A077587 -id:A077587 - cp's OEIS frontend

A114462 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k ascents of length 2 starting at an even level (0<=k<=floor(n/2)).

1, 1, 1, 1, 2, 3, 6, 7, 1, 18, 19, 5, 54, 59, 18, 1, 166, 191, 65, 7, 522, 631, 242, 34, 1, 1670, 2123, 906, 154, 9, 5418, 7247, 3395, 680, 55, 1, 17786, 25011, 12746, 2932, 300, 11, 58974, 87071, 47931, 12414, 1540, 81, 1, 197226, 305275, 180439, 51878, 7552

Offset: 0

Emeric Deutsch, Nov 29 2005

Row n has 1+floor(n/2) terms. Row sums are the Catalan numbers (A000108). Sum(kT(n,k), k=0..floor(n/2)) = binomial(2n-3,n-1)-binomial(2n-4,n) = A077587(n-2) (n>=2). Column 0 yields A114464.

			T(4,1) = 7 because we have (UU)DDUDUD, UD(UU)DDUD, UDUD(UU)DD, (UU)DUDDUD,
UD(UU)DUDD, (UU)DUDUDD and (UU)DUUDDD, where U=(1,1), D=(1,-1) (the ascents of length 2 starting at an even level are shown between parentheses; note that the last path has an ascent of length 2 that starts at an odd level).
Triangle starts:
1;
1;
1,   1;
2,   3;
6,   7,  1;
18, 19,  5;
54, 59, 18, 1;

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A077587, A000108, A114463, A114464, A114465, A102402.

G:= 1/2/z*(3*z^2+2*z^3*t+1-z^3*t^2-3*z^2*t-z^3+t*z-z -sqrt(1+20*z^3*t-18*z^5*t^2+15*z^4*t^2+18*z^5*t+6*z^5*t^3-2*z^4*t^3-12*z^2*t -12*z^3 -6*z-24*z^4*t-8*z^3*t^2+z^6-6*z^5+11*z^4 +z^2*t^2+6*z^6*t^2 -4*z^6*t^3 -4*z^6*t+z^6*t^4+2*t*z +11*z^2)): Gser:=simplify(series(G,z=0,17)): P[0]:=1: for n from 1 to 14 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 14 do seq(coeff(t*P[n],t^j),j=1..1+floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0,
      `if`(t=2, z, 1), expand(b(x-1, y-1, min(3, t+1))+
      `if`(t=2 and irem(y, 2)=0, z, 1)*b(x-1, y+1, 0))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0$2)):
seq(T(n), n=0..14);  # Alois P. Heinz, Mar 12 2014

b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x==0, If[t==2, z, 1], Expand[ b[x-1, y-1, Min[3, t+1]] + If[t==2 && Mod[y, 2]==0, z, 1]*b[x-1, y+1, 0]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Mar 31 2015, after Alois P. Heinz *)

G.f.: G(t,z) satisfies zG^2-(1-z+tz-3tz^2+3z^2-z^3-t^2z^3+2tz^3)G+1-z+z^2+tz-tz^2=0.

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.

A274883 Triangle read by rows, T(n,k) = 2^kbinomial(n,k)A057977(n-k) for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 3, 6, 12, 8, 2, 24, 24, 32, 16, 10, 20, 120, 80, 80, 32, 5, 120, 120, 480, 240, 192, 64, 35, 70, 840, 560, 1680, 672, 448, 128, 14, 560, 560, 4480, 2240, 5376, 1792, 1024, 256, 126, 252, 5040, 3360, 20160, 8064, 16128, 4608, 2304, 512

Offset: 0

Peter Luschny, Jul 14 2016

			Triangle starts:
                       1;
                      1, 2;
                    1, 4, 4;
                  3, 6, 12, 8;
               2, 24, 24, 32, 16;
            10, 20, 120, 80, 80, 32;
         5, 120, 120, 480, 240, 192, 64;
     35, 70, 840, 560, 1680, 672, 448, 128;
14, 560, 560, 4480, 2240, 5376, 1792, 1024, 256;

Cf. A000079 (T(n,n)), A057977 (T(n,0)), A077587 (row sum).

Cf. A189912. Row reversed A091894 is a subtriangle.

T := (n,k) -> 2^k*binomial(n,k)*((n-k)!/floor((n-k)/2)!^2)/(floor((n-k)/2)+1);
seq(seq(T(n,k), k=0..n), n=0..9);

cp's OEIS Frontend

A114462 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k ascents of length 2 starting at an even level (0<=k<=floor(n/2)).

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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).

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A274883 Triangle read by rows, T(n,k) = 2^kbinomial(n,k)A057977(n-k) for n>=0 and 0<=k<=n.

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cp's OEIS Frontend

A114462 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k ascents of length 2 starting at an even level (0<=k<=floor(n/2)).

Views

Author

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Comments

Examples

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Maple

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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

A274883 Triangle read by rows, T(n,k) = 2^k*binomial(n,k)*A057977(n-k) for n>=0 and 0<=k<=n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

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).

A274883 Triangle read by rows, T(n,k) = 2^kbinomial(n,k)A057977(n-k) for n>=0 and 0<=k<=n.