Real-rooted atlas

A038301 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(16 + 11\,x + 11\,n\right)\,P_{n}(x) = \left(540 + 1408\,x + 484\,x^{2} + 220\,n + 484\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 2160 + 5632\,x + 1936\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
02160563219369728
16080028512037171285184802816
21568000107008002509056021807104374809662914560
3384000003449600001177088000183997440011993907201649162244764729344
49088000001013760000045534720000103583744000121438310400633278300167256313856352187318272
5209920000002799104000001561190400000467489792000079759482880007480599920640325082860748831927780966425563645345792
6476160000000738918400000049264230400000183179673600000411391016960000561506759475200438861862010880163470238547968140482236252161829587348619264
71064960000000018855936000000014630584320000006502878412800000181347876864000003258216854323200037059446125363200248270424794726408091776808124416618121839509504129478489286901760
82355200000000004685824000000000414830592000000002145819033600000007153166254080000001595861316403200000238935902650368000023294508993085440001365487336370995200395597977286082560271973609384181769079256848778919936
9516096000000000011399168000000000011339694080000000006692600217600000000259644103065600000006924264933949440000012873281285652480000016520710983254016000014093177940816691200073432874533729075200191469421006463959041196683881290399744631800984524552142848
1011223040000000000027249868800000000003009380352000000000019957861580800000000088342322872320000000027418417283727360000006093353141875507200000971013216974929920000010903669248947650560000826799772527912550400038772557753808951705609190532208310270033925265409077677758873643681889966544218226688
112424832000000000000641957888000000000007793462476800000000005737885204480000000000285397420605440000000001010636173659340800000002613889114382008320000004979139995932557312000006942744501370748928000006929887567108951244800004729294698859659788288002016173003198065488691204380820352627895382835223167799941782139043843003425083105094355910656
12521011200000000000001493696512000000000000197723029504000000000001600257628569600000000008836343214899200000000003516096221859020800000000103758647162358988800000002300222420656167321600000038339377968680691302400000475192290316042371072000004268810741339113966796800026484050313614094814412800103496880830834028419481602075834874783679658328064101938319743841411792896205517389334486959700049920
131114112000000000000000343867392000000000000004929198489600000000000004349906649088000000000000264042508713984000000000001166397304366694400000000003867705844044922880000000009782958161022418944000000001897683497041338040320000000281155438436991736217600000031362691160858796490752000002561286444803468380078080000145662276724877521479270400052544570267961891351429120097860786954087755321180160448528606872902211888742414004196694415864359796342784
142372403200000000000000078433484800000000000000012104132198400000000000000115671857889280000000000000765583570239488000000000000371771852269289472000000000013685728371235880960000000000388980702029660815360000000008609003181699728670720000000014844102021301133115392000000019793342865964218229719040000020072122342949629754081280000015026213809513681163124736000078881725241779827016466432000264224124776036939367186432004592932934378518649740722176197352587024076973231046656950737950171172051122527404032
Technical details
Node id
n-540fd34dbc661863 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e1e7aa9d2da54c7d

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:             2160,               5632,                1936;
  1:            60800,             285120,              371712,               85184;
  2:          1568000,           10700800,            25090560,            21807104,              3748096;
  3:         38400000,          344960000,          1177088000,          1839974400,           1199390720,            164916224;
  4:        908800000,        10137600000,         45534720000,        103583744000,         121438310400,          63327830016,            7256313856;
  5:      20992000000,       279910400000,       1561190400000,       4674897920000,        7975948288000,        7480599920640,         3250828607488,          319277809664;
  6:     476160000000,      7389184000000,      49264230400000,     183179673600000,      411391016960000,      561506759475200,       438861862010880,       163470238547968,        14048223625216;
  7:   10649600000000,    188559360000000,    1463058432000000,    6502878412800000,    18134787686400000,    32582168543232000,     37059446125363200,     24827042479472640,      8091776808124416,      618121839509504;
  8:  235520000000000,   4685824000000000,   41483059200000000,  214581903360000000,   715316625408000000,  1595861316403200000,   2389359026503680000,   2329450899308544000,   1365487336370995200,   395597977286082560,    27197360938418176;
  9: 5160960000000000, 113991680000000000, 1133969408000000000, 6692600217600000000, 25964410306560000000, 69242649339494400000, 128732812856524800000, 165207109832540160000, 140931779408166912000, 73432874533729075200, 19146942100646395904, 1196683881290399744;
%F Formula
The row polynomials satisfy (11*n+16+11*x)*P(n,x) = (220*n+540+(484*n+1408)*x+484*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 2160 + 5632*x + 1936*x^2.
%t Mathematica
P[0, x_] := 2160 + 5632*x + 1936*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((220*n + 540 + (484*n + 1408)*x + 484*x^2)*P[n - 1, x])/(11*n + 16 + 11*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2160 + 5632*x + 1936*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((220*n + 540 + (484*n + 1408)*x + 484*x^2)*v[i - 1])/(11*n + 16 + 11*x)); 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

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.