Real-rooted atlas

A038322 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(19 + 24\,n x\right)\,P_{n}(x) = \left(361 + 912\,x + 576\,x^{2} + 456\,n x + 576\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1194867
2361182417283913
36859519849849655296212635
4130321131692837428484202496165888011051473
524760993127704011852352019961856015759360047775744557264563
6470458817131165123377920320758550528089828352005446434816133772083227490578841
78938717391580741601689852680512252218050560398239027200362187915264177916870656366917713921333807603339
81698356304134324674777622762679063047667428737024151330830336001835085437337613521682169856557714925158499067782758463877373610145
9322687697779733689923371255605973139712218521719005184517551439749120784499024461824770735883681792476846261010432169405908516864264180754022403027243871898659
1061310662578011548900949339201320641862068160593130380156928016389128925388800298109629295493123660995447488512030200263197327360160935613091020805019434326425600697437190619136142233709178126569
111164902588982193237202984118928306682387969161601549553118159974404893325636294656001038415208712634368153029609705019801615779637520603545601121184771200778240524530887111475200145764372839399424182601737180282886634725070178264443
122213314919066161670983891253741446992358445696884483925534566005268480139459780634397696003382266679806866227258151251687907524608719551470939521679366390753195844435968039864347420472115200166171385036915343364163319607710449664474764516668735488307596227764974716017
1342052983462257059138110850949728446415701023055337549696969607037803301314563827396201855136768001044274837390369947648205190845241616551116829621535553676975800323157032078747151368192246162345321415311360013681444034706029936645141699715522405335041172668356171776655361227083673851500953614185668933028214622355
147990066857828841212825960488663674364834803934439331568492823446861095970740879361018087389693466380288030864122971759822897152682259560428375032463361125618351039725080412161399617554911237106565121309583677109929456435209098160283079509907865645589737477631993970688155964891370846295162883264042572444992536576315535801847528816640651222103102988207607097
Technical details
Node id
n-e5750a3d58a1c7a2 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-58fdb67dc64c435b

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:           19,            48;
  2:          361,          1824,           1728;
  3:         6859,         51984,          98496,           55296;
  4:       130321,       1316928,        3742848,         4202496,         1658880;
  5:      2476099,      31277040,      118523520,       199618560,       157593600,        47775744;
  6:     47045881,     713116512,     3377920320,      7585505280,      8982835200,      5446434816,      1337720832;
  7:    893871739,   15807416016,    89852680512,    252218050560,    398239027200,    362187915264,    177916870656,     36691771392;
  8:  16983563041,  343246747776,  2276267906304,   7667428737024,  15133083033600,  18350854373376,  13521682169856,   5577149251584,    990677827584;
  9: 322687697779, 7336899233712, 55605973139712, 218521719005184, 517551439749120, 784499024461824, 770735883681792, 476846261010432, 169405908516864, 26418075402240;
%F Formula
The row polynomials satisfy (19+24*n*x)*P(n,x) = (361+(456*n+912)*x+(576*n+576)*x^2)*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[Together[((361 + (456*n + 912)*x + (576*n + 576)*x^2)*P[n - 1, x])/(19 + 24*n*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] = ((361 + (456*n + 912)*x + (576*n + 576)*x^2)*v[i - 1])/(19 + 24*n*x)); 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.