A038331 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(144 + 300\,x + 100\,x^{2} + 180\,n x + 100\,n x^{2} + 25\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1728 + 6480\,x + 5100\,x^{2} + 1125\,x^{3} + 2160\,n x + 2700\,n x^{2} + 750\,n x^{3} + 300\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 144 + 480\,x + 225\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	144	480	225								849
1	1728	8640	8100	2000							20468
2	20736	138240	194400	96000	15625						465001
3	248832	2073600	3888000	2880000	937500	112500					10140432
4	2985984	29859840	69984000	69120000	33750000	8100000	765625				214565449
5	35831808	418037760	1175731200	1451520000	945000000	340200000	64312500	5000000			4435633268
6	429981696	5733089280	18811699200	27869184000	22680000000	10886400000	3087000000	480000000	31640625		90008994801
7	5159780352	77396705280	290237644800	501645312000	489888000000	293932800000	111132000000	25920000000	3417187500	195312500	1798924742432

All 15 rows

Technical details

Node id: n-731c6b3b4289d4ac · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-399776af7ffddb1e

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:          144,            480,            225;
  1:         1728,           8640,           8100,            2000;
  2:        20736,         138240,         194400,           96000,           15625;
  3:       248832,        2073600,        3888000,         2880000,          937500,          112500;
  4:      2985984,       29859840,       69984000,        69120000,        33750000,         8100000,         765625;
  5:     35831808,      418037760,     1175731200,      1451520000,       945000000,       340200000,       64312500,        5000000;
  6:    429981696,     5733089280,    18811699200,     27869184000,     22680000000,     10886400000,     3087000000,      480000000,      31640625;
  7:   5159780352,    77396705280,   290237644800,    501645312000,    489888000000,    293932800000,   111132000000,    25920000000,    3417187500,     195312500;
  8:  61917364224,  1031956070400,  4353564672000,   8599633920000,   9797760000000,   7054387200000,  3333960000000,  1036800000000,  205031250000,   23437500000,   1181640625;
  9: 743008370688, 13621820129280, 63852281856000, 141893959680000, 184757760000000, 155196518400000, 88016544000000, 34214400000000, 9021375000000, 1546875000000, 155976562500, 7031250000;

%F Formula

The row polynomials satisfy (144+(180*n+300)*x+(25*n^2+100*n+100)*x^2)*P(n,x) = (1728+(2160*n+6480)*x+(300*n^2+2700*n+5100)*x^2+(125*n^2+750*n+1125)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 480*x + 225*x^2.

%t Mathematica

P[0, x_] := 144 + 480*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1728 + (2160*n + 6480)*x + (300*n^2 + 2700*n + 5100)*x^2 + (125*n^2 + 750*n + 1125)*x^3)*P[n - 1, x])/(144 + (180*n + 300)*x + (25*n^2 + 100*n + 100)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 144 + 480*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1728 + (2160*n + 6480)*x + (300*n^2 + 2700*n + 5100)*x^2 + (125*n^2 + 750*n + 1125)*x^3)*v[i - 1])/(144 + (180*n + 300)*x + (25*n^2 + 100*n + 100)*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)*12^(i-j)*5^j.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - 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
  - Parent-row provers of polytool polytool ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

A038271 → basis A007318 → (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.