Real-rooted atlas

A038220 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 + 6\,n x\right)\,P_{n}(x) = \left(25 + 60\,x + 36\,x^{2} + 30\,n x + 36\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
151217
225120108253
312590016208643509
462560001620017280648046585
531253750013500021600016200046656600281
61562522500010125002160000243000013996803265927569397
7781251312500708750018900000283500002449440011430720223948893892733
839062575000004725000015120000028350000032659200022861440089579520151165441149743089
91953125421875003037500001134000000255150000036741600003429216000201553920068024448010077696013933327265
1097656252343750001898437500810000000021262500000367416000004286520000033592320000170061120005038848000665127936167414286061
1148828125128906250011601562500556875000001670625000003367980000004715172000004618944000003117787200001385683200003658203648043535646721997181694277
1224414062570312500006960937500037125000000012529687500002886840000000471517200000055427328000004676680800000277136640000010974610944002612138803202829817036823680868660713
131220703125380859375004113281250002413125000000904921875000023455575000000437837400000006004627200000060796850400000450347040000002377832371200084894511104001839381073920182849716224279320125528169
14610351562520507812500023994140625001535625000000063344531250000182432250000000383107725000000600462720000000709296588000000630485856000000416120664960000198087192576000643783375872001279948013568011754624614403279657653673445
Technical details
Node id
n-b8540c42f24ca890 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-572def99d14f8fb6

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       5,       12;
  2:      25,      120,       108;
  3:     125,      900,      1620,        864;
  4:     625,     6000,     16200,      17280,       6480;
  5:    3125,    37500,    135000,     216000,     162000,      46656;
  6:   15625,   225000,   1012500,    2160000,    2430000,    1399680,     326592;
  7:   78125,  1312500,   7087500,   18900000,   28350000,   24494400,   11430720,    2239488;
  8:  390625,  7500000,  47250000,  151200000,  283500000,  326592000,  228614400,   89579520,  15116544;
  9: 1953125, 42187500, 303750000, 1134000000, 2551500000, 3674160000, 3429216000, 2015539200, 680244480, 100776960;
%F Formula
The row polynomials satisfy (5+6*n*x)*P(n,x) = (25+(30*n+60)*x+(36*n+36)*x^2)*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[((25 + (30*n + 60)*x + (36*n + 36)*x^2)*P[n - 1, x])/(5 + 6*n*x)]];
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] = ((25 + (30*n + 60)*x + (36*n + 36)*x^2)*v[i - 1])/(5 + 6*n*x)); 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.