Real-rooted atlas

A038326 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1121123
2144264121529
3172847524356133112167
420736760321045446388814641279841
52488321140480209088019166408784601610516436343
6298598416422912376358404599936031624560115956721771561148035889
735831808229920768632282112965986560885487680487018224148811124194871713404825447
84299816963153199104101165137921854694195221251704320155845831687142933952187076841621435888178310985281
95159780352425681879041560833556483338449551364590368133124207837455362571456222721010214944642315075914823579476911801152661463
1061917364224567575838720234125033472057230563737609180736266240100988098928647714368668160404085977856013890455488802829537229202593742460141426511213649
117430083706887492001071104343383382425609443043016704017312245530624022217381764300820365933283942413334837269248061118004150720186749457127203423740047332285311670611952809757913927
1289161004482569807710493081649447207069286415108868826726403116204195512320457043282008473648878239881461763840433133543424220024814942592089639739421056024650928340790441084880567984313842837672121914624432020321
131069932053790721275002364100608701251300255334423569835369693184540142060555468808912343999165235210892864887868620899851261472129024686477422620887043495949837421184012818482737211008320462068430275248959482676847634522712143931504036361936467383
14128391846454886416476953628377088981751820357468163599756674644049929074386617331875841663637546510843904228750162645241036823964302753310965761922136783338483712117463914537351782453837627496286233617945875832095411241125965448551984579981564018040837974983358324111592836324538749809
Technical details
Node id
n-b0b0fff0ab69f594 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-60cc3cc9b4461a16

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:         12,          11;
  2:        144,         264,          121;
  3:       1728,        4752,         4356,         1331;
  4:      20736,       76032,       104544,        63888,        14641;
  5:     248832,     1140480,      2090880,      1916640,       878460,       161051;
  6:    2985984,    16422912,     37635840,     45999360,     31624560,     11595672,      1771561;
  7:   35831808,   229920768,    632282112,    965986560,    885487680,    487018224,    148811124,     19487171;
  8:  429981696,  3153199104,  10116513792,  18546941952,  21251704320,  15584583168,   7142933952,   1870768416,   214358881;
  9: 5159780352, 42568187904, 156083355648, 333844955136, 459036813312, 420783745536, 257145622272, 101021494464, 23150759148, 2357947691;
%F Formula
The row polynomials satisfy P(n,x) = (12+11*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[(12 + 11*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] = (12 + 11*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. [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.