Real-rooted atlas

A013622 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(8 + 15\,n x\right)\,P_{n}(x) = \left(64 + 240\,x + 225\,x^{2} + 120\,n x + 225\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
183038
2644806751219
35125760162001350035972
44096614402592004320002531251009861
5327686144003456000864000010125000455625027424418
626214458982404147200013824000024300000021870000079734375727306759
72097152550502404644864001935360000453600000061236000004465125000136687500018948593792

All 15 rows

Technical details
Node id
n-44786784ea9cbc8c · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-74b7dcf206762b10

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:         8,         30;
  2:        64,        480,         675;
  3:       512,       5760,       16200,        13500;
  4:      4096,      61440,      259200,       432000,        253125;
  5:     32768,     614400,     3456000,      8640000,      10125000,       4556250;
  6:    262144,    5898240,    41472000,    138240000,     243000000,     218700000,      79734375;
  7:   2097152,   55050240,   464486400,   1935360000,    4536000000,    6123600000,    4465125000,    1366875000;
  8:  16777216,  503316480,  4954521600,  24772608000,   72576000000,  130636800000,  142884000000,   87480000000,   23066015625;
  9: 134217728, 4529848320, 50960793600, 297271296000, 1045094400000, 2351462400000, 3429216000000, 3149280000000, 1660753125000, 384433593750;
%F Formula
The row polynomials satisfy (8+15*n*x)*P(n,x) = (64+(120*n+240)*x+(225*n+225)*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[((64 + (120*n + 240)*x + (225*n + 225)*x^2)*P[n - 1, x])/(8 + 15*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] = ((64 + (120*n + 240)*x + (225*n + 225)*x^2)*v[i - 1])/(8 + 15*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

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.