Real-rooted atlas

A038271 → x ↦ x/(x+1) → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(49 - 84\,x + 252\,n x + 144\,n^{2} x^{2}\right)\,P_{n}(x) = \left(343 + 1764\,x + 3024\,x^{2} + 1728\,x^{3} + 1764\,n x + 5040\,n x^{2} + 3456\,n x^{3} + 1008\,n^{2} x^{2} + 1728\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
174855
24967212962017
33437056272162764862263
42401658563810247741445184001741825
51680757624044452801354752018144000895795245687799
61176494840416466754401896652803810240003762339841463132161144869985
7823543395300644574193122323399680622339200092177326087169347584229323571227724880503
857648013162405124269246912260220764168712748800017206434201620074173235212842119987234828517376653796608257
9403536072490394032384232222082732318023681097806348800271001338675242155763793924045267795968219419659468851597803520015093024313015
102824752491936973136033620319432027323180236801280774073600037940187414528737725866393609438958190592076796880814080361184624640007492001071104342405614469601
11197732674314914693147228764051069602629856097792014088514809600048689907181977611360978342461441816999451688960197111994089472013905608048640005768840824750081069932053790727655765629806775
121384128720111389402039682416180289846424545323579392014792940550080005842788861837312159053696794460163052559078837452841393518758789120389357025361920002422913146395033689874292518420481506820975755264169076414191378177
1396889010407863696321342419988400579638422336244457246721495730655619200066461723303399424206769805832798208462971460290347008753362041409961984885787232698368000734950321073160192408928030958813184137120708793729024209706682542981123694721541474812023
146782230728496510941499350416323860473371361989956324372889614658160425068160072369432041479372825329301214517780486481600444064858112123049133430293790721736142976088801280018006282866292424704133583156779878973446718914730892722176205512548892121497628888165452349440080000267749646389345
Technical details
Node id
n-df3f3321552d1634 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-697af679532967a7

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:        7,         48;
  2:       49,        672,        1296;
  3:      343,       7056,       27216,        27648;
  4:     2401,      65856,      381024,       774144,        518400;
  5:    16807,     576240,     4445280,     13547520,      18144000,       8957952;
  6:   117649,    4840416,    46675440,    189665280,     381024000,     376233984,     146313216;
  7:   823543,   39530064,   457419312,   2323399680,    6223392000,    9217732608,    7169347584,    2293235712;
  8:  5764801,  316240512,  4269246912,  26022076416,   87127488000,  172064342016,  200741732352,  128421199872,   34828517376;
  9: 40353607, 2490394032, 38423222208, 273231802368, 1097806348800, 2710013386752, 4215576379392, 4045267795968, 2194196594688, 515978035200;
%F Formula
The row polynomials satisfy (49+(252*n-84)*x+144*n^2*x^2)*P(n,x) = (343+(1764*n+1764)*x+(1008*n^2+5040*n+3024)*x^2+(1728*n^2+3456*n+1728)*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[((343 + (1764*n + 1764)*x + (1008*n^2 + 5040*n + 3024)*x^2 + (1728*n^2 + 3456*n + 1728)*x^3)*P[n - 1, x])/(49 + (252*n - 84)*x + 144*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] = ((343 + (1764*n + 1764)*x + (1008*n^2 + 5040*n + 3024)*x^2 + (1728*n^2 + 3456*n + 1728)*x^3)*v[i - 1])/(49 + (252*n - 84)*x + 144*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

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.