A038305 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(100 + 150\,x + 36\,x^{2} + 90\,n x + 36\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1000 + 2700\,x + 1530\,x^{2} + 243\,x^{3} + 900\,n x + 810\,n x^{2} + 162\,n x^{3} + 90\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 100 + 240\,x + 81\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	100	240	81								421
1	1000	3600	2430	432							7462
2	10000	48000	48600	17280	2025						125905
3	100000	600000	810000	432000	101250	8748					2051998
4	1000000	7200000	12150000	8640000	3037500	524880	35721				32588101
5	10000000	84000000	170100000	151200000	70875000	18370800	2500470	139968			507186238
6	100000000	960000000	2268000000	2419200000	1417500000	489888000	100018800	11197440	531441		7766335681
7	1000000000	10800000000	29160000000	36288000000	25515000000	11022480000	3000564000	503884800	47829690	1968300	117339726790

All 15 rows

Technical details

Node id: n-b618acefa39fed0e · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-09fd8ade2935dd5d

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:          100,           240,            81;
  1:         1000,          3600,          2430,           432;
  2:        10000,         48000,         48600,         17280,          2025;
  3:       100000,        600000,        810000,        432000,        101250,          8748;
  4:      1000000,       7200000,      12150000,       8640000,       3037500,        524880,         35721;
  5:     10000000,      84000000,     170100000,     151200000,      70875000,      18370800,       2500470,       139968;
  6:    100000000,     960000000,    2268000000,    2419200000,    1417500000,     489888000,     100018800,     11197440,      531441;
  7:   1000000000,   10800000000,   29160000000,   36288000000,   25515000000,   11022480000,    3000564000,    503884800,    47829690,     1968300;
  8:  10000000000,  120000000000,  364500000000,  518400000000,  425250000000,  220449600000,   75014100000,  16796160000,  2391484500,   196830000,   7144929;
  9: 100000000000, 1320000000000, 4455000000000, 7128000000000, 6682500000000, 4041576000000, 1650310200000, 461894400000, 87687765000, 10825650000, 785942190, 25509168;

%F Formula

The row polynomials satisfy (100+(90*n+150)*x+(9*n^2+36*n+36)*x^2)*P(n,x) = (1000+(900*n+2700)*x+(90*n^2+810*n+1530)*x^2+(27*n^2+162*n+243)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 100 + 240*x + 81*x^2.

%t Mathematica

P[0, x_] := 100 + 240*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1000 + (900*n + 2700)*x + (90*n^2 + 810*n + 1530)*x^2 + (27*n^2 + 162*n + 243)*x^3)*P[n - 1, x])/(100 + (90*n + 150)*x + (9*n^2 + 36*n + 36)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 100 + 240*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1000 + (900*n + 2700)*x + (90*n^2 + 810*n + 1530)*x^2 + (27*n^2 + 162*n + 243)*x^3)*v[i - 1])/(100 + (90*n + 150)*x + (9*n^2 + 36*n + 36)*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)*10^(i-j)*3^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

Related ?

Same fingerprint ?

A038291 → basis A013610 → (θ+1)²

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.