A038330 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 + 15\,x + 4\,x^{2} + 9\,n x + 4\,n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(108 + 324\,x + 204\,x^{2} + 36\,x^{3} + 108\,n x + 108\,n x^{2} + 24\,n x^{3} + 12\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 144 + 384\,x + 144\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	144	384	144								672
1	1728	6912	5184	1024							14848
2	20736	110592	124416	49152	6400						311296
3	248832	1658880	2488320	1474560	384000	36864					6291456
4	2985984	23887872	44789760	35389440	13824000	2654208	200704				123731968
5	35831808	334430208	752467968	743178240	387072000	111476736	16859136	1048576			2382364672
6	429981696	4586471424	12039487488	14269022208	9289728000	3567255552	809238528	100663296	5308416		45097156608
7	5159780352	61917364224	185752092672	256842399744	200658124800	96315899904	29132587008	5435817984	573308928	26214400	841813590016

All 15 rows

Technical details

Node id: n-70ef400185a6f340 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-1e916cb7d7a3299c

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,            384,            144;
  1:         1728,           6912,           5184,           1024;
  2:        20736,         110592,         124416,          49152,           6400;
  3:       248832,        1658880,        2488320,        1474560,         384000,          36864;
  4:      2985984,       23887872,       44789760,       35389440,       13824000,        2654208,         200704;
  5:     35831808,      334430208,      752467968,      743178240,      387072000,      111476736,       16859136,       1048576;
  6:    429981696,     4586471424,    12039487488,    14269022208,     9289728000,     3567255552,      809238528,     100663296,       5308416;
  7:   5159780352,    61917364224,   185752092672,   256842399744,   200658124800,    96315899904,    29132587008,    5435817984,     573308928,     26214400;
  8:  61917364224,   825564856320,  2786281390080,  4403012567040,  4013162496000,  2311581597696,   873977610240,  217432719360,   34398535680,   3145728000,   126877696;
  9: 743008370688, 10897456103424, 40865460387840, 72649707356160, 75676778496000, 50854795149312, 23073008910336, 7175279738880, 1513535569920, 207618048000, 16747855872, 603979776;

%F Formula

The row polynomials satisfy (9+(9*n+15)*x+(n^2+4*n+4)*x^2)*P(n,x) = (108+(108*n+324)*x+(12*n^2+108*n+204)*x^2+(4*n^2+24*n+36)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 384*x + 144*x^2.

%t Mathematica

P[0, x_] := 144 + 384*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((108 + (108*n + 324)*x + (12*n^2 + 108*n + 204)*x^2 + (4*n^2 + 24*n + 36)*x^3)*P[n - 1, x])/(9 + (9*n + 15)*x + (n^2 + 4*n + 4)*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 + 384*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((108 + (108*n + 324)*x + (12*n^2 + 108*n + 204)*x^2 + (4*n^2 + 24*n + 36)*x^3)*v[i - 1])/(9 + (9*n + 15)*x + (n^2 + 4*n + 4)*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)*4^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 ?

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.