A002057 -id:A002057 - cp's OEIS frontend

A236843 Triangle read by rows related to the Catalan transform of the Fibonacci numbers.

1, 1, 1, 2, 3, 1, 5, 9, 4, 1, 14, 28, 14, 6, 1, 42, 90, 48, 27, 7, 1, 132, 297, 165, 110, 35, 9, 1, 429, 1001, 572, 429, 154, 54, 10, 1, 1430, 3432, 2002, 1638, 637, 273, 65, 12, 1, 4862, 11934, 7072, 6188, 2548, 1260, 350, 90, 13, 1, 16796, 41990, 25194, 23256, 9996, 5508, 1700, 544, 104, 15, 1

Offset: 0

Philippe Deléham, Feb 01 2014

Row sums are A109262(n+1).

			Triangle begins:
    1;
    1,   1;
    2,   3,   1;
    5,   9,   4,   1;
   14,  28,  14,   6,  1;
   42,  90,  48,  27,  7, 1;
  132, 297, 165, 110, 35, 9, 1;
Production matrix is:
  1...1
  1...2...1
  0...1...1...1
  0...1...1...2...1
  0...0...0...1...1...1
  0...0...0...1...1...2...1
  0...0...0...0...0...1...1...1
  0...0...0...0...0...1...1...2...1
  0...0...0...0...0...0...0...1...1...1
  0...0...0...0...0...0...0...1...1...2...1
  0...0...0...0...0...0...0...0...0...1...1...1
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Columns: A000108 (k=0), A000245 (k=1), A002057 (k=2), A003517 (k=3), A000588 (k=4), A001392 (k=5), A003519 (k=6), A090749 (k=7), A000590 (k=8).
Cf. A026674, A026726, A032766, A109262.
Cf. A000045, A039599, A106566.

F:=Factorial;
A236843:= func< n,k | (1/4)*(6*k+5-(-1)^k)*F(2*n-Floor(k/2))/(F(n-k)*F(n+Floor((k+1)/2)+1)) >;
[A236843(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 13 2022

T[n_, k_]:= (1/4)*(6*k+5-(-1)^k)*(2*n-Floor[k/2])!/((n-k)!*(n+Floor[(k+1)/2]+1)!);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 13 2022 *)

T(n, k) = (1/4)*(6*k + 5 - (-1)^k)*(2*n - (k\2))!/((n-k)!*(n + (k+1)\2 + 1)!) \\ Andrew Howroyd, Jan 04 2023

F=factorial
def A236843(n,k): return (1/2)*(3*k+2+(k%2))*F(2*n-(k//2))/(F(n-k)*F(n+((k+1)//2)+1))
flatten([[A236843(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 13 2022

G.f. for the column k (with zeros omitted): C(x)^A032766(k+1) where C(x) is g.f. for Catalan numbers (A000108).
Sum_{k=0..n} T(n,k) = A109262(n+1).
Sum_{k=0..n} T(n+k,2k) = A026726(n).
Sum_{k=0..n} T(n+1+k,2k+1) = A026674(n+1).
T(n, k) = (1/4)*(6*k + 5 - (-1)^k)*(2*n - floor(k/2))!/((n-k)!*(n + floor((k+1)/2) + 1)!). - G. C. Greubel, Jun 13 2022

A279004 Irregular triangle read by rows: generalized Catalan triangle C_3(n,k).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 1, 3, 6, 9, 9, 1, 4, 10, 19, 28, 28, 1, 5, 15, 34, 62, 90, 90, 1, 6, 21, 55, 117, 207, 297, 297, 1, 7, 28, 83, 200, 407, 704, 1001, 1001, 1, 8, 36, 119, 319, 726, 1430, 2431, 3432, 3432

Offset: 0

N. J. A. Sloane, Dec 07 2016

The main diagonal is A000245, the third convolution of the Catalan numbers. See Tedford 2011. Also see A002057 for a similarly constructed triangle related to the fourth convolution of the Catalan numbers. - Peter Bala, Apr 14 2017

			Triangle begins:
1,1,1,
1,2,3,3,
1,3,6,9,9,
1,4,10,19,28,28,
1,5,15,34,62,90,90,
1,6,21,55,117,207,297,297,
1,7,28,83,200,407,704,1001,1001,
1,8,36,119,319,726,1430,2431,3432,3432,
...

Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
S. J. Tedford, Combinatorial interpretations of convolutions of the Catalan numbers, Integers 11 (2011) #A3

Cf. A009766, A000108, A000245, A002057.

c[m_][0, k_] /; k <= m-1 = 1;
c[m_][n_, k_] /; 0 <= k <= m+n-1 := c[m][n, k] = c[m][n-1, k]+c[m][n, k-1];
c[][, _] = 0;
Table[c[3][n, k], {n, 0, 7}, {k, 0, n+2}] // Flatten (* Jean-François Alcover, Oct 07 2018 *)

A360143 a(n) = Sum_{k=0..n} binomial(2n+2k,n-k).

