A032349 -id:A032349 - cp's OEIS frontend

A370099 a(n) = Sum_{k=0..n} binomial(2n,k) binomial(3*n-k-1,n-k).

1, 4, 32, 292, 2816, 28004, 284000, 2919620, 30316544, 317222212, 3339504032, 35329425124, 375282559232, 4000059761572, 42760427177696, 458259268924292, 4921911787962368, 52965710906750084, 570951048018417440, 6164049197776406180, 66639047280436354816

Offset: 0

Seiichi Manyama, Feb 10 2024

Cf. A370098, A370102.

Cf. A032349, A103885.

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

a(n) = [x^n] ( (1+x)^2/(1-x)^2 )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x*(1-x)^2/(1+x)^2 ).
a(n) = 2 * A103885(n) for n >= 1. - Peter Bala, Sep 16 2024
From Seiichi Manyama, Aug 09 2025: (Start)
a(n) = [x^n] (1-x)^(n-1)/(1-2*x)^(2*n).
a(n) = Sum_{k=0..n} 2^k * binomial(2*n,k) * binomial(n-1,n-k).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(2*n+k-1,k) * binomial(n-1,n-k). (End)

A379244 G.f. A(x) satisfies A(x) = ( (1 + xA(x)^3)/(1 - xA(x)) )^2.

Original entry on oeis.org

1, 4, 40, 540, 8400, 141876, 2528760, 46815116, 891483808, 17350187364, 343578992328, 6900588813564, 140230648164720, 2878066866407316, 59571280942854808, 1242093725341221996, 26064579113472078144, 550041399791036747460, 11665771061882347813224, 248527169321049466503132

Offset: 0

Seiichi Manyama, Dec 18 2024

nonn

Cf. A032349, A371675.

Cf. A364167, A379174.

a(n) = sum(k=0, n, binomial(2*n+4*k+2, k)*binomial(3*n+3*k+1, n-k)/(n+2*k+1));

G.f.: B(x)^2 where B(x) is the g.f. of A364167.

a(n) = Sum_{k=0..n} binomial(2*n+4*k+2,k) * binomial(3*n+3*k+1,n-k)/(n+2*k+1).

A108436 Number of returns to the x-axis in all paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1).

Original entry on oeis.org

2, 14, 106, 862, 7378, 65550, 599002, 5594942, 53181730, 512784142, 5003410762, 49312114334, 490192537586, 4909102791694, 49482525122490, 501626536004734, 5111038278845506, 52312236295906830, 537605889306476074

Offset: 1

Emeric Deutsch, Jun 04 2005

nonn

			a(2)=14 because there are 10 paths from (0,0) to (6,0) (see A027307): u(d)u(d), u(d)Ud(d), uud(d), uUdd(d), Ud(d)u(d), Ud(d)Ud(d), Udud(d), UdUdd(d), Uudd(d) and UUddd(d), the fourteen returns to the x-axis being shown between parentheses.

Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.

Cf. A027307, A108435, A032349.

a:=n->2+(1/n)*sum((3*n-j)*2^(n-j)*binomial(n,j)*binomial(2*n,n-j-1)/(n+j+2),j=0..n-2): seq(a(n),n=1..21);

a(n) = Sum_{k=1..n} k*A108435(k).
a(n) = A032349(n+1) - A027307(n).
a(n) = 2 + (1/n)*Sum_{j=0..n-2} (3n-j)*2^(n-j)*binomial(n, j)*binomial(2n, n-j-1)/(n+j+2).
G.f.: A^2-A, where A=1+zA^2+zA^3 or, equivalently, A=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3 (the g.f. of A027307).
D-finite with recurrence 3*(n+1)*(2*n+1)*a(n) +3*(-34*n^2+18*n-1)*a(n-1) +(394*n^2-1197*n+908)*a(n-2) +2*(-4*n^2+51*n-113)*a(n-3) -2*(2*n-7)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 24 2022

A110682 A convolution triangle of numbers based on A027307.

Original entry on oeis.org

1, 2, 1, 10, 4, 1, 66, 24, 6, 1, 498, 172, 42, 8, 1, 4066, 1360, 326, 64, 10, 1, 34970, 11444, 2706, 536, 90, 12, 1, 312066, 100520, 23526, 4672, 810, 120, 14, 1, 2862562, 911068, 211546, 42024, 7410, 1156, 154, 16, 1

Offset: 0

Philippe Deléham, Sep 15 2005

Triangle T(n,k) for A(x)^k = Sum_{n>=k} T(n,k)*x^n, where o.g.f. A(x) satisfies A(x) = (1+x*A(x)^2)/(1-x*A(x)^2). - Vladimir Kruchinin, Mar 16 2011

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Columns: A027307, A032349, A033296.

T[n_, k_] := (k/(2*n - k))*Sum[Binomial[2*n - k, n - k - j]*Binomial[2*n - k + j - 1, 2*n - k - 1], {j, 0, n - k}]; Table[T[n, k], {n, 0, 25}, {k, 1, n}] // Flatten (* G. C. Greubel, Sep 05 2017 *)

for(n=0,25, for(k=1,n, print1((k/(2*n-k))*sum(i=0,n-k, binomial(2*n-k,n-k-i)*binomial(2*n-k+i-1,2*n-k-1)), ", "))) \\ G. C. Greubel, Sep 05 2017

T(0, 0) = 1; T(n, k) = 0 if k<0 or if k>n; T(n, k) = Sum_{j, j>=0} T(n-1, k-1+j)*A006318(j).
Sum_{k, k>=0} T(n, k) = A108442(n+1).
T(n,k) = k/(2*n-k)*Sum_{i=0,n-k} binomial(2*n-k,n-k-i)*binomial(2*n-k+i-1,2*n-k-1), n >= k > 0. - Vladimir Kruchinin, Mar 16 2011

A108438 Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and having abscissa of the first peak equal to k.

