Real-rooted atlas

A038226 → basis A007318 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(19 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 361 + 912\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
03619125761849
1685925992328321382479507
2130321658464124761610506243317763418801
32476099156385203950784049904640315187207962624147008443
4470458813565582561125973440189637632017965670409077391361911029766321363049
5893871739790370800829950893504630545126407964780544060364652544254166958084586471424271818611107
616983563041171623373888758755968768191685718425630266166067203058475728896193166888140869714365644811007531417611688200277601
73226876977793668449616856185353243799045463042975129610351028794982413074983741030411010512624025659605782626304188228787240962641807540224502592611936843

All 15 rows

Technical details
Node id
n-6f599efa913bc710 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f3eacc80e11c3732

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             361,              912,               576;
  1:            6859,            25992,             32832,             13824;
  2:          130321,           658464,           1247616,           1050624,            331776;
  3:         2476099,         15638520,          39507840,          49904640,          31518720,            7962624;
  4:        47045881,        356558256,        1125973440,        1896376320,        1796567040,          907739136,          191102976;
  5:       893871739,       7903708008,       29950893504,       63054512640,       79647805440,        60364652544,        25416695808,         4586471424;
  6:     16983563041,     171623373888,      758755968768,     1916857184256,     3026616606720,      3058475728896,      1931668881408,       697143656448,       110075314176;
  7:    322687697779,    3668449616856,    18535324379904,    54630429751296,   103510287949824,    130749837410304,    110105126240256,     59605782626304,     18822878724096,     2641807540224;
  8:   6131066257801,   77445047466960,   440213954022720,  1482825950392320,  3277825785077760,   4968493821591552,   5229993496412160,   3775032899665920,   1788173478789120,   501943432642560,    63403380965376;
  9: 116490258898219, 1618601492059464, 10222746265638720, 38738827953999360, 97866512725893120, 173069201452105728, 218613728150028288, 197245469007544320, 124576085688975360, 52453088711147520, 13251306621763584, 1521681143169024;
%F Formula
The row polynomials satisfy P(n,x) = (19+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 912*x + 576*x^2.
%t Mathematica
P[0, x_] := 361 + 912*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(19 + 24*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 361 + 912*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (19 + 24*x)*v[i - 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.