A013702 -id:A013702 - cp's OEIS frontend

A013698 a(n) = binomial(3*n+2, n-1).

1, 8, 55, 364, 2380, 15504, 100947, 657800, 4292145, 28048800, 183579396, 1203322288, 7898654920, 51915526432, 341643774795, 2250829575120, 14844575908435, 97997533741800, 647520696018735, 4282083008118300

Offset: 1

Joachim.Rosenthal(AT)nd.edu (Joachim Rosenthal), Emeric Deutsch

Degree of variety K_{2,n}^1. Also number of double-rises (or odd-level peaks) in all generalized {(1,2),(1,-1)}-Dyck paths of length 3(n+1).
Number of dissections of a convex (2n+2)-gon by n-2 noncrossing diagonals into (2j+2)-gons, 1<=j<=n-1.
a(n) is the number of lattice paths from (0,0) to (3n+1,n-1) avoiding two consecutive up-steps. - Shanzhen Gao, Apr 20 2010

Robert Israel, Table of n, a(n) for n = 1..1000
M. S. Ravi et al., Dynamic Pole Assignment and Schubert Calculus, SIAM J. Control Optimization, 34 (1996), 813-832, esp. p. 825.
Daniel W. Stasiuk, An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads, Master's Thesis, University of Saskatchewan-Saskatoon (2018).

Cf. A013699 (q=2), A013700 (q=3), A013701 (q=4), A013702 (q=5).

A column of triangle A102537. Cf. A001764, A004319, A005809, A006013, A025174, A045721, A117671, A165817, A236194.

List([1..25], n-> Binomial(3*n+2, n-1)) # G. C. Greubel, Mar 21 2019

[Binomial(3*n+2, n-1): n in [1..25]]; // Vincenzo Librandi, Aug 10 2015

seq(binomial(3*n+2,n-1), n=0..30); # Robert Israel, Aug 09 2015

Table[Binomial[3*n+2, n-1], {n, 25}] (* Arkadiusz Wesolowski, Apr 02 2012 *)

first(m)=vector(m,n,binomial(3*n+2, n-1)); /* Anders Hellström, Aug 09 2015 */

[binomial(3*n+2, n-1) for n in (1..25)] # G. C. Greubel, Mar 21 2019

G.f.: g/((g-1)^3*(3*g-1)) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 09 2011
a(n) = Sum_{k=0..n-1} binomial(2*n+k+2,k). - Arkadiusz Wesolowski, Apr 02 2012
D-finite with recurrence 2*(2*n+3)*(n+1)*a(n) -n*(67*n+34)*a(n-1) +30*(3*n-1)*(3*n-2)*a(n-2)=0. - R. J. Mathar, Feb 05 2013
a(n+1) = (3*n+5)*(3*n+4)*(3*n+3)*a(n)/((2*n+5)*(2*n+4)*n). - Robert Israel, Aug 09 2015
With offset 0, the o.g.f. equals f(x)*g(x)^5, where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k,n). Cf. A045721 (k = 1), A025174 (k = 2), A004319 (k = 3), A236194 (k = 4), A165817 (k = -1), A117671 (k = -2). - Peter Bala, Nov 04 2015

A013699 Degree of variety K_{2,n}^2.

Original entry on oeis.org

1, 32, 610, 9842, 147798, 2145600, 30664890, 435668420, 6186432967, 88066807556, 1258885297696, 18084694597452, 261164661944060, 3791317346771584, 55316720239735242, 810944384733610356

Offset: 1

Joachim.Rosenthal(AT)nd.edu (Joachim Rosenthal)

Number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 4n+4 steps with all values less than or equal to n+1 (see A080934).

M. S. Ravi et al., Dynamic pole assignment and Schubert calculus, SIAM J. Control Optimization, 34 (1996), 813-832, esp. p. 825.

Cf. A013698 (q=1), A013700 (q=3), A013701 (q=4), A013702 (q=5).

K(n,q=2)=(2*n+n*q+2*q)!*sum(j=0,q,((q-2*j)*(n+2)+1)/(n+j*(n+2))!/(n+1+(q-j)*(n+2))!)

