Real-rooted atlas

A038305 → x ↦ x+1 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(16 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 256 + 96\,x + 9\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0256969361
140962304432276859
2655364915213824172881130321
310485769830403686406912064802432476099
41677721618874368884736022118403110402332872947045881
526843545635232153619818086461931520116121601306368816482187893871739
64294967296644245094442278584321585446912371589120557383685225472279936656116983563041
76871947673611596411699286973087744380507258881070176665620065812482508226562015539294478419683322687697779
810995116277762061584302080173946175488086973087744028538044416064210599936100329062401074954240755827203149280590496131066257801
9175921860444163628388371660834016140984320191340793036807175279738880188351093145635315829964847297986560443418624027713664010392624177147116490258898219
102814749767106566333186975989766531099068989444081936918118401722067137331205166201411993611301065588736181624268390421284093952017736744960997691904340122245314412213314919066161
1145035996273704961097752409171558412349714603180032849042878968627239798884951654401343212367118336335803091779584629630797086728854183084032922310737920691733053443537271296110539728159432342052983462257059
1272057594037927936189151184349560832230528005926027264172896004444520448891495022917058563343106335938969694024865698283522014818550677504330556168470528413195210588163873705099264264116256768123804495363571283524782969799006685782884121
131152921504606846976324259173170675712042559016478651187203457920088890408960194508005000085504080234552062535270425073297519542272060444556520325120113333543475609601652780842352640185937844764672158469754060809904359628804285540224011479125601434890715181127029874798299
1418446744073709551616553402322211286548487782220156096217088068094426365841899520414950410666849075201867276848000820838464187641650028216321719311829911470080362667339121950720604445565203251207933348043292672811365140791296633879016243203656994324480146932807680367332019243046721288441413567621167681
Technical details
Node id
n-af83e2dbf104af1b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-e2c46986e009fe93

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:            256,             96,              9;
  1:           4096,           2304,            432,             27;
  2:          65536,          49152,          13824,           1728,            81;
  3:        1048576,         983040,         368640,          69120,          6480,           243;
  4:       16777216,       18874368,        8847360,        2211840,        311040,         23328,          729;
  5:      268435456,      352321536,      198180864,       61931520,      11612160,       1306368,        81648,        2187;
  6:     4294967296,     6442450944,     4227858432,     1585446912,     371589120,      55738368,      5225472,      279936,       6561;
  7:    68719476736,   115964116992,    86973087744,    38050725888,   10701766656,    2006581248,    250822656,    20155392,     944784,     19683;
  8:  1099511627776,  2061584302080,  1739461754880,   869730877440,  285380444160,   64210599936,  10032906240,  1074954240,   75582720,   3149280,    59049;
  9: 17592186044416, 36283883716608, 34016140984320, 19134079303680, 7175279738880, 1883510931456, 353158299648, 47297986560, 4434186240, 277136640, 10392624, 177147;
%F Formula
The row polynomials satisfy P(n,x) = (16+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 96*x + 9*x^2.
%t Mathematica
P[0, x_] := 256 + 96*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(16 + 3*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] = 256 + 96*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 3*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.