Real-rooted atlas

A013620 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 - 45\,x + 135\,n x + 81\,n^{2} x^{2}\right)\,P_{n}(x) = \left(125 + 675\,x + 1215\,x^{2} + 729\,x^{3} + 675\,n x + 2025\,n x^{2} + 1458\,n x^{3} + 405\,n^{2} x^{2} + 729\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
153641
2253607291114
31252700109351166425424
462518000109350233280164025525280
5312511250091125029160004100625212576410169264
615625675000683437529160000615093756377292026040609188007904
77812539375004784062525515000071760937511160261009114213153061100163358173056
83906252250000031893750020412000007176093750148803480001822842630012244400640348678440158399081216
919531251265625002050312500153090000006458484375016740391500027342639450027549901440015690529804538742048900994049342720
109765625703125000128144531251093500000005382070312501674039150000341782993125045916502400003922632451125193710244500042190091252116626239504896
114882812538671875007831054687575178125000042287695312501534535887500037596129243750631351908000007191492827062553270317237500232045501886554518872583696274048124542976
122441406252109375000046986328125050118750000003171577148437513153164750000037596129243750075762228960000010787239240593751065406344750000696136505659650271132355021760477305916652894461463798349824
1312207031251142578125002776464843750325771875000002290583496093751068694635937500349106914406250082075748040000001402341101277187517312853102187500150829576226257508811801538207200310248845824378549820570235248471863583500857344
146103515625615234375000161960449218752073093750000001603408447265625831206939062500030546855010546875820757480400000001636064618156718752423799434306250002639517583959506252056087025581680001085870960385324753487439916467388051472783023662251146917847352238080
Technical details
Node id
n-c2d97b40275a2c81 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-241b495e7a2888a4

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:       5,        36;
  2:      25,       360,        729;
  3:     125,      2700,      10935,       11664;
  4:     625,     18000,     109350,      233280,      164025;
  5:    3125,    112500,     911250,     2916000,     4100625,      2125764;
  6:   15625,    675000,    6834375,    29160000,    61509375,     63772920,     26040609;
  7:   78125,   3937500,   47840625,   255150000,   717609375,   1116026100,    911421315,    306110016;
  8:  390625,  22500000,  318937500,  2041200000,  7176093750,  14880348000,  18228426300,  12244400640,   3486784401;
  9: 1953125, 126562500, 2050312500, 15309000000, 64584843750, 167403915000, 273426394500, 275499014400, 156905298045, 38742048900;
%F Formula
The row polynomials satisfy (25+(135*n-45)*x+81*n^2*x^2)*P(n,x) = (125+(675*n+675)*x+(405*n^2+2025*n+1215)*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[((125 + (675*n + 675)*x + (405*n^2 + 2025*n + 1215)*x^2 + (729*n^2 + 1458*n + 729)*x^3)*P[n - 1, x])/(25 + (135*n - 45)*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] = ((125 + (675*n + 675)*x + (405*n^2 + 2025*n + 1215)*x^2 + (729*n^2 + 1458*n + 729)*x^3)*v[i - 1])/(25 + (135*n - 45)*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

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.