A257612 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(4 - 24\,x - 12\,x^{2} + 8\,n - 32\,n x + 8\,n x^{2} + 32\,n^{2} x + 48\,n^{2} x^{2} - 32\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(8\,x - 8\,x^{3} + 16\,n x - 16\,n x^{2} - 32\,n^{2} x^{2} + 32\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-4 + 4\,x - 16\,n x + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(-4\,x + 4\,x^{2} + 8\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

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	48	12													64
3	8	368	552	32												960
4	16	2432	11040	4864	80											18432
5	32	15168	161472	215296	37920	192										430080
6	64	92416	2024256	6028288	3373760	277248	448									11796480
7	128	557824	23292288	132978176	166222720	46584576	1952384	1024								371589120
8	256	3354624	254678016	2544934912	5984017920	3817402368	594248704	13418496	2304							13212057600
9	512	20145152	2697738240	44457926656	177698032640	213237639168	77801371648	7193968640	90653184	5120						523197480960
10	1024	120909824	28004785152	730320502784	4643447203840	9382456123392	6500826085376	1460641005568	84014355456	604549120	11264					22830435532800
11	2048	725544960	286939711488	11494037299200	110969068523520	351289587474432	409837852053504	177550509637632	25861583923200	956465704960	3990497280	24576				1088250760396800
12	4096	4353458176	2915106447360	175454800904192	2485939818639360	11723257391284224	21311568306782208	15631009855045632	4474691673550848	438637002260480	10688723640320	26120749056	53248			56254193152819200
13	8192	26121158656	29451453087744	2619613173710848	53081019778457600	359338007208738816	964414784671285248	1102188325338611712	539007010813108224	106162039556915200	7203936227704832	117805812350976	169787531264	114688		3134162189942784000
14	16384	156727836672	296473617776640	38480940221333504	1092988047632056320	10325930660489920512	39328525354066722816	66131299520316702720	50565246883800072192	17209884434149867520	2404573704790523904	115442820664000512	1284719010365440	1097094856704	245760	187213954812582297600

Technical details

Node id: n-9c65a2e16bd170ad · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-2e37e4aadfbd49d5

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,       48,         12;
  3:   8,      368,        552,          32;
  4:  16,     2432,      11040,        4864,           80;
  5:  32,    15168,     161472,      215296,        37920,          192;
  6:  64,    92416,    2024256,     6028288,      3373760,       277248,         448;
  7: 128,   557824,   23292288,   132978176,    166222720,     46584576,     1952384,       1024;
  8: 256,  3354624,  254678016,  2544934912,   5984017920,   3817402368,   594248704,   13418496,     2304;
  9: 512, 20145152, 2697738240, 44457926656, 177698032640, 213237639168, 77801371648, 7193968640, 90653184, 5120;

%F Formula

The row polynomials satisfy (2*n-1)*P(n,x) = (8*n+4+(32*n^2-32*n-24)*x-(32*n^3-48*n^2-8*n+12)*x^2)*P(n-2,x) + ((16*n+8)*x-(32*n^2+16*n)*x^2+(32*n^2-8)*x^3)*d/dx P(n-2,x) - (4-(16*n^2-16*n+4)*x)*P(n-1,x) + ((8*n-4)*x-(8*n-4)*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[((8*n + 4 + (32*n^2 - 32*n - 24)*x - (32*n^3 - 48*n^2 - 8*n + 12)*x^2)*P[n - 2, x] + ((16*n + 8)*x - (32*n^2 + 16*n)*x^2 + (32*n^2 - 8)*x^3)*D[P[n - 2, x], x] - (4 - (16*n^2 - 16*n + 4)*x)*P[n - 1, x] + ((8*n - 4)*x - (8*n - 4)*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
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] = ((8*n + 4 + (32*n^2 - 32*n - 24)*x - (32*n^3 - 48*n^2 - 8*n + 12)*x^2)*v[i - 2] + ((16*n + 8)*x - (32*n^2 + 16*n)*x^2 + (32*n^2 - 8)*x^3)*deriv(v[i - 2], x) - (4 - (16*n^2 - 16*n + 4)*x)*v[i - 1] + ((8*n - 4)*x - (8*n - 4)*x^2)*deriv(v[i - 1], x))/(2*n - 1)); 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 2.?
  - 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 2.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

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.