Real-rooted atlas

A038259 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(36 - 30\,x + 90\,n x + 25\,n^{2} x^{2}\right)\,P_{n}(x) = \left(216 + 540\,x + 450\,x^{2} + 125\,x^{3} + 540\,n x + 750\,n x^{2} + 250\,n x^{3} + 150\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
162026
236240225501
32162160405020008426
4129617280486004800015625130801
577761296004860007200004687501125001924626
646656933120437400086400008437500405000076562527246901
72799366531840367416009072000011812500085050000321562505000000374604626
816796164478976029393280087091200014175000001360800000771750000240000000316406255033004801
910077696302330880226748160078382080001530900000018370800000138915000006480000000170859375019531250066373304426
10604661762015539200170061120006718464000015309000000022044960000020837250000012960000000051257812500117187500001181640625861937060501
11362797056133025587201247114880005542732800001443420000000242494560000027505170000002138400000000112767187500038671875000077988281250703125000011049342880026
12217678233687071293440897922713600443418624000012990780000000249422976000003300620400000030792960000000202980937500009281250000000280757812500050625000000041259765625140088030270001
1313060694016565963407360636708833280034586652672000112586760000000243187401600000367783416000000400308480000000316650262500000180984375000000729970312500001974375000000032182617187502392578125001759231760987426
147836416409636569943244804456961832960026411625676800094572878400000022697490816000003861725868000000480370176000000044331036750000003040537500000000153293765625000055282500000000013516699218750020097656250000137329101562521909368497889301
Technical details
Node id
n-f8c750a328bdc1fd · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2945f948d8e5ebc1

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:        6,        20;
  2:       36,       240,        225;
  3:      216,      2160,       4050,       2000;
  4:     1296,     17280,      48600,      48000,       15625;
  5:     7776,    129600,     486000,     720000,      468750,      112500;
  6:    46656,    933120,    4374000,    8640000,     8437500,     4050000,      765625;
  7:   279936,   6531840,   36741600,   90720000,   118125000,    85050000,    32156250,    5000000;
  8:  1679616,  44789760,  293932800,  870912000,  1417500000,  1360800000,   771750000,  240000000,   31640625;
  9: 10077696, 302330880, 2267481600, 7838208000, 15309000000, 18370800000, 13891500000, 6480000000, 1708593750, 195312500;
%F Formula
The row polynomials satisfy (36+(90*n-30)*x+25*n^2*x^2)*P(n,x) = (216+(540*n+540)*x+(150*n^2+750*n+450)*x^2+(125*n^2+250*n+125)*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[((216 + (540*n + 540)*x + (150*n^2 + 750*n + 450)*x^2 + (125*n^2 + 250*n + 125)*x^3)*P[n - 1, x])/(36 + (90*n - 30)*x + 25*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] = ((216 + (540*n + 540)*x + (150*n^2 + 750*n + 450)*x^2 + (125*n^2 + 250*n + 125)*x^3)*v[i - 1])/(36 + (90*n - 30)*x + 25*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.