Real-rooted atlas

A038320 → basis A013609 → D

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(34 + 24\,x + 17\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 408 + 288\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
0408288696
11040414688518430276
2235824499392352512829441170672
3501126014149440149817607050240124416042436860
4102229704360810720509379840359562240126904320179159041476802728
52027555796858729513615154050240142626355207550807040213199257625082265649965158964
6393925126081946453564164121901665284849296076803423032524801449754951683411188121634398535681655988125664
77533818036284254391361664105108492464641483884599500813093099407360739375025356826095589130245262975959044643802316854026612599788

All 15 rows

Technical details
Node id
n-5e42752da9a07882 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-8b732aeffa021ba5

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             408,              288;
  1:           10404,            14688,             5184;
  2:          235824,           499392,           352512,             82944;
  3:         5011260,         14149440,         14981760,           7050240,           1244160;
  4:       102229704,        360810720,        509379840,         359562240,         126904320,          17915904;
  5:      2027555796,       8587295136,      15154050240,       14262635520,        7550807040,        2131992576,        250822656;
  6:     39392512608,     194645356416,     412190166528,      484929607680,      342303252480,      144975495168,      34111881216,       3439853568;
  7:    753381803628,    4254391361664,   10510849246464,    14838845995008,    13093099407360,     7393750253568,    2609558913024,     526297595904,     46438023168;
  8:  14230545179640,   90405816435360,  255263481699840,   420433969858560,   445165379850240,   314234385776640,  147875005071360,   44735295651840,   7894463938560,    619173642240;
  9: 266111194859268, 1878431963712480, 5966783884733760, 11231593194792960, 13874321005332480, 11752366028046336, 6913156487086080, 2788500095631360, 738132378255360, 115785471098880, 8173092077568;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (17*n+34+(12*n+24)*x)*P(n-1,x) for n >= 1, with P(0,x) = 408 + 288*x.
%t Mathematica
P[0, x_] := 408 + 288*x;
P[n_, x_] := P[n, x] = Expand[Together[((17*n + 34 + (12*n + 24)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 408 + 288*x; for(n = 1, N, my(i = n + 1); v[i] = ((17*n + 34 + (12*n + 24)*x)*v[i - 1])/(n + 1)); 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

Proof chains ? JSON

Same fingerprint ?

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.