A071716 -id:A071716 - cp's OEIS frontend

A005807 Sum of adjacent Catalan numbers.

2, 3, 7, 19, 56, 174, 561, 1859, 6292, 21658, 75582, 266798, 950912, 3417340, 12369285, 45052515, 165002460, 607283490, 2244901890, 8331383610, 31030387440, 115948830660, 434542177290, 1632963760974, 6151850548776

Offset: 0

N. J. A. Sloane

nonn

The aerated sequence has Hankel transform F(n+2)*F(n+3) (A001654(n+2)). - Paul Barry, Nov 04 2008

			G.f. = 2 + 3*x+ 7*x^2 + 19*x^3 + 56*x^4 + 174*x^5 + 561*x^6 + 1859*x^7 + ...

D. E. Knuth, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Daniel Birmajer, Juan B. Gil, Michael D. Weiner, On rational Dyck paths and the enumeration of factor-free Dyck words, arXiv:1606.02183 [math.CO], (7-June-2016); see p. 9
Aleksandar Cvetkovic, Predrag Rajkovic and Milos Ivkovic, Catalan Numbers, the Hankel Transform and Fibonacci Numbers, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.3
Sergio Falcon, Catalan transform of the K-Fibonacci sequence, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832; http://dx.doi.org/10.4134/CKMS.2013.28.4.827.
Manuel Flores, Yuta Kimura, Baptiste Rognerud, Combinatorics of quasi-hereditary structures, arXiv:2004.04726 [math.RT], 2020.
Guo-Niu Han, Enumeration of Standard Puzzles
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 431
Qi Wang, Tau-tilting finite simply connected algebras, arXiv:1910.01937 [math.RT], 2019.
Fumitaka Yura, Hankel Determinant Solution for Elliptic Sequence, arXiv:1411.6972 [nlin.SI], (25-November-2014); see p. 7.

Cf. A000108.

Cf. A071716, A000778, A060941, A274052, A274244.

[((5*n+4)*Factorial(2*n))/(Factorial(n)*Factorial(n+2)): n in [0..30] ];  // Vincenzo Librandi, Aug 19 2011

A005807List := proc(m) local A, P, n; A := [2,3]; P := [2,3];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-2]]);
A := [op(A), P[-1]] od; A end: A005807List(25); # Peter Luschny, Mar 26 2022

a[n_]:=Binomial[2*n, n]*(5*n+4)/(n+1)/(n+2); (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)
a[ n_] := If[ n < 0, 0, CatalanNumber[n] + CatalanNumber[n + 1]]; (* Michael Somos, Jan 17 2015 *)
Total/@Partition[CatalanNumber[Range[0,30]],2,1] (* Harvey P. Dale, Jun 21 2025 *)

{a(n) = if( n<0, 0, binomial(2*n, n) * (5*n+4) / ((n+1) * (n+2)))};

from _future_ import division
A005807_list, b = [], 2
for n in range(10**3):
    A005807_list.append(b)
    b = b*(4*n+2)*(5*n+9)//((n+3)*(5*n+4)) # Chai Wah Wu, Jan 28 2016

[catalan_number(i)+catalan_number(i+1) for i in range(0,25)] # Zerinvary Lajos, May 17 2009

