Real-rooted atlas

A013620 → basis A008288 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-7 + 3\,x\right)\,P_{n-2}(x) + \left(10 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18311
273579139
3674792270271763
462299762518411078122363
55757211278581612272434212243283667
653211712509191147473525069126603153097293598219
7491816613488762149945588747244289348254383453946218745642179
845456841142482036186132312132223050558655381439143222117861858956561578955451
9420141248148152404722242736081864380402961311888316384542665460368640837629856196837343852147
10388321459315214660485258118281452494946043015211870470611463767416423057802913805713273282921060815904993154256107
1135891157194154686004044292636919790320552246745225831043980107451232020352503021967979875299122298352212101108469677821771471181629920803
12331729069789155968053240632553883944723986222488245318833246803517548385998326817044905821884369129213681447302749625173264386159722285196353144114988571946011
1330660525975321561686397670135648525115540482624350980214322283157746027038717910225121571622860043979374380451948834210276162890465319195879704761564030863744017401594323190124915614931
14283384224867971554442210421313852271659214175714325340925315666559612571533973368010399112031803609561377699128574382121734074671687455111308181769636834089175560104233550094579124074277347829692411669615210059
Technical details
Node id
n-419be61a3eb0b2df · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-fcbcaad3a652ee32

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         8,          3;
  2:        73,         57,          9;
  3:       674,        792,        270,         27;
  4:      6229,       9762,       5184,       1107,        81;
  5:     57572,     112785,      81612,      27243,      4212,       243;
  6:    532117,    1250919,    1147473,     525069,    126603,     15309,      729;
  7:   4918166,   13488762,   14994558,    8747244,   2893482,    543834,    53946,    2187;
  8:  45456841,  142482036,  186132312,  132223050,  55865538,  14391432,  2211786,  185895,   6561;
  9: 420141248, 1481524047, 2224273608, 1864380402, 961311888, 316384542, 66546036, 8640837, 629856, 19683;
%F Formula
The row polynomials satisfy P(n,x) = -(7-3*x)*P(n-2,x) + (10+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(7 - 3*x)*P[n - 2, x] + (10 + 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] = 1; v[2] = 8 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(7 - 3*x)*v[i - 2] + (10 + 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

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.