Real-rooted atlas

A038322 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(24 + 19\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1241943
25769123611849
3138243283225992685979507
4331776105062412476166584641303213418801
57962624315187204990464039507840156385202476099147008443
6191102976907739136179656704018963763201125973440356558256470458816321363049
7458647142425416695808603646525447964780544063054512640299508935047903708008893871739271818611107
811007531417669714365644819316688814083058475728896302661660672019168571842567587559687681716233738881698356304111688200277601
92641807540224188228787240965960578262630411010512624025613074983741030410351028794982454630429751296185353243799043668449616856322687697779502592611936843
106340338096537650194343264256017881734787891203775032899665920522999349641216049684938215915523277825785077760148282595039232044021395402272077445047466960613106625780121611482313284249
11152168114316902413251306621763584524530887111475201245760856889753601972454690075443202186137281500282881730692014521057289786651272589312038738827953999360102227462656387201618601492059464116490258898219929293739471222707
1236520347436056576346943300642537472151064895488104857639864347420472115207100836884271595520899439338674402099283073216697010749445637111133011443712278919561268795392098138364150131712023307861485656281633549194562687072221331491906616139959630797262576401
1387648833846535782490205258167059742724284749762935337779212437676395187299942424616234532141531136035078134208301681868837026919442096219750429312977891659507302417404580623172832460876547924037102735360242401759450825328645233674351779183232690554254748642232420529834622570591718264124282290785243
14210357201231685877762331458980317851811841199729933621894578176379914478980266616422482710548027995544616961309583677109929456435215551306165680412295168140702293879965635051529746565148976786178048514402049529330381619220361747793869327605765861715273992685219841160131147977718949761412980244331837182479900668578288412173885357344138503765449
Technical details
Node id
n-a48b231db9ea7677 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-f3eacc80e11c3732

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:            24,             19;
  2:           576,            912,            361;
  3:         13824,          32832,          25992,            6859;
  4:        331776,        1050624,        1247616,          658464,          130321;
  5:       7962624,       31518720,       49904640,        39507840,        15638520,         2476099;
  6:     191102976,      907739136,     1796567040,      1896376320,      1125973440,       356558256,       47045881;
  7:    4586471424,    25416695808,    60364652544,     79647805440,     63054512640,     29950893504,     7903708008,      893871739;
  8:  110075314176,   697143656448,  1931668881408,   3058475728896,   3026616606720,   1916857184256,   758755968768,   171623373888,   16983563041;
  9: 2641807540224, 18822878724096, 59605782626304, 110105126240256, 130749837410304, 103510287949824, 54630429751296, 18535324379904, 3668449616856, 322687697779;
%F Formula
The row polynomials satisfy P(n,x) = (24+19*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[(24 + 19*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] = (24 + 19*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.