, 30}] (* Vincenzo Librandi, Aug 11 2018 *)

D-finite with recurrence a(n) = a(n-1)*(4n^2+6n-4)/(n^2+3n) = A078817(n, 2) = 5*A007946(n)/(2n+1) = 30*A000984(n)/(n+3).
From Amiram Eldar, Feb 16 2023: (Start)
Sum_{n>=0} 1/a(n) = 4*Pi/(135*sqrt(3)) + 7/45.
Sum_{n>=0} (-1)^n/a(n) = 9/125 - 32*log(phi)/(375*sqrt(5)), where phi is the golden ratio (A001622). (End)

A078819 a(n) = 140*C(2n,n)/(n+4).

Original entry on oeis.org

35, 56, 140, 400, 1225, 3920, 12936, 43680, 150150, 523600, 1847560, 6584032, 23661365, 85652000, 312018000, 1142971200, 4207562730, 15557374800, 57750861000, 215145084000, 804104751450, 3014244096864, 11329763650800, 42691863032000, 161238018415500, 610258100044320

Offset: 0

Henry Bottomley, Dec 07 2002

			a(5)=140*C(10,5)/9=3920

Cf. A000108, A000984, A001622, A038629, A078817, A078818, A078820.

Table[140*Binomial[2*n, n]/(n + 4), {n, 0, 30}] (* Amiram Eldar, Feb 16 2023 *)

D-finite with recurrence a(n) = a(n-1)*(4n^2+10n-6)/(n^2+4n) = A078817(n, 3) = 7*A078820(n)/(2n+1) = 140*A000984(n)/(n+4).
From Amiram Eldar, Feb 16 2023: (Start)
Sum_{n>=0} 1/a(n) = Pi/(126*sqrt(3)) + 3/70.
Sum_{n>=0} (-1)^n/a(n) = 37/1750 - 3*log(phi)/(125*sqrt(5)), where phi is the golden ratio (A001622). (End)

A352680 Array read by ascending antidiagonals. A family of Catalan-like sequences. A(n, k) = [x^k] ((n - 1)x + 1)(1 - sqrt(1 - 4x))/(2x).

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 2, 3, 1, 3, 3, 5, 9, 1, 4, 4, 7, 14, 28, 1, 5, 5, 9, 19, 42, 90, 1, 6, 6, 11, 24, 56, 132, 297, 1, 7, 7, 13, 29, 70, 174, 429, 1001, 1, 8, 8, 15, 34, 84, 216, 561, 1430, 3432, 1, 9, 9, 17, 39, 98, 258, 693, 1859, 4862, 11934, 1, 10, 10, 19, 44, 112, 300, 825, 2288, 6292, 16796, 41990

Offset: 0

Peter Luschny, Mar 27 2022

			Array starts:
n\k 0, 1,  2,  3,  4,   5,   6,    7,    8,     9, ...
------------------------------------------------------
[0] 1, 0,  1,  3,  9,  28,  90,  297, 1001,  3432, ... A071724
[1] 1, 1,  2,  5, 14,  42, 132,  429, 1430,  4862, ... A000108
[2] 1, 2,  3,  7, 19,  56, 174,  561, 1859,  6292, ... A071716
[3] 1, 3,  4,  9, 24,  70, 216,  693, 2288,  7722, ... A038629
[4] 1, 4,  5, 11, 29,  84, 258,  825, 2717,  9152, ... A352681
[5] 1, 5,  6, 13, 34,  98, 300,  957, 3146, 10582, ...
[6] 1, 6,  7, 15, 39, 112, 342, 1089, 3575, 12012, ...
[7] 1, 7,  8, 17, 44, 126, 384, 1221, 4004, 13442, ...
[8] 1, 8,  9, 19, 49, 140, 426, 1353, 4433, 14872, ...
[9] 1, 9, 10, 21, 54, 154, 468, 1485, 4862, 16302, ...
.
Seen as a triangle:
[0] 1;
[1] 1, 0;
[1] 1, 1, 1;
[2] 1, 2, 2,  3;
[3] 1, 3, 3,  5,  9;
[4] 1, 4, 4,  7, 14, 28;
[5] 1, 5, 5,  9, 19, 42,  90;
[6] 1, 6, 6, 11, 24, 56, 132, 297;

Rows: A071724, A000108, A071716, A038629, A352681.
Diagonals: A077587 (main), A271823.
Compare A352682 for a similar array based on the Bell numbers.

