Real-rooted atlas

A038326 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(529 - 276\,x + 828\,n x + 144\,n^{2} x^{2}\right)\,P_{n}(x) = \left(12167 + 19044\,x + 9936\,x^{2} + 1728\,x^{3} + 19044\,n x + 16560\,n x^{2} + 3456\,n x^{3} + 3312\,n^{2} x^{2} + 1728\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1234871
2529220812964033
312167761768942427648205415
42798412336064411350425436165184009791425
5643634367161840157684320146257920596160008957952446114375
61480358891853666784544010904067278643204113504000123619737614631321619665690625
734048254474974005870417517151108827079653888022075804800099513888768235564277762293235712845234534375
87831098528113074529716485371926340032996531263078410154870208000610351851110421671913553924219553710083482851737635605366890625
91801152661463338303456413921588555360552323438032857620484204116266112003158570829496321495362035220484367238089932872095030968325159780352001475493095234375
1041426511213649864553277502240456709666158792011296393675038720161157790200960001452942581568307285983317025177603348215868948480829092856135680118674948096000749200107110460316482337890625
1195280975791392721873197920806672128386161709082640357248449973099520582470298869184000612657455227969536435075584147398656211774653710991360699201642007756801501238093414400018954762709893121069932053790722437373625927734375
1221914624432020321548818420558421952354345806317068086410955619132508385280200952253109868480002415620823470279884820013476870780338176116899608848467230724824491329853521920138113904594124800026157572539652505629530124684623872150682097575526497522924588837890625
135040363619364673831367472564558068030496317632808003052576327573012062000719872667608040887229728000902838282772017106944854861369194760159232582549717428195033088288504581525240610816103240143684108288000260703806311869972484414753640351268864450539471750823936209706682542981123868628186780615234375
141159283632453874980933871243522130608137625845231470147485774569588955443996748345344214969789165687972416003230154745028772315955234408170110089096409088267972870016969715220481548307920852124611379266486652532565737472002098665640810553278464473850224064369524736725368549518826536966752555177883992064288881654523494400152311605959636962890625
Technical details
Node id
n-3af8c990f1653979 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e4d74122052cd055

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:            23,             48;
  2:           529,           2208,            1296;
  3:         12167,          76176,           89424,           27648;
  4:        279841,        2336064,         4113504,         2543616,          518400;
  5:       6436343,       67161840,       157684320,       146257920,        59616000,         8957952;
  6:     148035889,     1853666784,      5440109040,      6727864320,      4113504000,      1236197376,       146313216;
  7:    3404825447,    49740058704,    175171511088,    270796538880,    220758048000,     99513888768,     23556427776,     2293235712;
  8:   78310985281,  1307452971648,   5371926340032,   9965312630784,  10154870208000,   6103518511104,   2167191355392,   421955371008,   34828517376;
  9: 1801152661463, 33830345641392, 158855536055232, 343803285762048, 420411626611200, 315857082949632, 149536203522048, 43672380899328, 7209503096832, 515978035200;
%F Formula
The row polynomials satisfy (529+(828*n-276)*x+144*n^2*x^2)*P(n,x) = (12167+(19044*n+19044)*x+(3312*n^2+16560*n+9936)*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[((12167 + (19044*n + 19044)*x + (3312*n^2 + 16560*n + 9936)*x^2 + (1728*n^2 + 3456*n + 1728)*x^3)*P[n - 1, x])/(529 + (828*n - 276)*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] = ((12167 + (19044*n + 19044)*x + (3312*n^2 + 16560*n + 9936)*x^2 + (1728*n^2 + 3456*n + 1728)*x^3)*v[i - 1])/(529 + (828*n - 276)*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.