A316649 → basis A013611

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(-2\,n + n^{2}\right)\,P_{n}(x) = \left(-8 - 32\,x - 32\,x^{2} + 12\,n + 48\,n x + 48\,n x^{2} - 4\,n^{2} - 16\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(12 + 56\,x + 64\,x^{2} - 30\,n - 140\,n x - 160\,n x^{2} + 12\,n^{2} + 56\,n^{2} x + 64\,n^{2} x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 3 + 20\,x + 32\,x^{2}\)

\(P_{2}(x) = 26 + 300\,x + 1264\,x^{2} + 2304\,x^{3} + 1536\,x^{4}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	20	32													55
2	26	300	1264	2304	1536											5430
3	252	4128	27696	97472	189952	194560	81920									595980
4	2568	54480	499488	2586048	8274176	16762880	21012480	14909440	4587520							68689080
5	26928	701376	8143488	55522304	246258432	742712320	1543192576	2182053888	2010251264	1089994752	264241152					8143098480
6	287648	8881600	124736512	1053918720	5968017920	23867730944	69144309760	146238898176	224164577280	242936709120	176733290496	77510737920	15502147584			983254243680
7	3112896	111150080	1830950144	18446121984	127000011776	632239382528	2347403980800	6604617531392	14153422733312	22993104273408	27880379645952	24472631377920	14702679687168	5412464099328	921270484992	120267604543680

All 15 rows

Technical details

Node id: n-7f43c706f4df7398 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-dd4f339d9dffb950

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:         1;
  1:         3,          20,           32;
  2:        26,         300,         1264,          2304,           1536;
  3:       252,        4128,        27696,         97472,         189952,          194560,            81920;
  4:      2568,       54480,       499488,       2586048,        8274176,        16762880,         21012480,         14909440,           4587520;
  5:     26928,      701376,      8143488,      55522304,      246258432,       742712320,       1543192576,       2182053888,        2010251264,        1089994752,          264241152;
  6:    287648,     8881600,    124736512,    1053918720,     5968017920,     23867730944,      69144309760,     146238898176,      224164577280,      242936709120,       176733290496,        77510737920,        15502147584;
  7:   3112896,   111150080,   1830950144,   18446121984,   127000011776,    632239382528,    2347403980800,    6604617531392,    14153422733312,    22993104273408,     27880379645952,     24472631377920,     14702679687168,      5412464099328,       921270484992;
  8:  34013312,  1378752768,  26055394816,  304753390592,  2469639309312,  14705116819456,   66561157505024,  233661014458368,   643012724064256,  1391968241319936,   2362839624843264,   3112476623241216,   3119427398664192,   2299836606971904,   1176462409334784,    373114546421760,    55276229099520;
  9: 374416128, 16985318400, 362135973888, 4822043394048, 44942276960256, 311378589548544, 1662228653998080, 6991486209490944, 23491160237604864, 63530400071548928, 138637613209747456, 243568896680919040, 341992616897806336, 378603380005666816, 323231741831544832, 205372687416033280, 91457591946772480, 25476199811645440, 3341140958904320;

%F Formula

The row polynomials satisfy (n^2-2*n)*P(n,x) = -(4*n^2-12*n+8+(16*n^2-48*n+32)*x+(16*n^2-48*n+32)*x^2)*P(n-2,x) + (12*n^2-30*n+12+(56*n^2-140*n+56)*x+(64*n^2-160*n+64)*x^2)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 20*x + 32*x^2, P(2,x) = 26 + 300*x + 1264*x^2 + 2304*x^3 + 1536*x^4.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 20*x + 32*x^2;
P[2, x_] := 26 + 300*x + 1264*x^2 + 2304*x^3 + 1536*x^4;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 12*n + 8 + (16*n^2 - 48*n + 32)*x + (16*n^2 - 48*n + 32)*x^2)*P[n - 2, x] + (12*n^2 - 30*n + 12 + (56*n^2 - 140*n + 56)*x + (64*n^2 - 160*n + 64)*x^2)*P[n - 1, x])/(n^2 - 2*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 20*x + 32*x^2; v[3] = 26 + 300*x + 1264*x^2 + 2304*x^3 + 1536*x^4; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 - 12*n + 8 + (16*n^2 - 48*n + 32)*x + (16*n^2 - 48*n + 32)*x^2)*v[i - 2] + (12*n^2 - 30*n + 12 + (56*n^2 - 140*n + 56)*x + (64*n^2 - 160*n + 64)*x^2)*v[i - 1])/(n^2 - 2*n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows in which T(n,k) is the number of length k chains from (0,0) to (n,n) of the poset [n] X [n] order…?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003