Real-rooted atlas

A038324 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 10\,n x\right)\,P_{n}(x) = \left(121 + 220\,x + 100\,x^{2} + 110\,n x + 100\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112031
2121440300861
3133172609900400022491
41464110648021780017600050000564921
5161051146410039930004840000275000060000013808151
61771561193261206588450010648000090750000396000007000000330812181
7194871712480185401014621300204974000023292500001524600000539000000800000007804717011
8214358881311794736014881112400360754240005124350000044721600000237160000007040000000900000000181909942641
92357947691385845985802104614468005952444960001014621300000110685960000078262800000034848000000089100000000100000000004198337389071
102593742460147158953820028938448935009353842080000186013905000002435091120000021522270000000127776000000004900500000000110000000000011000000000096107885636301
11285311670611570623341222038906136901500141476861460000321538321500000491076709200000520838934000000386522400000000197653500000000665500000000001331000000000012000000000002185064408144331
123138428376721684748009466405135610070998002074993968080000530538230475000092603036592000001145845654800000010204191360000000652256550000000029282000000000008784600000000001584000000000001300000000000049389127576453161
133452271214393181599137794746066762930922974002967241374354400084296629953250000165527927908200000234079898052000000243199894080000000186545373300000000104683150000000000418732600000000001132560000000000018590000000000001400000000000001110729954219382791
14379749833583241966635940030068085679094684483300415413792409616000129816810128005000028323667664292000004506038037501000000535039766976000000047879979147000000003224241020000000000161212051000000000058138080000000000014314300000000000021560000000000000150000000000000024870052815998233221
Technical details
Node id
n-8b6b127f2b492ad8 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-d4975513c6489427

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:         11,          20;
  2:        121,         440,          300;
  3:       1331,        7260,         9900,         4000;
  4:      14641,      106480,       217800,       176000,         50000;
  5:     161051,     1464100,      3993000,      4840000,       2750000,        600000;
  6:    1771561,    19326120,     65884500,    106480000,      90750000,      39600000,      7000000;
  7:   19487171,   248018540,   1014621300,   2049740000,    2329250000,    1524600000,    539000000,     80000000;
  8:  214358881,  3117947360,  14881112400,  36075424000,   51243500000,   44721600000,  23716000000,   7040000000,   900000000;
  9: 2357947691, 38584598580, 210461446800, 595244496000, 1014621300000, 1106859600000, 782628000000, 348480000000, 89100000000, 10000000000;
%F Formula
The row polynomials satisfy (11+10*n*x)*P(n,x) = (121+(110*n+220)*x+(100*n+100)*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[((121 + (110*n + 220)*x + (100*n + 100)*x^2)*P[n - 1, x])/(11 + 10*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] = ((121 + (110*n + 220)*x + (100*n + 100)*x^2)*v[i - 1])/(11 + 10*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.