A302644 a(n) = (n+2)*(n+1)*(n^6-12*n^5+70*n^4-210*n^3+409*n^2-378*n+360)/720.

1, 2, 6, 20, 70, 252, 896, 2976, 8955, 24310, 60038, 136500, 289016, 575680, 1087920, 1964384, 3408789, 5712426, 9282070, 14674100, 22635690, 34153988, 50514256, 73368000, 104812175, 147480606, 204648822, 280353556, 379528220, 508155720, 673440032, 883998016

Offset: 0

Author

Bryan T. Ek, Apr 10 2018

Keywords

• nonn
• easy

Comments

The limit as q->1^- of the unimodal polynomial [q^(4k+3)(1-q^n)( q-q^n)-q^(3k)q(1+q)( 1-q^n)( q-q^2+q^5-q^n)-q^(2k)(q^(nk)(q^(2n)(q^(9)- q^(8)-q^(7)+q^(6)+ q^(5)-q^(3)+q)-q^n(q^(10)-q^(8)+q^(6)+q^(5))+q^(10))-q^(2n)+q^n(q^(5)+q^(4)-q^(2)+1)-q^(9)+q^(7)-q^(5)-q^(4)+q^(3)+q^(2)-q)+q^k q^(nk)q^(3) ( 1+q ) ( 1-q^n ) ( q^5-q^n+q^(n+3)-q^(n+4))-q^(nk)q^6(1-q^n)( q-q^n)]/[(1-q)^2(1-q^2)^2(1-q^3)(1-q^(n-1))(1-q^n)q^(2k+1)] after making the simplification k=n. The unimodal polynomial is from O'Hara's proof of unimodality of q-binomials after making the restriction to partitions of size <=3. See G_3(n,k) from arXiv:1711.11252.

As the size restriction s increases, G_s->G_infinity=G: the q-binomials. Then substituting k=n and q=1 yields the central binomial coefficients: A000984.

Examples

			For n=4, G_3(4,4)=q^16+q^15+2*q^14+3*q^13+5*q^12+5*q^11+7*q^10+7*q^9+8*q^8+7*q^7+7*q^6+5*q^5+5*q^4+3*q^3+2*q^2+q+1 (using the formula in the comments). Then substituting q=1 yields 70.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Bryan Ek, q-Binomials and related symmetric unimodal polynomials, arXiv:1711.11252 [math.CO], 2017-2018.
• Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Cf. A000984, A002522, A302612, A302645, A302646.

Programs

PARI

Vec((1 - 7*x + 24*x^2 - 46*x^3 + 64*x^4 - 36*x^5 + 56*x^6) / (1 - x)^9 + O(x^40)) \\ Colin Barker, Apr 11 2018

Formula

a(n) = (n+2)*(n+1)*(n^6-12*n^5+70*n^4-210*n^3+409*n^2-378*n+360)/720.

From Colin Barker, Apr 11 2018: (Start)

G.f.: (1 - 7*x + 24*x^2 - 46*x^3 + 64*x^4 - 36*x^5 + 56*x^6) / (1 - x)^9.

a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>8.

(End)

Extensions

More terms from Colin Barker, Apr 11 2018