Real-rooted atlas

A038221 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + 2\,x + 2\,n\right)\,P_{n}(x) = \left(6 + 24\,x + 24\,x^{2} + 12\,n + 24\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1181230
2180432144756
3151264807776172817496
4116647257615552012441620736384912
5855366998402177280311040018662402488328188128
66065286158592251942405225472055987200268738562985984170061120
74199040509483522586608647054387201097349120940584960376233984358318083469246848

All 15 rows

Technical details
Node id
n-abb390907d328d80 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4be2d3b20c0d29e0

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:        18,         12;
  2:       180,        432,         144;
  3:      1512,       6480,        7776,        1728;
  4:     11664,      72576,      155520,      124416,        20736;
  5:     85536,     699840,     2177280,     3110400,      1866240,       248832;
  6:    606528,    6158592,    25194240,    52254720,     55987200,     26873856,      2985984;
  7:   4199040,   50948352,   258660864,   705438720,   1097349120,    940584960,    376233984,     35831808;
  8:  28553472,  403107840,  2445520896,  8277147648,  16930529280,  21069103104,  15049359360,   5159780352,   429981696;
  9: 191476224, 3083774976, 21767823360, 88038752256, 223482986496, 365699432448, 379243855872, 232190115840, 69657034752, 5159780352;
%F Formula
The row polynomials satisfy (2*n-1+2*x)*P(n,x) = (12*n+6+(24*n+24)*x+24*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[((12*n + 6 + (24*n + 24)*x + 24*x^2)*P[n - 1, x])/(2*n - 1 + 2*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] = ((12*n + 6 + (24*n + 24)*x + 24*x^2)*v[i - 1])/(2*n - 1 + 2*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.