Real-rooted atlas

A013622 → x ↦ x/(x+1) → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(11 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1111627
2121352256729
3133158088448409619683
4146418518418585618022465536531441
51610511171280340736049561603604480104857614348907
6177156115460896562214401090355201189478406920601616777216387420489
719487171198414832865810176209893376030529945602664431616129184563226843545610460353203
821435888124943578881269854924836941234176671658803207815666073656841207808236223201284294967296282429536481
92357947691308676788641795937679366095303639041329884430336193437735321618757598576641169304846336425201762304687194767367625597484987
102593742460137727163056024694143091209578334289920243812145561604255630177075251583396085760428745110323202338609692672075591424409601099511627776205891132094649
1128531167061145649867297763319990348928014487230613504042144670875648085821875237683212483181852753921296953958727680943239242711040457328117678080133040906960896175921860444165559060566555523
1231384283767215477984075731243823872605849621247938233139206953870694481920161835536162488322746300007605862434239584510410752311268950094643202012243717783552087806998594191362322168557862912281474976710656150094635296999121
13345227121439316527931023579685697103438760448303845516733890561104892788123238402892810208904478725610298586966261768160434308314562568902291972706795527193771291076198404185466932989788161660350518871982084025092166962380845035996273704964052555153018976267
143797498335832417733087520240544731128274640924164253837234274467841701534893709787136494991969079210803210799824779910053888179529554782920376322284921606328077516822156815576514691072161140476920106844168523132663542841344309932096856103321669355434261505638472057594037927936109418989131512359209
Technical details
Node id
n-780fe707feb8bee2 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-89a932e93fe55d02

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:         11,          16;
  2:        121,         352,          256;
  3:       1331,        5808,         8448,         4096;
  4:      14641,       85184,       185856,       180224,         65536;
  5:     161051,     1171280,      3407360,      4956160,       3604480,       1048576;
  6:    1771561,    15460896,     56221440,    109035520,     118947840,      69206016,      16777216;
  7:   19487171,   198414832,    865810176,   2098933760,    3052994560,    2664431616,    1291845632,     268435456;
  8:  214358881,  2494357888,  12698549248,  36941234176,   67165880320,   78156660736,   56841207808,   23622320128,   4294967296;
  9: 2357947691, 30867678864, 179593767936, 609530363904, 1329884430336, 1934377353216, 1875759857664, 1169304846336, 425201762304, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (11+16*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[(11 + 16*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] = (11 + 16*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

Candidate equalities ?

Same fingerprint ?

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.