Original entry on oeis.org

1, 1, 4, 3, 2, 1, 24, 18, 13, 7, 3, 1, 172, 130, 96, 55, 28, 12, 4, 1, 1360, 1034, 772, 458, 249, 119, 50, 18, 5, 1, 11444, 8738, 6568, 3982, 2244, 1137, 526, 219, 80, 25, 6, 1, 100520, 76994, 58140, 35770, 20624, 10836, 5293, 2383, 981, 365, 119, 33, 7, 1

Offset: 1

Emeric Deutsch, Jun 04 2005

Row n contains 2n terms. Row sums yield A027307. T(n,1)=A032349(n-1).

			T(2,3) = 2 because we have Uuddd and uUddd.
Triangle begins:
1,1;
4,3,2,1;
24,18,13,7,3,1;
172,130,96,55,28,12,4,1;

Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.

Cf. A027307, A032349.

A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: G:=1/(1-t^2*z*A-t*z*A^2)-1: Gserz:=simplify(series(G,z=0,10)): for n from 1 to 8 do P[n]:=sort(coeff(Gserz,z^n)) od: > for n from 1 to 8 do seq(coeff(P[n],t^k),k=1..2*n) od; # yields sequence in triangular form

G.f.: G = G(t,z) = 1/(1-t^2zA-tzA^2)-1, where A=1+zA^2+zA^3=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3 (the g.f. of A027307).

A137842 Number of paths from (0,0) if n is even, or from (2,1) if n is odd, to (3n,0) that stay in first quadrant (but may touch horizontal axis) and where each step is (2,1), (1,2) or (1,-1).

Original entry on oeis.org

1, 1, 2, 4, 10, 24, 66, 172, 498, 1360, 4066, 11444, 34970, 100520, 312066, 911068, 2862562, 8457504, 26824386, 80006116, 255680170, 768464312, 2471150402, 7474561164, 24161357010, 73473471344, 238552980386, 728745517972

Offset: 0

Paul Barry, Feb 13 2008

Row sums of the inverse of the Riordan array (1/(1+x^2),x(1-x^2)/(1+x^2)).

a(n) is the maximum number of distinct sets that can be obtained as complete parenthesizations of “S_1 union S_2 intersect S_3 union S_4 intersect S_5 union ... S_{n+1}”, where the total of n union and intersection operations alternate, starting with a union, and S_1, S_2, ... , S_{n+1} are sets. - Alexander Burstein, Nov 22 2023

Cf. A084078. [From R. J. Mathar, Feb 28 2009]

G.f.: (1+v^2)/(1-v), where v=2*sqrt(x^2+3)*sin(asin(x(x^2+18)/((x^2+3)^(3/2)))/3)/3-x/3; a(2n)=A027307(n); a(2n+1)=A032349(n+1).

A257532 Triangle, read by rows, T(n,k)=k/nSum_{i=0..n-k} C(2n,n-k-i)C(2n+i-1,i).

Original entry on oeis.org

1, 4, 1, 24, 8, 1, 172, 64, 12, 1, 1360, 536, 120, 16, 1, 11444, 4672, 1156, 192, 20, 1, 100520, 42024, 11088, 2096, 280, 24, 1, 911068, 387456, 106908, 22016, 3420, 384, 28, 1, 8457504, 3643448, 1038984, 227408, 39120, 5192, 504, 32, 1

Offset: 1

Vladimir Kruchinin, Apr 28 2015

			1;
4, 1;
24, 8, 1;
172, 64, 12, 1;
1360, 536, 120, 16, 1;

Cf. A027307. First column = A032349.

T(n,k):=(k*sum(binomial(2*n,n-k-i)*binomial(2*n+i-1,i),i,0,n-k))/n;

G.f.: 1/(1-x*B(x)^2*y)-1, where B(x) is g.f. of A027307.

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

cp's OEIS Frontend

A370099 a(n) = Sum_{k=0..n} binomial(2*n,k) * binomial(3*n-k-1,n-k).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A379244 G.f. A(x) satisfies A(x) = ( (1 + x*A(x)^3)/(1 - x*A(x)) )^2.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A108436 Number of returns to the x-axis in all paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A110682 A convolution triangle of numbers based on A027307.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A108438 Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and having abscissa of the first peak equal to k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A137842 Number of paths from (0,0) if n is even, or from (2,1) if n is odd, to (3n,0) that stay in first quadrant (but may touch horizontal axis) and where each step is (2,1), (1,2) or (1,-1).

Views

Author

Keywords

Comments

Crossrefs

Formula

A257532 Triangle, read by rows, T(n,k)=k/n*Sum_{i=0..n-k} C(2*n,n-k-i)*C(2*n+i-1,i).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maxima

Formula

A370099 a(n) = Sum_{k=0..n} binomial(2n,k) binomial(3*n-k-1,n-k).

A379244 G.f. A(x) satisfies A(x) = ( (1 + xA(x)^3)/(1 - xA(x)) )^2.

A257532 Triangle, read by rows, T(n,k)=k/nSum_{i=0..n-k} C(2n,n-k-i)C(2n+i-1,i).