a(n) = C(n)+C(n+1) = ((5*n+4)*(2*n)!)/(n!*(n+2)!).
G.f. A(x) satisfies x^2*A(x)^2 + (x-1)*A(x) + (x+2) = 0. - Michael Somos, Sep 11 2003
G.f.: (1-x - (1+x)*sqrt(1-4*x)) / (2*x^2) = (4+2*x) / (1-x + (1+x)*sqrt(1-4*x)). a(n)*(n+2)*(5*n-1) = a(n-1)*2*(2*n-1)*(5*n+4), n>0. - Michael Somos, Sep 11 2003
a(n) ~ 5*Pi^(-1/2)*n^(-3/2)*2^(2*n)*{1 - 93/40*n^-1 + 625/128*n^-2 - 10227/1024*n^-3 + 661899/32768*n^-4 ...}. - Joe Keane (jgk(AT)jgk.org), Sep 13 2002
G.f.: c(x)*(1+c(x))= (-1 +(1+x)*c(x))/x with the g.f. c(x) of A000108 (Catalan).
a(n) = binomial(2*n,n)/(n+1)*hypergeom([-1,n+1/2],[n+2],-4). - Peter Luschny, Aug 15 2012
D-finite with recurrence (n+2)*a(n) + (-3*n-2)*a(n-1) + 2*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Dec 02 2012
0 = a(n)*(+16*a(n+1) + 38*a(n+2) - 18*a(n+3)) + a(n+1)*(-14*a(n+1) + 19*a(n+2) - 7*a(n+3)) + a(n+2)*(+a(n+2) + a(n+3)) for all n>=0. - Michael Somos, Jan 17 2015
0 = a(n)^2*(+368*a(n+1) - 182*a(n+2)) + a(n)*a(n+1)*(-306*a(n+1) + 317*a(n+2)) + a(n)*a(n+2)*(-77*a(n+2)) + a(n+1)^2*(-14*a(n+1) - 6*a(n+2)) + a(n+1)*a(n+2)*(+8*a(n+2)) for all n>=0. - Michael Somos, Jan 17 2015
E.g.f.: (BesselI(0,2*x) - (x - 1)*BesselI(1,2*x)/x)*exp(2*x). - Ilya Gutkovskiy, Jun 08 2016
G.f. with 1 prepended: Let E(x) = exp( Sum_{n >= 1} binomial(5*n,2*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/5) = ( x/series reversion of x*D(x)^5 )^(1/5), where D(x) = 1 + 2*x + 23*x^2 + 371*x^3 + ... is the o.g.f. for A060941 .... Cf. A274052 and A274244. - Peter Bala, Jan 01 2020

More terms from Joe Keane (jgk(AT)jgk.org), Feb 08 2000

Asymptotic series corrected and extended by Michael Somos, Sep 11 2003

A115127 Second (k=2) triangle of numbers related to totally asymmetric exclusion process (case alpha=1, beta=1).

Original entry on oeis.org

3, 6, 7, 10, 16, 19, 15, 30, 47, 56, 21, 50, 95, 146, 174, 28, 77, 170, 311, 471, 561, 36, 112, 280, 586, 1043, 1562, 1859, 45, 156, 434, 1015, 2044, 3564, 5291, 6292, 55, 210, 642, 1652, 3682, 7204, 12363, 18226, 21658, 66, 275, 915, 2562, 6230, 13392, 25623

Offset: 2

Wolfdieter Lang, Jan 13 2006

This is the second floor (k=2) of a pyramid of numbers, called X(1,1,k=2,n,m) with n>=m+1>=2. One could use offset n>=1 and add a zero main diagonal.
The column sequences give for n>=m+1 and m=1..7: A000217, A005581, A024191, A115129, A115130, A115132, A115133.
The diagonal sequences give for M:=n-m=1..3: A071716, A071726, A115134.

			[3];[6,7];[10,16,19];[15,30,47,56];...
Main diagonal (n-m=1) example: a(3,2)= 7 = 5 + 2 because
A115126(3,2)=5 and A115126(2,2)=2.
Subdiagonal (n-m>1) example: a(4,2)= 16 = 9 + 7 because
A115126(4,2)=9 and a(3,2)=7.

W. Lang: First 10 rows.

Row sums give A115128.

a(n,m)= b(n,m) + b(n-1,m) with b(n,m):=A115126(n,m) if n=m+1 (main diagonal), A115126(n,m) + a(n,-1,m) if n>m+1 (subdiagonals) and 0 if n

A167422 Expansion of (1+x)*c(x), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 2, 3, 7, 19, 56, 174, 561, 1859, 6292, 21658, 75582, 266798, 950912, 3417340, 12369285, 45052515, 165002460, 607283490, 2244901890, 8331383610, 31030387440, 115948830660, 434542177290, 1632963760974, 6151850548776

Offset: 0

Paul Barry, Nov 03 2009

Hankel transform is A167423.

