A257614 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 30\,n + 25\,n^{2}\right)\,P_{n}(x) = \left(-36 - 2784\,x - 576\,x^{2} - 30\,n + 1420\,n x - 960\,n x^{2} + 150\,n^{2} + 1800\,n^{2} x + 10025\,n^{2} x^{2} - 1000\,n^{3} x - 14750\,n^{3} x^{2} + 5625\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(645\,x - 2706\,x^{2} - 711\,x^{3} + 100\,n x + 5\,n x^{2} - 1860\,n x^{3} - 500\,n^{2} x + 1700\,n^{2} x^{2} + 13025\,n^{2} x^{3} + 1000\,n^{3} x^{2} - 14750\,n^{3} x^{3} + 2500\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(525\,x^{2} - 390\,x^{3} - 135\,x^{4} + 125\,n x^{2} + 325\,n x^{3} - 450\,n x^{4} - 625\,n^{2} x^{2} - 2000\,n^{2} x^{3} + 2625\,n^{2} x^{4} + 2500\,n^{3} x^{3} - 2500\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(36 - 108\,x - 45\,n + 540\,n x - 25\,n^{2} - 900\,n^{2} x + 500\,n^{3} x\right)\,P_{n-1}(x) + \left(-60\,x - 72\,x^{2} - 175\,n x + 60\,n x^{2} + 250\,n^{2} x + 400\,n^{2} x^{2} - 500\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 2 + 8\,x\)

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	8														10
2	4	112	36													152
3	8	976	2196	128												3308
4	16	7376	52704	29504	400											90000
5	32	53040	914580	1625920	331500	1152										2926224
6	64	374736	13600440	55281280	37779000	3372624	3136									110411280
7	128	2631344	185970492	1471710080	2299547000	743881968	32233964	8192								4735983168
8	256	18438352	2421102672	33945654976	101180068000	76377723696	13181558992	295013632	20736							227419581312
9	512	129111472	30588224868	714809975488	3658043815300	5267583094032	2189105549932	217516265728	2614507308	51200						12080390595840
10	1024	903876560	379259732520	14163795041280	116217462711000	284449487077728	227786226913560	56655180165120	3413337592680	22596914000	123904					703088250149376
11	2048	6327348912	4646701736460	269102575729920	3375628580466000	13035436832821536	17742677911340424	8641609165992960	1362331789632720	51630019294000	191402304588	294912				44483261306964480
12	4096	44291909328	56500720660224	4963001532499520	91923685302528000	531810285253409952	1135531386325787136	945440507117173248	297832740380190720	31018759578122000	759620799987456	1594508735808	692224			3039338125811699712
13	8192	310044381104	683689085294004	89594203784636992	2386791501698549500	19918422188938962144	63119561228103861768	82441875889768309248	44816449925112664824	9547166006794198000	677556166121317252	10939025364704064	13099375101644	1605632		223009053234193594368
14	16384	2170312863184	8247520214692056	1592381958315287936	59788942094869844600	699335447936309808048	3158495761100334245808	6100698815842854884352	5221186870390348447152	1942598466489749466800	289378479739170047864	14331437624837591424	154870101809217496	106345330296016	3686400	17487569829314460395520

Technical details

Node id: n-6f20c4def6bc959e · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-957858e7426cd59f

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,         8;
  2:   4,       112,          36;
  3:   8,       976,        2196,          128;
  4:  16,      7376,       52704,        29504,           400;
  5:  32,     53040,      914580,      1625920,        331500,          1152;
  6:  64,    374736,    13600440,     55281280,      37779000,       3372624,          3136;
  7: 128,   2631344,   185970492,   1471710080,    2299547000,     743881968,      32233964,         8192;
  8: 256,  18438352,  2421102672,  33945654976,  101180068000,   76377723696,   13181558992,    295013632,      20736;
  9: 512, 129111472, 30588224868, 714809975488, 3658043815300, 5267583094032, 2189105549932, 217516265728, 2614507308, 51200;

%F Formula

The row polynomials satisfy (25*n^2-30*n+9)*P(n,x) = (150*n^2-30*n-36-(1000*n^3-1800*n^2-1420*n+2784)*x+(5625*n^4-14750*n^3+10025*n^2-960*n-576)*x^2)*P(n-2,x) - ((500*n^2-100*n-645)*x-(1000*n^3+1700*n^2+5*n-2706)*x^2-(2500*n^4-14750*n^3+13025*n^2-1860*n-711)*x^3)*d/dx P(n-2,x) - ((625*n^2-125*n-525)*x^2-(2500*n^3-2000*n^2+325*n-390)*x^3+(2500*n^3-2625*n^2+450*n+135)*x^4)*d^2/dx^2 P(n-2,x) - (25*n^2+45*n-36-(500*n^3-900*n^2+540*n-108)*x)*P(n-1,x) + ((250*n^2-175*n-60)*x-(500*n^3-400*n^2-60*n+72)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((150*n^2 - 30*n - 36 - (1000*n^3 - 1800*n^2 - 1420*n + 2784)*x + (5625*n^4 - 14750*n^3 + 10025*n^2 - 960*n - 576)*x^2)*P[n - 2, x] - ((500*n^2 - 100*n - 645)*x - (1000*n^3 + 1700*n^2 + 5*n - 2706)*x^2 - (2500*n^4 - 14750*n^3 + 13025*n^2 - 1860*n - 711)*x^3)*D[P[n - 2, x], x] - ((625*n^2 - 125*n - 525)*x^2 - (2500*n^3 - 2000*n^2 + 325*n - 390)*x^3 + (2500*n^3 - 2625*n^2 + 450*n + 135)*x^4)*D[P[n - 2, x], {x, 2}] - (25*n^2 + 45*n - 36 - (500*n^3 - 900*n^2 + 540*n - 108)*x)*P[n - 1, x] + ((250*n^2 - 175*n - 60)*x - (500*n^3 - 400*n^2 - 60*n + 72)*x^2)*D[P[n - 1, x], x])/(25*n^2 - 30*n + 9)]];
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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((150*n^2 - 30*n - 36 - (1000*n^3 - 1800*n^2 - 1420*n + 2784)*x + (5625*n^4 - 14750*n^3 + 10025*n^2 - 960*n - 576)*x^2)*v[i - 2] - ((500*n^2 - 100*n - 645)*x - (1000*n^3 + 1700*n^2 + 5*n - 2706)*x^2 - (2500*n^4 - 14750*n^3 + 13025*n^2 - 1860*n - 711)*x^3)*deriv(v[i - 2], x) - ((625*n^2 - 125*n - 525)*x^2 - (2500*n^3 - 2000*n^2 + 325*n - 390)*x^3 + (2500*n^3 - 2625*n^2 + 450*n + 135)*x^4)*deriv(deriv(v[i - 2], x), x) - (25*n^2 + 45*n - 36 - (500*n^3 - 900*n^2 + 540*n - 108)*x)*v[i - 1] + ((250*n^2 - 175*n - 60)*x - (500*n^3 - 400*n^2 - 60*n + 72)*x^2)*deriv(v[i - 1], x))/(25*n^2 - 30*n + 9)); 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.