A075502 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-x^{2}\right)\,P_{n-2}(x) + \left(-7\,x^{2}\right)\,P_{n-2}'(x) + \left(7 + 2\,x\right)\,P_{n-1}(x) + \left(7\,x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 49 + 42\,x + 3\,x^{2}\)

\(P_{1}(x) = 343 + 686\,x + 126\,x^{2} + 4\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	49	42	3								94
1	343	686	126	4							1159
2	2401	10290	3675	280	5						16651
3	16807	148862	92610	12740	525	6					271550
4	117649	2117682	2168103	480200	34300	882	7				4918823
5	823543	29882846	48706686	16336404	1800750	78204	1372	8			97629813
6	5764801	420006930	1067664675	522361560	83446755	5445468	158466	2016	9		2104850680
7	40353607	5891626622	23050968570	16049676580	3573588375	328845762	14117880	294000	2835	10	48949474241

All 15 rows

Technical details

Node id: n-5d7d7e644f711df6 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-1b31ab8c99ad484d

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:         49,            42,              3;
  1:        343,           686,            126,              4;
  2:       2401,         10290,           3675,            280,             5;
  3:      16807,        148862,          92610,          12740,           525,            6;
  4:     117649,       2117682,        2168103,         480200,         34300,          882,           7;
  5:     823543,      29882846,       48706686,       16336404,       1800750,        78204,        1372,          8;
  6:    5764801,     420006930,     1067664675,      522361560,      83446755,      5445468,      158466,       2016,        9;
  7:   40353607,    5891626622,    23050968570,    16049676580,    3573588375,    328845762,    14117880,     294000,     2835,     10;
  8:  282475249,   82563479922,   492907779903,   480125569000,  145137688850,  18099828054,  1075429509,   32598720,   509355,   3850,   11;
  9: 1977326743, 1156453669406, 10474908597846, 14100726305204, 5679976071000, 934358004888, 73812512004, 3054590616, 68762925, 835450, 5082, 12;

%F Formula

The row polynomials satisfy P(n,x) = -x^2*P(n-2,x) - 7*x^2*d/dx P(n-2,x) + (7+2*x)*P(n-1,x) + 7*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 49 + 42*x + 3*x^2, P(1,x) = 343 + 686*x + 126*x^2 + 4*x^3.

%t Mathematica

P[0, x_] := 49 + 42*x + 3*x^2;
P[1, x_] := 343 + 686*x + 126*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - 7*x^2*D[P[n - 2, x], x] + (7 + 2*x)*P[n - 1, x] + 7*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 49 + 42*x + 3*x^2; v[2] = 343 + 686*x + 126*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = -x^2*v[i - 2] - 7*x^2*deriv(v[i - 2], x) + (7 + 2*x)*v[i - 1] + 7*x*deriv(v[i - 1], x)); 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: Stirling2 triangle with scaled diagonals (powers of 7).?
  - 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

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.