A395972 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 + x + 2\,n x + n^{2} x\right)\,P_{n}(x) = \left(2 + 6\,x + 4\,x^{2} + 12\,n x + 16\,n x^{2} + 2\,n^{2} x + 16\,n^{2} x^{2} + 4\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-2\,x^{2} - 6\,x^{3} - 8\,n x^{3} - 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 4 + 32\,x + 24\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
1	4	32	24								60
2	8	144	432	192							776
3	16	512	3456	6144	1920						12048
4	32	1600	19200	76800	96000	23040					216672
5	64	4608	86400	614400	1728000	1658880	322560				4414912
6	128	12544	338688	3763200	18816000	40642560	31610880	5160960			100344960
7	256	32768	1204224	19267584	150528000	578027520	1011548160	660602880	92897280		2514108672
8	512	82944	3981312	86704128	975421440	5852528640	18207866880	26754416640	15049359360	1857945600	68788307456

All 15 rows

Technical details

Node id: n-143090c9b4a3eff8 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-84f985e00599cd78

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 1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:    4,     32,       24;
   2:    8,    144,      432,        192;
   3:   16,    512,     3456,       6144,        1920;
   4:   32,   1600,    19200,      76800,       96000,        23040;
   5:   64,   4608,    86400,     614400,     1728000,      1658880,        322560;
   6:  128,  12544,   338688,    3763200,    18816000,     40642560,      31610880,       5160960;
   7:  256,  32768,  1204224,   19267584,   150528000,    578027520,    1011548160,     660602880,       92897280;
   8:  512,  82944,  3981312,   86704128,   975421440,   5852528640,   18207866880,   26754416640,    15049359360,     1857945600;
   9: 1024, 204800, 12441600,  353894400,  5419008000,  46820229120,  227598336000,  594542592000,   752467968000,   371589120000,   40874803200;
  10: 2048, 495616, 37171200, 1338163200, 26763264000, 314735984640, 2203151892480, 8992456704000, 20233027584000, 22481141760000, 9891702374400, 980995276800;

%F Formula

The row polynomials satisfy (1+(n^2+2*n+1)*x)*P(n,x) = (2+(2*n^2+12*n+6)*x+(4*n^3+16*n^2+16*n+4)*x^2)*P(n-1,x) - (2*x^2+(2*n^2+8*n+6)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 4 + 32*x + 24*x^2.

%t Mathematica

P[1, x_] := 4 + 32*x + 24*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (2*n^2 + 12*n + 6)*x + (4*n^3 + 16*n^2 + 16*n + 4)*x^2)*P[n - 1, x] - (2*x^2 + (2*n^2 + 8*n + 6)*x^3)*D[P[n - 1, x], x])/(1 + (n^2 + 2*n + 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 4 + 32*x + 24*x^2; for(n = 2, N, my(i = n); v[i] = ((2 + (2*n^2 + 12*n + 6)*x + (4*n^3 + 16*n^2 + 16*n + 4)*x^2)*v[i - 1] - (2*x^2 + (2*n^2 + 8*n + 6)*x^3)*deriv(v[i - 1], x))/(1 + (n^2 + 2*n + 1)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
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: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).?
  - 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

Related ?

Same fingerprint ?

A144084 → θ+1

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.