Original entry on oeis.org

1, 3, 13, 59, 271, 1250, 5775, 26696, 123423, 570576, 2637306, 12187755, 56312089, 260134905, 1201493926, 5548533913, 25619837773, 118283258215, 546041467522, 2520515546083, 11633752319476, 53693477980816, 247798435809211, 1143547904185879, 5277058908767419

Offset: 0

Seiichi Manyama, Jan 27 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A001700, A032443, A108080, A360144.

Cf. A000108, A000984, A002057.

A360143 := proc(n)
    add(binomial(2*n+2*k,n-k),k=0..n) ;
end proc:
seq(A360143(n),n=0..70) ;# R. J. Mathar, Mar 12 2023

Table[Sum[Binomial[2n+2k,n-k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Jul 23 2025 *)

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

my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x*(2/(1+sqrt(1-4*x)))^4)))

G.f.: 1 / ( sqrt(1-4*x) * (1 - x * c(x)^4) ), where c(x) is the g.f. of A000108.
D-finite with recurrence +n*(n-7)*a(n) -(7*n-4)*(n-7)*a(n-1) +4*(n^2-13*n+17)*a(n-2) +(35*n^2-217*n+304)*a(n-3) -2*(n-2)*(7*n-29)*a(n-4) +4*(n-2)*(2*n-9)*a(n-5)=0. - R. J. Mathar, Mar 12 2023
a(n) = binomial(2*n, n)*hypergeom([1, -n, 1/2+n, 1+n], [(1+n)/3, (2+n)/3, 1+n/3], -4/27). - Stefano Spezia, Jun 17 2025

A000908 Atom-rooted polyenoids with n edges with symmetry class C_s.

Original entry on oeis.org

0, 0, 1, 4, 14, 47, 164, 565, 1982, 6977, 24850, 89082, 321855, 1169853, 4276923, 15713799, 57998270, 214934984, 799473752, 2983682702, 11169374372, 41929478873, 157807392886, 595340271682, 2250901007539, 8527699269192, 32369066434276

Offset: 0

E. K. Lloyd (E.K.Lloyd(AT)soton.ac.uk)

nonn

B. N. Cyvin, E. Brendsdal, J. Brunvoll and S. J. Cyvin, Isomers of polyenes attached to benzene, Croatica Chemica Acta, 68 (1995), 63-73, C(x).
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]

Cf. A000912, A000913, A000935, A000936, A000941, A000942, A000947, A000948, A000953, A003446, A063786.

U0 := (1-sqrt(1-4*x))/2/x ;
V0 := 1+x*subs(x=x^2,U0) ;
C := ( subs(x=x^2,U0)^3 -3*subs(x=x^4,U0)*subs(x=x^2,V0) -subs(x=x^6,U0) +3*subs(x=x^6,V0) )/6 ; # (19)
taylor(%,x=0,60) ;
L := gfun[seriestolist](%) ;
seq(op(2*i+1,L),i=0..(nops(L)-1)/2) ; # R. J. Mathar, Jul 26 2019

u0[x_] := (1 - Sqrt[1 - 4 x])/(2 x); v0[x_] := 1 + x u0[x^2];
gf = Simplify[(u0[x]^3 - 3 u0[x^2] v0[x] - u0[x^3] + 3 v0[x^3])/6]
CoefficientList[gf + O[x]^30, x] (* Andrey Zabolotskiy, Feb 08 2023 *)

a(n) = A003446(n+1) - u((n-3)/6) - (u(n/3) - u((n-3)/6))/2 - (u(n/2) + (u((n+1)/2) - u((n-3)/6))) for n > 0 where u(n) = binomial(2*n, n)/(n+1) if n is an integer and 0 otherwise. - Sean A. Irvine, Oct 05 2015

More terms from Sean A. Irvine, Oct 05 2015

A000935 Number of free planar polyenoids with 2n nodes and symmetry point group C_{2h}.

Original entry on oeis.org

0, 1, 2, 7, 20, 63, 191, 598, 1870, 5906

Offset: 1

E. K. Lloyd (E.K.Lloyd(AT)soton.ac.uk)

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]

a(8)-a(10) and improved title by Sean A. Irvine, Oct 15 2015

A000947 Number of free nonplanar polyenoids with n nodes and symmetry point group C_{2v}.

