Real-rooted atlas

A038284 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(32 + 36\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1024 + 2304\,x + 1296\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01024230412964624
13276811059212441646656314432
21048576471859279626245971968167961621381376
333554432188743680424673280477757440268738560604661761453933568
41073741824724775731220384317440305764761602579890176011609505792217678233698867482624
534359738368270582939648913217421312171228266496019263179980801300264648704487599243264783641640966722988818432
610995116277769895604649984389639433093128766887244595212328435187712011095591668940862412703137792200612260085762821109907456457163239653376
73518437208883235624176739942416030879532974084208105877405696710117866812211279888260016373765991619501228032288881654523494481247965334732810155995666841631087100296429568

All 15 rows

Technical details
Node id
n-6f2d8ef928c9c388 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-cb7716efe3e0f2cd

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:              1024,               2304,                1296;
  1:             32768,             110592,              124416,               46656;
  2:           1048576,            4718592,             7962624,             5971968,              1679616;
  3:          33554432,          188743680,           424673280,           477757440,            268738560,             60466176;
  4:        1073741824,         7247757312,         20384317440,         30576476160,          25798901760,          11609505792,           2176782336;
  5:       34359738368,       270582939648,        913217421312,       1712282664960,        1926317998080,        1300264648704,         487599243264,          78364164096;
  6:     1099511627776,      9895604649984,      38963943309312,      87668872445952,      123284351877120,      110955916689408,       62412703137792,       20061226008576,        2821109907456;
  7:    35184372088832,    356241767399424,    1603087953297408,    4208105877405696,     7101178668122112,     7988826001637376,     5991619501228032,     2888816545234944,      812479653347328,     101559956668416;
  8:  1125899906842624,  12666373951979520,   64123518131896320,  192370554395688960,   378729528966512640,   511284864104792064,   479329560098242560,   308140431491727360,   129996744535572480,   32499186133893120,    3656158440062976;
  9: 36028797018963968, 445856363109679104, 2507942042491944960, 8464304393410314240, 19044684885173207040, 29995378694147801088, 33744801030916276224, 27116357971272007680, 15252951358840504320, 5719856759565189120, 1286967770902167552, 131621703842267136;
%F Formula
The row polynomials satisfy P(n,x) = (32+36*x)*P(n-1,x) for n >= 1, with P(0,x) = 1024 + 2304*x + 1296*x^2.
%t Mathematica
P[0, x_] := 1024 + 2304*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[(32 + 36*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] = 1024 + 2304*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = (32 + 36*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.