A013700 Degree of variety K_{2,n}^3.

Original entry on oeis.org

1, 128, 6765, 265720, 9112264, 290926848, 8916942687, 266668876540, 7853149169635, 228982270335000, 6632994268595136, 191292945772217856, 5500214758962096400, 157819424038439232000, 4521902974531722618723

Offset: 1

Joachim.Rosenthal(AT)nd.edu (Joachim Rosenthal)

M. S. Ravi et al., Dynamic pole assignment and Schubert calculus, SIAM J. Control Optimization, 34 (1996), 813-832, esp. p. 825.

Cf. A013698 (q=1), A013699 (q=2), A013701 (q=4), A013702 (q=5).

K(n,q=3)=(2*n+n*q+2*q)!*sum(j=0,q,((q-2*j)*(n+2)+1)/(n+j*(n+2))!/(n+1+(q-j)*(n+2))!)

Edited by Ralf Stephan, May 13 2003

A013701 Degree of variety K_{2,n}^4.

Original entry on oeis.org

1, 512, 75025, 7174454, 562110290, 39541748736, 2610763825782, 165745451110910, 10262482704258873, 625250747214775916, 37701606156514031251, 2258713106034310399852, 134810129909509070121060

Offset: 1

Joachim.Rosenthal(AT)nd.edu (Joachim Rosenthal)

Number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 6n+8 steps with all values less than or equal to n+1 (see A080934).

M. S. Ravi et al., Dynamic pole assignment and Schubert calculus, SIAM J. Control Optimization, 34 (1996), 813-832, esp. p. 825.

Cf. A013698 (q=1), A013699 (q=2), A013700 (q=3), A013702 (q=5).

K(n,q=4)=(2*n+n*q+2*q)!*sum(j=0,q,((q-2*j)*(n+2)+1)/(n+j*(n+2))!/(n+1+(q-j)*(n+2))!)

A082635 Square array read by antidiagonals: degree of the K(2,p)^q variety.

Original entry on oeis.org

1, 2, 1, 5, 8, 1, 14, 55, 32, 1, 42, 364, 610, 128, 1, 132, 2380, 9842, 6765, 512, 1, 429, 15504, 147798, 265720, 75025, 2048, 1, 1430, 100947, 2145600, 9112264, 7174454, 832040, 8192, 1, 4862, 657800, 30664890, 290926848, 562110290, 193710244

Offset: 1

Ralf Stephan, May 14 2003

Numbers are related to the dynamic pole assignment problem. "The variety K(m,p)^q can also be viewed as the parameterization of the space of rational curves of degree q of the Grassmann variety Grass(m,m+p)".

Also lim(n->inf, T(n+1,2i)/T(n,2i)) = 4^(i+1).

			Top left corner of array:
1,2,5,14,42,132,429,1430,... A000108 (Catalan numbers)
1,8,55,364,2380,15504,100947,...A013068 deg K(2,n)^1
1,32,610,9842,147798,2145600,...A013069 deg K(2,n)^2
1,128,6765,265720,9112264,... A013070 deg K(2,n)^3
1,512,75025,7174454,... A013071 deg K(2,n)^4

M. S. Ravi et al., Dynamic pole assignment and Schubert calculus, SIAM J. Control Optimization, 34 (1996), 813-832, esp. p. 825.

Cf. A013702.
Second column is A004171(q), third is A000045(5q).
T(n, 2i) = A080934((i+1)n+2i, n+1).

degK2(p, q)=(-1)^q*(2p+pq+2q)!*sum(j=0, q, ((q-2j)(p+2)+1)/(p+j(p+2))!/(p+1+(q-j)(p+2))!).

cp's OEIS Frontend

A013698 a(n) = binomial(3*n+2, n-1).

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A013699 Degree of variety K_{2,n}^2.

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A013700 Degree of variety K_{2,n}^3.

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A013701 Degree of variety K_{2,n}^4.

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A082635 Square array read by antidiagonals: degree of the K(2,p)^q variety.

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