Real-rooted atlas

A027467 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(16 - 4\,x + 12\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(256 + 192\,x + 48\,x^{2} + 4\,x^{3} + 192\,n x + 80\,n x^{2} + 8\,n x^{3} + 16\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1161632
2256512144912
34096122886912102424320
465536262144221184655366400620800
510485765242880589824026214405120003686415360000
61677721610066329614155776083886080245760003538944200704371200000
72684354561879048192317089382423488102409175040001981808642247884810485768806400000
8429496729634359738368676457349126012954214429360128000845571686414386462721342177285308416205824000000
96871947673661847529062413915694039041443109011456845571686400304405807104690550210569663676416764411904262144004751360000000
1010995116277761099511627776027831388078080329853488332802254857830400097409858273282762200842240515396075520611529523204194304000126877696108544000000000
111759218604441619351404648857654425825574912072567767433216056693568307200028573558426828897229469646848226774273228803587639869440369098752000223304744966039797762457600000000000
12281474976710656337769972052787210449758510383104154811237190860801360645639372800078373188827873283111343028699136870813209198592172206713733120236223201280002143725551616115964116992283534950455214080000000000
1345035996273704965854679515581644819759543365088051232200737335699046431446032554393600020377029095247052892451335709917184301881912522178567163799291297792122836064665600014863163824537612060268167168589752696832131533373441232076800000000000
14720575940379279361008806316530991104368844809481643622465572410574514421767043911292184166400507161613037259980825886373998776811529660221200709713922674485068751175685503055697018880083233717417410569005000231485446605230204518429463475650566039797760027328512000000000000
Technical details
Node id
n-cb3b4ec925637cfb · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-9e3a89ee8f65d0f5

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:          16,           16;
  2:         256,          512,           144;
  3:        4096,        12288,          6912,          1024;
  4:       65536,       262144,        221184,         65536,         6400;
  5:     1048576,      5242880,       5898240,       2621440,       512000,        36864;
  6:    16777216,    100663296,     141557760,      83886080,     24576000,      3538944,      200704;
  7:   268435456,   1879048192,    3170893824,    2348810240,    917504000,    198180864,    22478848,    1048576;
  8:  4294967296,  34359738368,   67645734912,   60129542144,  29360128000,   8455716864,  1438646272,  134217728,   5308416;
  9: 68719476736, 618475290624, 1391569403904, 1443109011456, 845571686400, 304405807104, 69055021056, 9663676416, 764411904, 26214400;
%F Formula
The row polynomials satisfy (16+(12*n-4)*x+n^2*x^2)*P(n,x) = (256+(192*n+192)*x+(16*n^2+80*n+48)*x^2+(4*n^2+8*n+4)*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[((256 + (192*n + 192)*x + (16*n^2 + 80*n + 48)*x^2 + (4*n^2 + 8*n + 4)*x^3)*P[n - 1, x])/(16 + (12*n - 4)*x + 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] = ((256 + (192*n + 192)*x + (16*n^2 + 80*n + 48)*x^2 + (4*n^2 + 8*n + 4)*x^3)*v[i - 1])/(16 + (12*n - 4)*x + 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.