A257614 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 5\,n\right)\,P_{n}(x) = \left(12 - 96\,x - 48\,x^{2} + 30\,n - 180\,n x - 10\,n x^{2} + 150\,n^{2} x + 225\,n^{2} x^{2} - 125\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(30\,x - 30\,x^{3} + 75\,n x - 50\,n x^{2} - 25\,n x^{3} - 125\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-12 + 18\,x - 5\,n - 60\,n x + 50\,n^{2} x\right)\,P_{n-1}(x) + \left(-15\,x + 15\,x^{2} + 25\,n x - 25\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 2 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	4														6
2	4	56	12													72
3	8	488	732	32												1260
4	16	3688	17568	7376	80											28728
5	32	26520	304860	406480	66300	192										804384
6	64	187368	4533480	13820320	7555800	562104	448									26659584
7	128	1315672	61990164	367927520	459909400	123980328	4604852	1024								1019729088
8	256	9219176	807034224	8486413744	20236013600	12729620616	1883079856	36876704	2304							44188260480
9	512	64555736	10196074956	178702493872	731608763060	877930515672	312729364276	27189533216	290500812	5120						2138711807232
10	1024	451938280	126419910840	3540948760320	23243492542200	47408247846288	32540889559080	7081897520640	379259732520	2259691400	11264					114323867513856
11	2048	3163674456	1548900578820	67275643932480	675125716093200	2172572805470256	2534668273048632	1080201145749120	151370198848080	5163001929400	17400209508	24576				6687946249560576
12	4096	22145954664	18833573553408	1240750383124880	18384737060505600	88635047542234992	162218769475112448	118180063389646656	33092526708910080	3101875957812200	69056436362496	132875727984	53248			424941815549002752
13	8192	155022190552	227896361764668	22398550946159248	477358300339709900	3319737031489827024	9017080175443408824	10305234486221038656	4979605547234740536	954716600679419800	61596015101937932	911585447058672	1007644238588	114688		29138867351931617280
14	16384	1085156431592	2749173404897352	398095489578821984	11957788418973968920	116555907989384968008	451213680157190606544	762587351980356860544	580131874487816494128	194259846648974946680	26307134521742731624	1194286468736465952	11913084754555192	7596095021144	245760	2144620637102167031808

Technical details

Node id: n-58613cbd6cc60a77 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-598bdb4e63e884bb

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:   1;
  1:   2,        4;
  2:   4,       56,          12;
  3:   8,      488,         732,           32;
  4:  16,     3688,       17568,         7376,           80;
  5:  32,    26520,      304860,       406480,        66300,          192;
  6:  64,   187368,     4533480,     13820320,      7555800,       562104,          448;
  7: 128,  1315672,    61990164,    367927520,    459909400,    123980328,      4604852,        1024;
  8: 256,  9219176,   807034224,   8486413744,  20236013600,  12729620616,   1883079856,    36876704,      2304;
  9: 512, 64555736, 10196074956, 178702493872, 731608763060, 877930515672, 312729364276, 27189533216, 290500812, 5120;

%F Formula

The row polynomials satisfy (5*n-3)*P(n,x) = (30*n+12+(150*n^2-180*n-96)*x-(125*n^3-225*n^2+10*n+48)*x^2)*P(n-2,x) + ((75*n+30)*x-(125*n^2+50*n)*x^2+(125*n^2-25*n-30)*x^3)*d/dx P(n-2,x) - (5*n+12-(50*n^2-60*n+18)*x)*P(n-1,x) + ((25*n-15)*x-(25*n-15)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((30*n + 12 + (150*n^2 - 180*n - 96)*x - (125*n^3 - 225*n^2 + 10*n + 48)*x^2)*P[n - 2, x] + ((75*n + 30)*x - (125*n^2 + 50*n)*x^2 + (125*n^2 - 25*n - 30)*x^3)*D[P[n - 2, x], x] - (5*n + 12 - (50*n^2 - 60*n + 18)*x)*P[n - 1, x] + ((25*n - 15)*x - (25*n - 15)*x^2)*D[P[n - 1, x], x])/(5*n - 3)]];
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] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((30*n + 12 + (150*n^2 - 180*n - 96)*x - (125*n^3 - 225*n^2 + 10*n + 48)*x^2)*v[i - 2] + ((75*n + 30)*x - (125*n^2 + 50*n)*x^2 + (125*n^2 - 25*n - 30)*x^3)*deriv(v[i - 2], x) - (5*n + 12 - (50*n^2 - 60*n + 18)*x)*v[i - 1] + ((25*n - 15)*x - (25*n - 15)*x^2)*deriv(v[i - 1], x))/(5*n - 3)); 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 Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
    
    ↑ Liu–Wang theorem (above)

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.