Real-rooted atlas

A038327 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(10 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(169 + 78\,x + 9\,x^{2} + 39\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
116319
2247969352
337182223432276400
4549254461613338172881114688
57997088238753346206669064802432031616
61151008314394744741487520077203001052332872935651584
716411150624171174315114481251904125105405301260441816482187620756992

All 15 rows

Technical details
Node id
n-d740dc352bf5ce5f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-27f46ac2dd1a0a19

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:          16,           3;
  2:         247,          96,           9;
  3:        3718,        2223,         432,          27;
  4:       54925,       44616,       13338,        1728,         81;
  5:      799708,      823875,      334620,       66690,       6480,        243;
  6:    11510083,    14394744,     7414875,     2007720,     300105,      23328,       729;
  7:   164111506,   241711743,   151144812,    51904125,   10540530,    1260441,     81648,     2187;
  8:  2321695129,  3938676144,  2900540916,  1209158496,  311424750,   50594544,   5041764,   279936,   6561;
  9: 32629228840, 62685768483, 53172127944, 26104868244, 8161819848, 1681693650, 227675448, 19446804, 944784, 19683;
%F Formula
The row polynomials satisfy (3*n+10+3*x)*P(n,x) = (39*n+169+(9*n+78)*x+9*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[((39*n + 169 + (9*n + 78)*x + 9*x^2)*P[n - 1, x])/(3*n + 10 + 3*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] = ((39*n + 169 + (9*n + 78)*x + 9*x^2)*v[i - 1])/(3*n + 10 + 3*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.