Real-rooted atlas

A038220 → Veronese 2 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-5 + 12\,x\right)\,P_{n-2}(x) + \left(6\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
133
2131225
363108171
43137441441201
51563468021608403
678132811621168172858825
73906316405217236836288411771
81953139374881265760463104207362882401
997656352734248701344466560055987220176803
1048828132929686057129120408672008812800248832141237625
11244140631611327963625490883262913281060646408211456988663371
1212207031387890623222412112482438961408108273024015377817629859846920643601
13610351563476074216813568115600173529008649881554176215438745611645337648444505203
14305175781325634765604807495121441188171331208314321075225150196736252912844835831808339111536425
Technical details
Node id
n-4390e032b20d4acd · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-5239f7bb38edc1ec

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      3;
  2:     13,      12;
  3:     63,     108;
  4:    313,     744,     144;
  5:   1563,    4680,    2160;
  6:   7813,   28116,   21168,    1728;
  7:  39063,  164052,  172368,   36288;
  8: 195313,  937488, 1265760,  463104,  20736;
  9: 976563, 5273424, 8701344, 4665600, 559872;
%F Formula
The row polynomials satisfy P(n,x) = -(5-12*x)*P(n-2,x) + 6*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[-(5 - 12*x)*P[n - 2, x] + 6*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = -(5 - 12*x)*v[i - 2] + 6*v[i - 1]); 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.