Original entry on oeis.org

1, 2, 4, 10, 15, 44, 56, 177, 212, 706, 792, 2714, 2961

Offset: 7

E. K. Lloyd (E.K.Lloyd(AT)soton.ac.uk)

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]

a(n) = A000063(n + 2) - A000936(n). - Sean A. Irvine, Oct 15 2015

a(16)-a(19) and title improved by Sean A. Irvine, Oct 16 2015

A000948 Number of free nonplanar polyenoids with n nodes and symmetry point group C_s.

Original entry on oeis.org

0, 3, 20, 99, 450, 1896, 7771, 30895, 121144, 468409, 1796584, 6841014, 25925062

Offset: 7

E. K. Lloyd (E.K.Lloyd(AT)soton.ac.uk)

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]

a(n) = A000131(n+2) - A000941(n). - Sean A. Irvine, Oct 15 2015

a(16)-a(19) and title improved by Sean A. Irvine, Oct 16 2015

A000953 Number of free nonplanar polyenoids with n nodes.

Original entry on oeis.org

1, 5, 24, 109, 465, 1943, 7827, 31095, 121356, 469235, 1797376, 6844290, 25928036

Offset: 7

E. K. Lloyd (E.K.Lloyd(AT)soton.ac.uk)

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]

a(n) = A000207(n) - A000942(n). - Sean A. Irvine, Oct 15 2015

a(16)-a(19) from Sean A. Irvine, Oct 15 2015

A050145 T(n,k)=M(2n,n-1,k-1), 0<=k<=n, n >= 0, array M as in A050144.

Original entry on oeis.org

0, 1, 0, 2, 1, 1, 5, 4, 5, 1, 14, 14, 20, 7, 1, 42, 48, 75, 35, 9, 1, 132, 165, 275, 154, 54, 11, 1, 429, 572, 1001, 637, 273, 77, 13, 1, 1430, 2002, 3640, 2548, 1260, 440, 104, 15, 1, 4862, 7072, 13260, 9996, 5508, 2244, 663, 135, 17, 1

Offset: 0

Clark Kimberling

First 7 columns of T are A000108, A002057, A000344, A000588, A001392, A000589, A000590.

			Rows: {0}; {1,0}; {2,1,1}; ...

A099364 An inverse Chebyshev transform of (1-x)^2.

Original entry on oeis.org

1, -2, 2, -4, 5, -10, 14, -28, 42, -84, 132, -264, 429, -858, 1430, -2860, 4862, -9724, 16796, -33592, 58786, -117572, 208012, -416024, 742900, -1485800, 2674440, -5348880, 9694845, -19389690, 35357670, -70715340, 129644790, -259289580, 477638700, -955277400, 1767263190, -3534526380

Offset: 0

Paul Barry, Oct 13 2004

Second binomial transform of the expansion of c(-x)^4 (i.e. of (-1)^n*4C(2n+3,n)/(n+4)). The g.f. is transformed to (1-x)^2 under the Chebyshev transformation A(x)->(1/(1+x^2))A(x/(1+x^2)).

G. Levy, Solution of second order recurrence equations (2010) PhD Thesis, Florida State University, page 2

Cf. A089408, A002057.

G.f.: (c(x^2)-1)(1-2x)/x^2 with c(x) the g.f. of A000108; a(n)=sum{k=0..n, (k+1)C(n, (n-k)/2)(-1)^k*C(2, k)(1+(-1)^(n-k))/(n+k+2)}; a(n)=sum{k=0..n, (k+1)C(n, (n-k)/2)b(k)(1+(-1)^(n-k))/(n+k+2)} where b(n)=0^n+sum{k=0..n, C(n, k)(-1)^(n-k)(-3k+k(k+1)/2)}; a(2n)=C(n+1); a(2n+1)=-2*C(n+1).

D-finite with recurrence: (n+4)*a(n) +2*a(n-1) -4*n*a(n-2)=0. - R. J. Mathar, Nov 09 2012

cp's OEIS Frontend

A236843 Triangle read by rows related to the Catalan transform of the Fibonacci numbers.

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A279004 Irregular triangle read by rows: generalized Catalan triangle C_3(n,k).

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A360143 a(n) = Sum_{k=0..n} binomial(2*n+2*k,n-k).

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A000908 Atom-rooted polyenoids with n edges with symmetry class C_s.

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A000935 Number of free planar polyenoids with 2n nodes and symmetry point group C_{2h}.

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A000947 Number of free nonplanar polyenoids with n nodes and symmetry point group C_{2v}.

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A000948 Number of free nonplanar polyenoids with n nodes and symmetry point group C_s.

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A000953 Number of free nonplanar polyenoids with n nodes.

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A050145 T(n,k)=M(2n,n-1,k-1), 0<=k<=n, n >= 0, array M as in A050144.

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A360143 a(n) = Sum_{k=0..n} binomial(2n+2k,n-k).