# Compare with the Julia function A352686Row.
function A352680Row(n, len)
    a = BigInt(n)
    P = BigInt[1]; T = BigInt[1]
    for k in 0:len-1
        T = push!(T, a)
        P = cumsum(push!(P, a))
        a = P[end]
    end
T end
for n in 0:9 println(A352680Row(n, 9)) end

for n from 0 to 9 do
    ogf := ((n - 1)*x + 1)*(1 - sqrt(1 - 4*x))/(2*x);
    ser := series(ogf, x, 12):
    print(seq(coeff(ser, x, k), k = 0..9)); od:
# Alternative:
alias(PS = ListTools:-PartialSums):
CatalanRow := proc(n, len) local a, k, P, R;
a := n; P := [1]; R := [1];
for k from 0 to len-1 do
    R := [op(R), a]; P := PS([op(P), a]); a := P[-1] od;
R end: seq(lprint(CatalanRow(n, 9)), n = 0..9);
# Recurrence:
A := proc(n, k) option remember: if k < 3 then [1, n, n + 1][k + 1] else
A(n, k-1)*((6 - 4*k)*(n - 3 + k*(3 + n)))/((1 + k)*(6 - k*(3 + n))) fi end:
seq(print(seq(A(n, k), k = 0..9)), n = 0..9);

T[n_, 0] := 1;
T[n_, k_] := (n - 1) CatalanNumber[k - 1] + CatalanNumber[k];
Table[T[n, k], {n, 0, 9}, {k, 0, 9}] // TableForm

A(n, k) = (n-1)*CatalanNumber(k-1) + CatalanNumber(k) for n >= 0 and k >= 1, A(n, 0) = 1. (Cf. A352682.)
D-finite with recurrence: A(n, k) = A(n, k-1)*((6 - 4*k)*(n - 3 + k*(3 + n)))/((1 + k)*(6 - k*(3 + n))) for k >= 3, otherwise 1, n, n + 1 for k = 0, 1, 2.
Given a list T let PS(T) denote the list of partial sums of T. Given two list S and T let [S, T] denote the concatenation of the lists. Further let P[end] denote the last element of the list P. Row n of the array A with length k can be computed by the following procedure:
     A = [n], P = [1], R = [1];
     Repeat k times: R = [R, A], P = PS([P, A]), A = [P[end]];
     Return R.

cp's OEIS Frontend

A078817 Table by antidiagonals giving variants on Catalan sequence: T(n,k)=C(2n,n)*C(2k,k)*(2k+1)/(n+k+1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A078818 a(n) = 30*binomial(2n,n)/(n+3).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

Formula

A078819 a(n) = 140*C(2n,n)/(n+4).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A352680 Array read by ascending antidiagonals. A family of Catalan-like sequences. A(n, k) = [x^k] ((n - 1)*x + 1)*(1 - sqrt(1 - 4*x))/(2*x).

Views

Author

Keywords

Examples

Crossrefs

Programs

Julia

Maple

Mathematica

Formula

A078817 Table by antidiagonals giving variants on Catalan sequence: T(n,k)=C(2n,n)C(2k,k)(2k+1)/(n+k+1).

A352680 Array read by ascending antidiagonals. A family of Catalan-like sequences. A(n, k) = [x^k] ((n - 1)x + 1)(1 - sqrt(1 - 4x))/(2x).