A153648 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-30 - 120\,x + 15\,n + 60\,n x\right)\,P_{n-2}(x) + \left(-4 - 8\,x + 2\,n + 4\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 10 + 20\,x\)

\(P_{2}(x) = 50 + 200\,x + 32\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	10	20														30
2	50	200	32													282
3	250	1500	2064	128												3942
4	1250	10000	22608	10432	512											44802
5	6250	62500	206176	237056	50432	2048										564462
6	31250	375000	1601472	2811776	1682688	236544	8192									6746922
7	156250	2187500	11545584	27955840	29592320	10260480	1085440	32768								82816182
8	781250	12500000	78363248	240358976	364954880	243399680	57528320	4898816	131072							1002916242
9	3906250	70312500	511160256	1906243584	3812580864	3876065280	1720565760	305528832	21823488	524288						12228711102
10	19531250	390625000	3229019232	14164307712	35145997824	48550742016	34412298240	11018502144	1561657344	96206848	2097152					148590984762
11	97656250	2148437500	19906692304	100507961472	298512554496	524581306368	521399967744	267503075328	65856405504	7756316672	420478976	8388608				1808699241222
12	488281250	11718750000	120279923088	686848461760	2376105384960	5080233830400	6570354155520	4850621448192	1886260101120	374080798720	37670092800	1824522240	33554432			21996519304482
13	2441406250	63476562500	714941480736	4556837576192	18007770622720	45444362065920	72757521530880	71279203713024	40213036597248	12360931082240	2043349565440	179683983360	7868514304	134217728		267631558918542
14	12207031250	341796875000	4191113057792	29492963387008	131095380030208	381689637176320	730661833482240	900569064603648	684874076454912	304360826601472	76540322775040	10820339630080	844447612928	33755758592	536870912	3255528301347402

Technical details

Node id: n-e1680309668a7c43 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-5e144e91eeb503a1

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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:       2;
  1:      10,       20;
  2:      50,      200,        32;
  3:     250,     1500,      2064,        128;
  4:    1250,    10000,     22608,      10432,        512;
  5:    6250,    62500,    206176,     237056,      50432,       2048;
  6:   31250,   375000,   1601472,    2811776,    1682688,     236544,       8192;
  7:  156250,  2187500,  11545584,   27955840,   29592320,   10260480,    1085440,     32768;
  8:  781250, 12500000,  78363248,  240358976,  364954880,  243399680,   57528320,   4898816,   131072;
  9: 3906250, 70312500, 511160256, 1906243584, 3812580864, 3876065280, 1720565760, 305528832, 21823488, 524288;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (15*n-30+(60*n-120)*x)*P(n-2,x) + (2*n-4+(4*n-8)*x)*P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 10 + 20*x, P(2,x) = 50 + 200*x + 32*x^2.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 10 + 20*x;
P[2, x_] := 50 + 200*x + 32*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((15*n - 30 + (60*n - 120)*x)*P[n - 2, x] + (2*n - 4 + (4*n - 8)*x)*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 10 + 20*x; v[3] = 50 + 200*x + 32*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((15*n - 30 + (60*n - 120)*x)*v[i - 2] + (2*n - 4 + (4*n - 8)*x)*v[i - 1])/(n - 2)); 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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + j*prime(j)*T(n-2, k-1) with j=3, read by rows.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? 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