A038271 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(49 - 35\,x + 105\,n x + 25\,n^{2} x^{2}\right)\,P_{n}(x) = \left(343 + 735\,x + 525\,x^{2} + 125\,x^{3} + 735\,n x + 875\,n x^{2} + 250\,n x^{3} + 175\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	7	20														27
2	49	280	225													554
3	343	2940	4725	2000												10008
4	2401	27440	66150	56000	15625											167616
5	16807	240100	771750	980000	546875	112500										2668032
6	117649	2016840	8103375	13720000	11484375	4725000	765625									40932864
7	823543	16470860	79413075	168070000	187578125	115762500	37515625	5000000								610633728
8	5764801	131766880	741188700	1882384000	2626093750	2160900000	1050437500	280000000	31640625							8910176256
9	40353607	1037664180	6670698300	19765032000	33088781250	34034175000	22059187500	8820000000	1993359375	195312500						127704563712
10	282475249	8070721400	58368610125	197650320000	386035781250	476478450000	386035781250	205800000000	69767578125	13671875000	1181640625					1803343233024
11	1977326743	62144554780	499375886625	1902384330000	4246393593750	6114806775000	5944951031250	3961650000000	1790701171875	526367187500	90986328125	7031250000				25148769435648
12	13841287201	474558418320	4194757447650	17755587080000	44587132734375	73377681300000	83229314437500	66555720000000	37604724609375	14738281250000	3821425781250	590625000000	41259765625			346984909111296
13	96889010407	3598734672260	34702084339650	161575842428000	450825453203125	834671124787500	1081981087687500	1009428420000000	684405987890625	335295898437500	115916582031250	26873437500000	3754638671875	239257812500		4743365438472192
14	678223072849	27128922913960	283400355440475	1439493868904000	4418089441390625	9088641136575000	13254268324171875	14131997880000000	11178631135546875	6571799609375000	2839956259765625	877865625000000	183977294921875	23447265625000	1373291015625	64320748633718784

Technical details

Node id: n-f2c6356ce072a734 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-580354feafd3742e

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:        7,         20;
  2:       49,        280,        225;
  3:      343,       2940,       4725,        2000;
  4:     2401,      27440,      66150,       56000,       15625;
  5:    16807,     240100,     771750,      980000,      546875,      112500;
  6:   117649,    2016840,    8103375,    13720000,    11484375,     4725000,      765625;
  7:   823543,   16470860,   79413075,   168070000,   187578125,   115762500,    37515625,    5000000;
  8:  5764801,  131766880,  741188700,  1882384000,  2626093750,  2160900000,  1050437500,  280000000,   31640625;
  9: 40353607, 1037664180, 6670698300, 19765032000, 33088781250, 34034175000, 22059187500, 8820000000, 1993359375, 195312500;

%F Formula

The row polynomials satisfy (49+(105*n-35)*x+25*n^2*x^2)*P(n,x) = (343+(735*n+735)*x+(175*n^2+875*n+525)*x^2+(125*n^2+250*n+125)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((343 + (735*n + 735)*x + (175*n^2 + 875*n + 525)*x^2 + (125*n^2 + 250*n + 125)*x^3)*P[n - 1, x])/(49 + (105*n - 35)*x + 25*n^2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((343 + (735*n + 735)*x + (175*n^2 + 875*n + 525)*x^2 + (125*n^2 + 250*n + 125)*x^3)*v[i - 1])/(49 + (105*n - 35)*x + 25*n^2*x^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 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 whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.