Real-rooted atlas

A038334 → basis A013609 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01296230410244624
14665612441611059232768314432
21679616597196879626244718592104857621381376
360466176268738560477757440424673280188743680335544321453933568
42176782336116095057922579890176030576476160203843174407247757312107374182498867482624
578364164096487599243264130026464870419263179980801712282664960913217421312270582939648343597383686722988818432
628211099074562006122600857662412703137792110955916689408123284351877120876688724459523896394330931298956046499841099511627776457163239653376
71015599566684168124796533473282888816545234944599161950122803279888260016373767101178668122112420810587740569616030879532974083562417673994243518437208883231087100296429568

All 15 rows

Technical details
Node id
n-b884b5b428819484 · 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:               1296,                2304,                1024;
  1:              46656,              124416,              110592,                32768;
  2:            1679616,             5971968,             7962624,              4718592,              1048576;
  3:           60466176,           268738560,           477757440,            424673280,            188743680,             33554432;
  4:         2176782336,         11609505792,         25798901760,          30576476160,          20384317440,           7247757312,           1073741824;
  5:        78364164096,        487599243264,       1300264648704,        1926317998080,        1712282664960,         913217421312,         270582939648,          34359738368;
  6:      2821109907456,      20061226008576,      62412703137792,      110955916689408,      123284351877120,       87668872445952,       38963943309312,        9895604649984,       1099511627776;
  7:    101559956668416,     812479653347328,    2888816545234944,     5991619501228032,     7988826001637376,     7101178668122112,     4208105877405696,     1603087953297408,     356241767399424,      35184372088832;
  8:   3656158440062976,   32499186133893120,  129996744535572480,   308140431491727360,   479329560098242560,   511284864104792064,   378729528966512640,   192370554395688960,   64123518131896320,   12666373951979520,   1125899906842624;
  9: 131621703842267136, 1286967770902167552, 5719856759565189120, 15252951358840504320, 27116357971272007680, 33744801030916276224, 29995378694147801088, 19044684885173207040, 8464304393410314240, 2507942042491944960, 445856363109679104, 36028797018963968;
%F Formula
The row polynomials satisfy P(n,x) = (36+32*x)*P(n-1,x) for n >= 1, with P(0,x) = 1296 + 2304*x + 1024*x^2.
%t Mathematica
P[0, x_] := 1296 + 2304*x + 1024*x^2;
P[n_, x_] := P[n, x] = Expand[(36 + 32*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] = 1296 + 2304*x + 1024*x^2; for(n = 1, N, my(i = n + 1); v[i] = (36 + 32*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.