A257616 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 12\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(-16 - 1600\,x - 240\,x^{2} - 24\,n + 720\,n x - 1296\,n x^{2} + 72\,n^{2} + 864\,n^{2} x + 6804\,n^{2} x^{2} - 432\,n^{3} x - 8424\,n^{3} x^{2} + 2916\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(336\,x - 1520\,x^{2} - 272\,x^{3} + 72\,n x - 108\,n x^{2} - 1968\,n x^{3} - 216\,n^{2} x + 864\,n^{2} x^{2} + 8316\,n^{2} x^{3} + 432\,n^{3} x^{2} - 7992\,n^{3} x^{3} + 1296\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(288\,x^{2} - 240\,x^{3} - 48\,x^{4} + 108\,n x^{2} + 324\,n x^{3} - 432\,n x^{4} - 324\,n^{2} x^{2} - 1296\,n^{2} x^{3} + 1620\,n^{2} x^{4} + 1296\,n^{3} x^{3} - 1296\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(16 - 64\,x - 12\,n + 288\,n x - 18\,n^{2} - 432\,n^{2} x + 216\,n^{3} x\right)\,P_{n-1}(x) + \left(-24\,x - 32\,x^{2} - 90\,n x + 108\,n^{2} x + 216\,n^{2} x^{2} - 216\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

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

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	128	36													168
3	8	1248	2808	128												4192
4	16	10624	78624	42496	400											132160
5	32	86656	1578816	2806784	541600	1152										5015040
6	64	697344	27172800	112271360	75480000	6276096	3136									221900800
7	128	5588480	430627968	3501414400	5470960000	1722511872	68458880	8192								11199569920
8	256	44730368	6506606592	94526234624	284489920000	212684027904	35424858112	715685888	20736							634392084480
9	512	357894144	95520798720	2330082115584	12123049651200	17457191497728	7135876478976	679259013120	7247356416	51200						39828584857600
10	1024	2863267840	1377554273280	54073491456000	453547916544000	1117260255854592	888953916426240	216293965824000	12397988459520	71581696000	123904					2743979533926400
11	2048	22906396672	19646531573760	1203919097856000	15509798367744000	60570470180634624	82443139968086016	39705083821424640	6094840432896000	218294795264000	692918499328	294912				205765909020672000
12	4096	183251730432	278246884368384	26034303322685440	497310937081344000	2920955870360567808	6265678637574537216	5192810436196564992	1611287436143554560	162714395766784000	3740874778730496	6597062295552	692224			16680817918423859200
13	8192	1466015055872	3923493895913472	551355056375201792	15208085904849920000	129276975321394937856	413028589797204197376	539465913204511604736	290873194473138610176	60832343619399680000	4169622613837463552	62775902334615552	61939136110592	1605632		1453472842792094924800
14	16384	11728123068416	55173006049591296	11501407945143353344	448822754612445593600	5363243257651735560192	24493102788333137215488	47473000361997021216768	40488598486836410499072	14897897937921487667200	2172302132324236673024	103512671506290180096	1036026446931214336	574678030352384	3686400	135453073584987045888000

Technical details

Node id: n-09788d19776fc9f2 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-34bff62b9cc31312

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,       128,          36;
  3:   8,      1248,        2808,           128;
  4:  16,     10624,       78624,         42496,            400;
  5:  32,     86656,     1578816,       2806784,         541600,           1152;
  6:  64,    697344,    27172800,     112271360,       75480000,        6276096,          3136;
  7: 128,   5588480,   430627968,    3501414400,     5470960000,     1722511872,      68458880,         8192;
  8: 256,  44730368,  6506606592,   94526234624,   284489920000,   212684027904,   35424858112,    715685888,      20736;
  9: 512, 357894144, 95520798720, 2330082115584, 12123049651200, 17457191497728, 7135876478976, 679259013120, 7247356416, 51200;

%F Formula

The row polynomials satisfy (9*n^2-12*n+4)*P(n,x) = (72*n^2-24*n-16-(432*n^3-864*n^2-720*n+1600)*x+(2916*n^4-8424*n^3+6804*n^2-1296*n-240)*x^2)*P(n-2,x) - ((216*n^2-72*n-336)*x-(432*n^3+864*n^2-108*n-1520)*x^2-(1296*n^4-7992*n^3+8316*n^2-1968*n-272)*x^3)*d/dx P(n-2,x) - ((324*n^2-108*n-288)*x^2-(1296*n^3-1296*n^2+324*n-240)*x^3+(1296*n^3-1620*n^2+432*n+48)*x^4)*d^2/dx^2 P(n-2,x) - (18*n^2+12*n-16-(216*n^3-432*n^2+288*n-64)*x)*P(n-1,x) + ((108*n^2-90*n-24)*x-(216*n^3-216*n^2+32)*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[((72*n^2 - 24*n - 16 - (432*n^3 - 864*n^2 - 720*n + 1600)*x + (2916*n^4 - 8424*n^3 + 6804*n^2 - 1296*n - 240)*x^2)*P[n - 2, x] - ((216*n^2 - 72*n - 336)*x - (432*n^3 + 864*n^2 - 108*n - 1520)*x^2 - (1296*n^4 - 7992*n^3 + 8316*n^2 - 1968*n - 272)*x^3)*D[P[n - 2, x], x] - ((324*n^2 - 108*n - 288)*x^2 - (1296*n^3 - 1296*n^2 + 324*n - 240)*x^3 + (1296*n^3 - 1620*n^2 + 432*n + 48)*x^4)*D[P[n - 2, x], {x, 2}] - (18*n^2 + 12*n - 16 - (216*n^3 - 432*n^2 + 288*n - 64)*x)*P[n - 1, x] + ((108*n^2 - 90*n - 24)*x - (216*n^3 - 216*n^2 + 32)*x^2)*D[P[n - 1, x], x])/(9*n^2 - 12*n + 4)]];
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] = ((72*n^2 - 24*n - 16 - (432*n^3 - 864*n^2 - 720*n + 1600)*x + (2916*n^4 - 8424*n^3 + 6804*n^2 - 1296*n - 240)*x^2)*v[i - 2] - ((216*n^2 - 72*n - 336)*x - (432*n^3 + 864*n^2 - 108*n - 1520)*x^2 - (1296*n^4 - 7992*n^3 + 8316*n^2 - 1968*n - 272)*x^3)*deriv(v[i - 2], x) - ((324*n^2 - 108*n - 288)*x^2 - (1296*n^3 - 1296*n^2 + 324*n - 240)*x^3 + (1296*n^3 - 1620*n^2 + 432*n + 48)*x^4)*deriv(deriv(v[i - 2], x), x) - (18*n^2 + 12*n - 16 - (216*n^3 - 432*n^2 + 288*n - 64)*x)*v[i - 1] + ((108*n^2 - 90*n - 24)*x - (216*n^3 - 216*n^2 + 32)*x^2)*deriv(v[i - 1], x))/(9*n^2 - 12*n + 4)); 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) = 6*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) = 6*x + 2.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A142458 → (θ+1)²

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.