Apparently a(n) = A071716(n) if n>1. - R. J. Mathar, Nov 12 2009

G. C. Greubel, Table of n, a(n) for n = 0..500
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2

Cf. A000108, A001450, A005807, A071716, A167423, A182959.

A167422List := proc(m) local A, P, n; A := [1, 2]; P := [1];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), A[-1]]);
A := [op(A), P[-1]] od; A end: A167422List(26); # Peter Luschny, Mar 24 2022

Table[If[n < 2, n + 1, Binomial[2 n, n]/(n + 1) + Binomial[2 (n - 1), n - 1]/n], {n, 0, 25}] (* Michael De Vlieger, Oct 05 2015 *)
CoefficientList[Series[(1 + t)*(1 - Sqrt[1 - 4*t])/(2*t), {t, 0, 50}], t] (* G. C. Greubel, Jun 12 2016 *)

a(n) = if (n<2, n+1, binomial(2*n, n)/(n+1) + binomial(2*(n-1), n-1)/n);
vector(50, n, a(n-1)) \\ Altug Alkan, Oct 04 2015

a(n) = Sum_{k=0..n} A000108(k)*C(1,n-k).
a(0)= 1, a(n) = A005807(n-1) for n>0. - Philippe Deléham, Nov 25 2009
(n+1)*a(n) +(-3*n+1)*a(n-1) +2*(-2*n+5)*a(n-2)=0, for n>2. - R. J. Mathar, Feb 10 2015
-(n+1)*(5*n-6)*a(n) +2*(5*n-1)*(2*n-3)*a(n-1)=0. - R. J. Mathar, Feb 10 2015
The o.g.f. A(x) satisfies [x^n] A(x)^(5*n) = binomial(5*n,2*n) = A001450(n). Cf. A182959. - Peter Bala, Oct 04 2015

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.

A000778 a(n) = Catalan(n) + Catalan(n+1) - 1.

Original entry on oeis.org

1, 2, 6, 18, 55, 173, 560, 1858, 6291, 21657, 75581, 266797, 950911, 3417339, 12369284, 45052514, 165002459, 607283489, 2244901889, 8331383609, 31030387439, 115948830659, 434542177289, 1632963760973, 6151850548775, 23229299473603, 87900903988155

Offset: 0

N. J. A. Sloane

nonn

T. D. Noe, Table of n, a(n) for n = 0..200
J. R. Stembridge, Some combinatorial aspects of reduced words in finite Coxeter groups, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1285-1332.

Cf. A000108.

Equals A005807(n) - 1. Cf. A071716.

Table[CatalanNumber[n] + CatalanNumber[n + 1] - 1, {n, 0, 200}] (* T. D. Noe, Jun 20 2012 *)

D-finite with recurrence (n+2)*a(n) +(-5*n-4)*a(n-1) +(3*n+4)*a(n-2) +(5*n-16)*a(n-3) +2*(-2*n+7)*a(n-4)=0. - R. J. Mathar, Jun 17 2020

A350584 Triangle read by rows, T(n, k) = [x^k] ((2x^3 - 3x^2 - x + 1)/(1 - x)^(n + 2)), for n >= 1 and 0 <= k < n.

Original entry on oeis.org

1, 1, 3, 1, 4, 7, 1, 5, 12, 19, 1, 6, 18, 37, 56, 1, 7, 25, 62, 118, 174, 1, 8, 33, 95, 213, 387, 561, 1, 9, 42, 137, 350, 737, 1298, 1859, 1, 10, 52, 189, 539, 1276, 2574, 4433, 6292, 1, 11, 63, 252, 791, 2067, 4641, 9074, 15366, 21658

Offset: 1

Peter Luschny, Mar 27 2022

			Triangle starts:
[1] [1]
[2] [1,  3]
[3] [1,  4,  7]
[4] [1,  5, 12,  19]
[5] [1,  6, 18,  37,  56]
[6] [1,  7, 25,  62, 118,  174]
[7] [1,  8, 33,  95, 213,  387,  561]
[8] [1,  9, 42, 137, 350,  737, 1298, 1859]
[9] [1, 10, 52, 189, 539, 1276, 2574, 4433, 6292]

A280891 (row sums), A135339 (alternating row sums), A005807 or A071716 (main diagonal).

