Real-rooted atlas

A038330 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(28 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 784 + 1344\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
078413445762704
121952564484838413824140608
26146562107392270950415482883317767311616
31721036873758720126443520108380160464486407962624380204032
44818903042478292992531062784060692889603901685760133772083219110297619770609664
513492928512809575710722081766113282973951590402549101363201310966415363745618329645864714241028071702528
637780199833625906422743047771926822912133233031249921427496763392097885492346884195092529152102736959897611007531417653459728531456
710578455953408816052316405762797893656248325595787312496647194583687495686166786017853443523877724487681294485694709762773897917235226418075402242779905883635712

All 15 rows

Technical details
Node id
n-9ca6bd6fc8b00a8f · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-8e9037e0ccb64a62

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:              784,              1344,                576;
  1:            21952,             56448,              48384,              13824;
  2:           614656,           2107392,            2709504,            1548288,              331776;
  3:         17210368,          73758720,          126443520,          108380160,            46448640,             7962624;
  4:        481890304,        2478292992,         5310627840,         6069288960,          3901685760,          1337720832,           191102976;
  5:      13492928512,       80957571072,       208176611328,       297395159040,        254910136320,        131096641536,         37456183296,         4586471424;
  6:     377801998336,     2590642274304,      7771926822912,     13323303124992,      14274967633920,       9788549234688,       4195092529152,      1027369598976,       110075314176;
  7:   10578455953408,    81605231640576,    279789365624832,    559578731249664,     719458368749568,     616678601785344,     352387772448768,    129448569470976,     27738979172352,      2641807540224;
  8:  296196766695424,  2538829428817920,   9792627796869120,  22383149249986560,   33574723874979840,   34534001699979264,   24667144071413760,  12081866483957760,   3883457084129280,    739706111262720,    63403380965376;
  9: 8293509467471872, 78195946407591936, 335125484603965440, 861751246124482560, 1477287850499112960, 1772745420598935552, 1519496074799087616, 930303719264747520, 398701593970606080, 113914741134458880, 19528241337335808, 1521681143169024;
%F Formula
The row polynomials satisfy P(n,x) = (28+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 784 + 1344*x + 576*x^2.
%t Mathematica
P[0, x_] := 784 + 1344*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(28 + 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] = 784 + 1344*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (28 + 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.