Real-rooted atlas

A038259 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 - 165\,x + 495\,n x + 225\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1331 + 5445\,x + 7425\,x^{2} + 3375\,x^{3} + 5445\,n x + 12375\,n x^{2} + 6750\,n x^{3} + 2475\,n^{2} x^{2} + 3375\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1116071
2121132020253466
31331217806682554000143936
4146413194401470150237600012656255445856
5161051439230026952750653400006960937527337500193792976
61771561579783604447203751437480000229710937518042750005581406256601475296
71948717174405562068486937752767149000058959140625694645875004297682812510935000000217619282816
821435888193538420801004475087004870182240001297101093750203762790000018909804375009622800000002075941406256992617505536
923579476911157537957401420614765900803580069600025682601656250504312905250006240235443750047632860000000205518199218753844335937500220119789683456
10259374246011414768614600195334530311251262768680800004708476970312501109488391550000171606474703125017465382000000001130350095703125422876953125000697746972656256813191808856576
112853116706111711870023666026261642408512519099376297100008138938762968750223746825629250004152876687815625052832780550000000455907871933593752558405566406250084427383691406251245564843750000207928272890064896
123138428376721205424402839920346653679792365028012418569080000134292489588984375421922585472300000913632871319437500139478540652000000015044959773808593751125698449218750000557220732363281250164414559375000000219271311035156256270077720540348416
1334522712143931244797413384238045064978373007450400577585537844000213375844569164062575418662153173625001866421437123993750033242385522060000000430285849530925781254024371955957031250026560854909316406250117556409953125000003135579747802734375381454233398437500187136184013558747136
14379749833583241289990782009020405783338891202622755608086197529816000328598800636512656251290497107954304250003592861266463687968757313324814853200000001104400347129376171875123950656243476562500010225929140086816406256034562377593750000002414396405808105468755874395194335937500065684083557128906255535452060117536301056
Technical details
Node id
n-6bf1f5b549d6cfd5 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d38d67bb1f56416c

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:         11,           60;
  2:        121,         1320,          2025;
  3:       1331,        21780,         66825,         54000;
  4:      14641,       319440,       1470150,       2376000,        1265625;
  5:     161051,      4392300,      26952750,      65340000,       69609375,       27337500;
  6:    1771561,     57978360,     444720375,    1437480000,     2297109375,     1804275000,      558140625;
  7:   19487171,    744055620,    6848693775,   27671490000,    58959140625,    69464587500,    42976828125,    10935000000;
  8:  214358881,   9353842080,  100447508700,  487018224000,  1297101093750,  2037627900000,  1890980437500,   962280000000,   207594140625;
  9: 2357947691, 115753795740, 1420614765900, 8035800696000, 25682601656250, 50431290525000, 62402354437500, 47632860000000, 20551819921875, 3844335937500;
%F Formula
The row polynomials satisfy (121+(495*n-165)*x+225*n^2*x^2)*P(n,x) = (1331+(5445*n+5445)*x+(2475*n^2+12375*n+7425)*x^2+(3375*n^2+6750*n+3375)*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[((1331 + (5445*n + 5445)*x + (2475*n^2 + 12375*n + 7425)*x^2 + (3375*n^2 + 6750*n + 3375)*x^3)*P[n - 1, x])/(121 + (495*n - 165)*x + 225*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] = ((1331 + (5445*n + 5445)*x + (2475*n^2 + 12375*n + 7425)*x^2 + (3375*n^2 + 6750*n + 3375)*x^3)*v[i - 1])/(121 + (495*n - 165)*x + 225*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.