# Compare the analogue algorithm for the Bell triangle in A046937.
A350584Triangle := proc(len) local A, P, T, n; A := [2]; P := [1]; T := [[1]];
for n from 1 to len-1 do P := ListTools:-PartialSums([op(P), A[-1]]);
A := P; T := [op(T), P] od; T end:
A350584Triangle(10): ListTools:-Flatten(%);
# Alternative:
ogf := n -> (2*x^3 - 3*x^2 - x + 1)/(1 - x)^(n + 2):
ser := n -> series(ogf(n), x, n):
row := n -> seq(coeff(ser(n), x, k), k = 0..n-1):
seq(row(n), n = 1..10);

A071717 Expansion of (1 + x^2C)C^2, where C = (1 - sqrt(1-4x))/(2x) is g.f. for Catalan numbers, A000108.

Original entry on oeis.org

1, 2, 6, 17, 51, 160, 519, 1727, 5863, 20228, 70720, 250002, 892126, 3209328, 11626385, 42378075, 155307615, 571925820, 2115257100, 7853744910, 29263124250, 109384710240, 410075910270, 1541481197334, 5808790935126

Offset: 0

N. J. A. Sloane, Jun 06 2002

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Stoyan Dimitrov, On permutation patterns with constrained gap sizes, arXiv:2002.12322 [math.CO], 2020.

Cf. A000108, A071716, A071718, A071719.

seq(coeff(series( ((1-x-3*x^2) -(1+x-x^2)*sqrt(1-4*x))/(2*x^2) , x, n+1), x, n), n = 0..30); # G. C. Greubel, May 30 2020

With[{$MaxExtraPrecision = 1000}, CoefficientList[Series[(1 + x^2*#)*#^2 &[(1 - (1 - 4 x)^(1/2))/(2 x)], {x, 0, 24}], x]] (* Michael De Vlieger, May 30 2020 *)

def A071717_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( ((1-x-3*x^2) -(1+x-x^2)*sqrt(1-4*x))/(2*x^2) ).list()
A071717_list(30) # G. C. Greubel, May 30 2020

Conjecture: (n+2)*a(n) +(-3*n-2)*a(n-1) +(-5*n+8)*a(n-2) +2*(2*n-7)*a(n-3)=0. - R. J. Mathar, Aug 25 2013

G.f.: ( (1 -x -3*x^2) - (1 +x -x^2)*sqrt(1-4*x) )/(2*x^2). - G. C. Greubel, May 30 2020

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cp's OEIS Frontend

A005807 Sum of adjacent Catalan numbers.

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A115127 Second (k=2) triangle of numbers related to totally asymmetric exclusion process (case alpha=1, beta=1).

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A167422 Expansion of (1+x)*c(x), c(x) the g.f. of A000108.

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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 - 4*x))/(2*x).

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A000778 a(n) = Catalan(n) + Catalan(n+1) - 1.

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A350584 Triangle read by rows, T(n, k) = [x^k] ((2*x^3 - 3*x^2 - x + 1)/(1 - x)^(n + 2)), for n >= 1 and 0 <= k < n.

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A071717 Expansion of (1 + x^2*C)*C^2, where C = (1 - sqrt(1-4*x))/(2*x) is g.f. for Catalan numbers, A000108.

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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).

A350584 Triangle read by rows, T(n, k) = [x^k] ((2x^3 - 3x^2 - x + 1)/(1 - x)^(n + 2)), for n >= 1 and 0 <= k < n.

A071717 Expansion of (1 + x^2C)C^2, where C = (1 - sqrt(1-4x))/(2x) is g.f. for Catalan numbers, A000108.