Real-rooted atlas

basis-map coefficients of A371080, shifted → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(6\,x + 3\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1369
2185436108
31625946482161620
419448100113406480129629160
529160133164218700165240583207776612360
652488025719124735584428652020412004898884665614696640
7110224805715943211487873611843042468584320225348483919104279936396809280
8264539520143796124831000462563531602592235660622495234227222926758430233088167961611904278400
971425670404041219081692329016400113953547520848179836003985287868811904278400219189888022674816010077696392841187200
10214277011200125522112672030123436368963972580524000322826079312017044942622405962456241281371826368001995383808016628198406046617614142282739200
1170711413696004270795924896010693866677788814916921911337613036808931696075617875412640299030711696648104500759168148157247744017459608320011972302848362797056551549026828800
1225456108930560015799133811801604106039759497728601172388874886455882665300908163504452050756800153021781458374447118045434803210196361374284815174893859840147857940172884894511104217678233623165059126809600
1399278824829184006314398840185984016961503090749235225909347021819206425401273800603500817020322916006009680705207073306816275573446070757126803663662059264120360254299084814871395982643212182362343424594261577728130606940161042427660706432000
14416971064282572800271161480777562368075026952285258378241189961593460902080012223095817562622720867261205275873408044108380720392468481641639715937020800451097941448943360913732887779712001346760157065523214039429773824009805316032512041141186150407836416409650036527713908736000
Technical details
Node id
n-d1061742df012f76 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-21d4cbc8ebff768c

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:          1;
  1:          3,           6;
  2:         18,          54,          36;
  3:        162,         594,         648,          216;
  4:       1944,        8100,       11340,         6480,        1296;
  5:      29160,      133164,      218700,       165240,       58320,        7776;
  6:     524880,     2571912,     4735584,      4286520,     2041200,      489888,       46656;
  7:   11022480,    57159432,   114878736,    118430424,    68584320,    22534848,     3919104,     279936;
  8:  264539520,  1437961248,  3100046256,   3531602592,  2356606224,   952342272,   229267584,   30233088,   1679616;
  9: 7142567040, 40412190816, 92329016400, 113953547520, 84817983600, 39852878688, 11904278400, 2191898880, 226748160, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3*n + 6*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3*n + 6*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 ?