Real-rooted atlas

A038334 → x ↦ x+1 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + 2\,x + 2\,n\right)\,P_{n}(x) = \left(180 + 112\,x + 16\,x^{2} + 40\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 720 + 448\,x + 64\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0720448641232
11760017280537651240768
24160005632002764805734440961317120
39600000166400001126400036864005734403276841796608
42176000004608000003993600001802240004423680055050242621441307987968
5486400000012185600000129024000007454720000252313600049545216051380224209715240478785536
6107520000000311296000000389939200000275251200000119275520000322961408005284823040469762048167772161241349423104
7235520000000077414400000001120665600000093585408000004954521600000171756748800038755368960054358179840422785843213421772837780199833600
851200000000000188416000000000309657600000000298844160000000187170816000000792723456000002290089984000044291850240005435817984003758096384010737418241142473242968064
91105920000000000450560000000000082903040000000009083289600000000657457152000000032942063616000001162661068800000287897026560000487210352640005315022028800330712481792858993459234358824936669184
102375680000000000010616832000000000021626880000000000026528972800000000021799895040000000012623177318400000052707301785600000159450660864000003454764318720000519691042816000510242114764802886218022912687194767361028395173966512128
115079040000000000002470707200000000000552075264000000000074973184000000000006897532928000000000453437816832000000021880174018560000007830799122432000002072858591232000003992172101632000054047868452864004824107266867202501388953190454975581388830652810991776038912
12108134400000000000005688524800000000000013835960320000000000020610809856000000000020992491520000000000015450473758720000000084641725808640000000350082784296960000001096311877140480000025795573579776000004471232753827840005503055697018880045025001157427202155042790440964398046511104910295599149712670720
1322937600000000000000012976128000000000000003413114880000000000000553438412800000000000061832429568000000000005038197964800000000000309009475174400000000014510010138624000000005251241764454400000001461749169520640000003095468829573120000048777084587212800005503055697018880004156153952993280018471795346636803518437208883226944749734831495053312
144849664000000000000000293601280000000000000008304721920000000000000014562623488000000000000017710029209600000000000015829101969408000000000010748155658240000000000056504589746176000000000232160162217984000000007468432731668480000000187103893698641920000036020000925941760000052028890226360320000541839330168012800037999121855938560015762598695796736281474976710656795234235417188989140992
Technical details
Node id
n-8c5c43d91328f213 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-b29ddf49d7bb6a39

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:              720,              448,               64;
  1:            17600,            17280,             5376,              512;
  2:           416000,           563200,           276480,            57344,             4096;
  3:          9600000,         16640000,         11264000,          3686400,           573440,            32768;
  4:        217600000,        460800000,        399360000,        180224000,         44236800,          5505024,           262144;
  5:       4864000000,      12185600000,      12902400000,       7454720000,       2523136000,        495452160,         51380224,         2097152;
  6:     107520000000,     311296000000,     389939200000,     275251200000,     119275520000,      32296140800,       5284823040,       469762048,       16777216;
  7:    2355200000000,    7741440000000,   11206656000000,    9358540800000,    4954521600000,    1717567488000,     387553689600,     54358179840,     4227858432,     134217728;
  8:   51200000000000,  188416000000000,  309657600000000,  298844160000000,  187170816000000,   79272345600000,   22900899840000,   4429185024000,   543581798400,   37580963840,   1073741824;
  9: 1105920000000000, 4505600000000000, 8290304000000000, 9083289600000000, 6574571520000000, 3294206361600000, 1162661068800000, 287897026560000, 48721035264000, 5315022028800, 330712481792, 8589934592;
%F Formula
The row polynomials satisfy (2*n+7+2*x)*P(n,x) = (40*n+180+(16*n+112)*x+16*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 720 + 448*x + 64*x^2.
%t Mathematica
P[0, x_] := 720 + 448*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((40*n + 180 + (16*n + 112)*x + 16*x^2)*P[n - 1, x])/(2*n + 7 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 720 + 448*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((40*n + 180 + (16*n + 112)*x + 16*x^2)*v[i - 1])/(2*n + 7 + 2*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

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.