Real-rooted atlas

A038323 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 - 99\,x + 297\,n x + 81\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1331 + 3267\,x + 2673\,x^{2} + 729\,x^{3} + 3267\,n x + 4455\,n x^{2} + 1458\,n x^{3} + 891\,n^{2} x^{2} + 729\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1113647
21217927291642
3133113068240571166450120
4146411916645292545132161640251412800
516105126353809702990141134409021375212576437760000
617715613478701616009933531049568029770537514030042426040609971200000
7194871714464333722465529759597704184076411046255401566324200512689330611001624262400000
821435888156123052483616110313210519593638416810430175015844594550488225583292269376814083486784401592384000000
9235794769169452277444511421315724173573295033633284651746503921537151224291144424863613334152296963451916556993874204890014197760000000
102593742460184886116876070320430912052727580350528061021861535250862738173269288006471683749048891891755520189855410634454261625379000421900912521335104000000000
11285311670611102712201419969454191267064541254652801736010548064636807501739855316093048193756614746725814789797256044807657501562256152578283354295005105001041504145188725836967808000000000000
12313842837672112325464170395212479532472525146050682410921280174043066507323753280870024632604842626455244279676390450647559582722526975515544529511344446758898000336930068739270659649118104787247730591665289179937280000000000
1334522712143931146878448030542816223392214282682865247584761743042765350945616366255864555169030781088707975857045705249305740433503388167227149974457354374055639716306035001606033327657189864264911944492284868254746081363274982057023524844107468800000000000
14379749833583241173994469205412242082002000832944191211346618666440256425864045624920402510034905511452669848167628535248129825872047262895370745395218549684934440542883124913703262225878006183228311480180961218932146483937286452556154482649717976723678162282536514727830236622592987392000000000000
Technical details
Node id
n-0ccca49e54dc7c79 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-4d60fe62fd169372

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:         11,          36;
  2:        121,         792,          729;
  3:       1331,       13068,        24057,         11664;
  4:      14641,      191664,       529254,        513216,        164025;
  5:     161051,     2635380,      9702990,      14113440,       9021375,       2125764;
  6:    1771561,    34787016,    160099335,     310495680,     297705375,     140300424,      26040609;
  7:   19487171,   446433372,   2465529759,    5977041840,    7641104625,    5401566324,    2005126893,     306110016;
  8:  214358881,  5612305248,  36161103132,  105195936384,  168104301750,  158445945504,   88225583292,   26937681408,   3486784401;
  9: 2357947691, 69452277444, 511421315724, 1735732950336, 3328465174650, 3921537151224, 2911444248636, 1333415229696, 345191655699, 38742048900;
%F Formula
The row polynomials satisfy (121+(297*n-99)*x+81*n^2*x^2)*P(n,x) = (1331+(3267*n+3267)*x+(891*n^2+4455*n+2673)*x^2+(729*n^2+1458*n+729)*x^3)*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[((1331 + (3267*n + 3267)*x + (891*n^2 + 4455*n + 2673)*x^2 + (729*n^2 + 1458*n + 729)*x^3)*P[n - 1, x])/(121 + (297*n - 99)*x + 81*n^2*x^2)]];
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] = ((1331 + (3267*n + 3267)*x + (891*n^2 + 4455*n + 2673)*x^2 + (729*n^2 + 1458*n + 729)*x^3)*v[i - 1])/(121 + (297*n - 99)*x + 81*n^2*x^2)); 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.