|
| |
|
|
A302646
|
|
Values of unimodal polynomial analogous to A302612 and A302644 arising from a partition size <= 5 restriction.
|
|
5
|
|
|
|
0, 1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 183755, 683046, 2443168, 8263360, 26184420, 77558760, 215182923, 561542454, 1385168400, 3245959640, 7260395142, 15567955260, 32124894880, 64016082000, 123566718600, 231661933776, 422854091037, 753068219386
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
Consider the unimodal polynomial from O'Hara's proof of unimodality of q-binomials after making the restriction to partitions of size <=5. See G_5(n,k) from arXiv:1711.11252. If we make the simplification k=n and take the limit as q->1^-, we obtain the listed polynomial.
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.
|
|
|
LINKS
|
Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).
|
|
|
FORMULA
|
a(n) = n*(n+3)*(n+2)*(n+1)*(n^10-50*n^9+1140*n^8-15420*n^7+136533*n^6-824370*n^5+3436190*n^4-9762880*n^3+18198936*n^2-20242080*n+10886400)/43545600.
G.f.: x*(1 - 13*x + 81*x^2 - 315*x^3 + 855*x^4 - 1701*x^5 + 2583*x^6 - 2961*x^7 + 2835*x^8 - 1365*x^9 + 2002*x^10) / (1 - x)^15.
a(n) = 15*a(n-1) - 105*a(n-2) + 455*a(n-3) - 1365*a(n-4) + 3003*a(n-5) - 5005*a(n-6) + 6435*a(n-7) - 6435*a(n-8) + 5005*a(n-9) - 3003*a(n-10) + 1365*a(n-11) - 455*a(n-12) + 105*a(n-13) - 15*a(n-14) + a(n-15) for n>14.
(End)
|
|
|
EXAMPLE
|
For n=6, G_5(6,6)=q^36+q^35+2*q^34+3*q^33+5*q^32+7*q^31+11*q^30+13*q^29+18*q^28+22*q^27+28*q^26+32*q^25+39*q^24+42*q^23+48*q^22+51*q^21+55*q^20+55*q^19+58*q^18+55*q^17+55*q^16+51*q^15+48*q^14+42*q^13+39*q^12+32*q^11+28*q^10+22*q^9+18*q^8+13*q^7+11*q^6+7*q^5+5*q^4+3*q^3+2*q^2+q+1 (using the formula in the referenced paper). Then substituting q=1 yields 924.
|
|
|
PROG
|
(PARI) concat(0, Vec(x*(1 - 13*x + 81*x^2 - 315*x^3 + 855*x^4 - 1701*x^5 + 2583*x^6 - 2961*x^7 + 2835*x^8 - 1365*x^9 + 2002*x^10) / (1 - x)^15 + O(x^40))) \\ Colin Barker, Apr 19 2018
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
0 prepended to the sequence and formulas adjusted accordingly by Colin Barker, Apr 19 2018
|
|
|
STATUS
|
approved
|
| |
|
|