Real-rooted atlas

A013611 → basis A007318 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 + 4\,n x\right)\,P_{n}(x) = \left(25 + 40\,x + 16\,x^{2} + 20\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
15813
2258048153
31256007202561701
4625400072005120128018225
53125250006000064000320006144190269
615625150000450000640000480000184320286721948617
7781258750003150000560000056000003225600100352013107219663317
8390625500000021000000448000005600000043008000200704005242880589824196101729
91953125281250001350000003360000005040000004838400003010560001179648002654208026214401937102445
109765625156250000843750000240000000042000000004838400000376320000019660800006635520001310720001153433618983603961
11488281258593750005156250000165000000003300000000044352000000413952000002703360000012165120000360448000063438848050331648184799573253
12244140625468750000030937500000110000000000247500000000380160000000413952000000324403200000182476800000720896000001903165440030198988802181038081788720397713
13122070312525390625000182812500000715000000000178750000000030888000000003843840000000351436800000023721984000001171456000000412352512000981467136001417674752093952409617228201725341
1461035156251367187500001066406250000455000000000012512500000000240240000000003363360000000035143680000000276756480000001640038400000072161689600002290089984000496186163200657666867204026531840165221278841385
Technical details
Node id
n-c6f185eccade38bd · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-d2174dc9221c8d54

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:       5,        8;
  2:      25,       80,        48;
  3:     125,      600,       720,       256;
  4:     625,     4000,      7200,      5120,      1280;
  5:    3125,    25000,     60000,     64000,     32000,      6144;
  6:   15625,   150000,    450000,    640000,    480000,    184320,     28672;
  7:   78125,   875000,   3150000,   5600000,   5600000,   3225600,   1003520,    131072;
  8:  390625,  5000000,  21000000,  44800000,  56000000,  43008000,  20070400,   5242880,   589824;
  9: 1953125, 28125000, 135000000, 336000000, 504000000, 483840000, 301056000, 117964800, 26542080, 2621440;
%F Formula
The row polynomials satisfy (5+4*n*x)*P(n,x) = (25+(20*n+40)*x+(16*n+16)*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[((25 + (20*n + 40)*x + (16*n + 16)*x^2)*P[n - 1, x])/(5 + 4*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] = ((25 + (20*n + 40)*x + (16*n + 16)*x^2)*v[i - 1])/(5 + 4*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

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.