Real-rooted atlas

A038281 → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(22 + 24\,x + 11\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 264 + 288\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0264288552
143569504518419044
26388820908822809682944584016
3878460383328062726404561920124416016790460
411595672632491201379980801505433608211456017915904463416696
5148811124974036448265646304038639462403161410560137952460825082265612435014676
618707684161428586790446753749504850068172809273470976060699082752220723937283439853568326863242912
7231507591482020429889287714368668161683134982144229518406656020030697308161092583489536340545503232464380231688457586410348
828295372292027780910977601212257933568030857474672640504940494643205508441759744040061394616320187300026777605108182548480619173642240216138319375560
93423740047332373498914254401833540124521605333934907699201018296664197120133304290585804812118571871436807554434413363203090450441830407492001071104081730920775685468299480201668
104108488056798449301856681580826891921826316808800992597703680192021656677171202932694392887705631993029740593152249296335640985601359798194405376049447207069286401078848154238976106993205379072137204605139605488
11489594826768476640924136860550438455448211633024139837993496847360343238711310443520599107568832774144762500542150803456712987519933218816486127854499921920235698353696931840771376430280867841530002836920729613909116699279363418681411395170076
1257998156401804088225193089710396853837627496286233621535050998514493445873195726867589120115328207000309022721677501192731767603218300013011619282944149727379185975951369074386617331875840395973234210845491211781021844289617922142003971689021441797485850368409684678108805326520344
1368354970044983380104396681523247344074026737807393571203230257649777174016096907729493315220480211435046167233208320345984621000927068160431357449559597383680411750292761433866240299454758371951902720163338959111973765120647956201435928985601767153276643442688029658516531078758402311053236187955202086710538416974965620
1480203164852780499213124154248636808960100220814262317450240473771121967318855680155052367189304352768037212568125433044664326765921477351462666240948986389031114244096010352578789430337208320878400624557725581312057495313607414765322242851007286318087536640103672992229748637696026099494547349307392040674536956908011520295814814232058265651193965209163119156544
Technical details
Node id
n-0dc59a93b8bc259f · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-60cc3cc9b4461a16

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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:           264,            288;
  1:          4356,           9504,            5184;
  2:         63888,         209088,          228096,           82944;
  3:        878460,        3833280,         6272640,         4561920,          1244160;
  4:      11595672,       63249120,       137998080,       150543360,         82114560,         17915904;
  5:     148811124,      974036448,      2656463040,      3863946240,       3161410560,       1379524608,        250822656;
  6:    1870768416,    14285867904,     46753749504,     85006817280,      92734709760,      60699082752,      22072393728,      3439853568;
  7:   23150759148,   202042988928,    771436866816,   1683134982144,    2295184066560,    2003069730816,    1092583489536,    340545503232,     46438023168;
  8:  282953722920,  2778091097760,  12122579335680,  30857474672640,   50494049464320,   55084417597440,   40061394616320,  18730002677760,   5108182548480,   619173642240;
  9: 3423740047332, 37349891425440, 183354012452160, 533393490769920, 1018296664197120, 1333042905858048, 1211857187143680, 755443441336320, 309045044183040, 74920010711040, 8173092077568;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (11*n+22+(12*n+24)*x)*P(n-1,x) for n >= 1, with P(0,x) = 264 + 288*x.
%t Mathematica
P[0, x_] := 264 + 288*x;
P[n_, x_] := P[n, x] = Expand[Together[((11*n + 22 + (12*n + 24)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 264 + 288*x; for(n = 1, N, my(i = n + 1); v[i] = ((11*n + 22 + (12*n + 24)*x)*v[i - 1])/(n + 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.